diff --git a/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/2301.00064v1.pdf.txt b/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/2301.00064v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7a252984c3e3393fdc0ce958ee4d75ce48f853f --- /dev/null +++ b/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/2301.00064v1.pdf.txt @@ -0,0 +1,1338 @@ +arXiv:2301.00064v1 [astro-ph.SR] 30 Dec 2022 +MNRAS 000, 1–11 (2022) +Preprint 3 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The orbital kinematics of 휂 Carinae over three periastra with a possible +detection of the elusive secondary’s motion +Emily Strawn1, Noel D. Richardson1,★ Anthony F. J. Moffat2, Nour Ibrahim1,3, +Alexis Lane1, Connor Pickett1, André-Nicolas Chené4, Michael F. Corcoran5,6, +Augusto Damineli7, Theodore R. Gull8,9, D. John Hillier10, Patrick Morris11, +Herbert Pablo12, Joshua D. Thomas13 Ian R. Stevens14, Mairan Teodoro9, Gerd Weigelt15 +1 Embry Riddle Aeronautical University, Department of Physics and Astronomy, 3700 Willow Creek Road, Prescott, AZ 86301, United States +2 Département de physique, Université de Montréal, Complexe des Sciences, 1375 Avenue Thérèse-Lavoie-Roux, Montréal (Qc), H2V 0B3, Canada +3 Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA +4 NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawai’i, 96720, USA +5 CRESST & X-ray Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA +6 The Catholic University of America, 620 Michigan Ave., N.E. Washington, DC 20064, USA +7 Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária, São Paulo, Brasil +8 Exoplanets & Stellar Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA +9 Space Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD 21218, USA +10 Department of Physics & Astronomy & Pittsburgh Particle Physics, Astrophysics, & Cosmology Center (PITT PACC), +University of Pittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260, USA +11 California Institute of Technology, IPAC, M/C 100-22, Pasadena, CA 91125, USA +12 American Association of Variable Star Observers, 49 Bay State Road, Cambridge, MA 02138, USA +13 Department of Physics, Clarkson University, 8 Clarkson Ave, Potsdam, NY 13699, USA +14 School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK +15 Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The binary 휂 Carinae is the closest example of a very massive star, which may have formed through a merger during its Great +Eruption in the mid-nineteenth century. We aimed to confirm and improve the kinematics using a spectroscopic data set taken +with the CTIO 1.5 m telescope over the time period of 2008–2020, covering three periastron passages of the highly eccentric +orbit. We measure line variability of H훼 and H훽, where the radial velocity and orbital kinematics of the primary star were +measured from the H훽 emission line using a bisector method. At phases away from periastron, we observed the He ii 4686 +emission moving opposite the primary star, consistent with a possible Wolf-Rayet companion, although with a seemingly narrow +emission line. This could represent the first detection of emission from the companion. +Key words: techniques: spectroscopic — stars: massive — stars: variables: S Doradus — stars: winds, outflows — binaries: +spectroscopic — stars: individual: 휂 Carinae +1 INTRODUCTION +The binary star system 휂 Carinae is known for being one of +the most massive and luminous binaries in our local galaxy +(Davidson & Humphreys 2012). The two stars are locked in a highly +eccentric orbit (Damineli 1996a; Damineli et al. 1997). Envelop- +ing these stars is the Homunculus nebula which was formed by +a large eruption in the mid-nineteenth century (e.g., Currie et al. +1996). The Great Eruption that formed the Homunculus nebula was +recently modeled to be the product of a binary merger in a triple sys- +tem leading to the current orbit (Portegies Zwart & van den Heuvel +2016; Hirai et al. 2021), supported by light echo observations (e.g., +Smith et al. 2018) and an extended central high-mass torus-like struc- +★ E-mail: noel.richardson@erau.edu +ture surrounding the central binary (Morris et al. 2017). In this sce- +nario, the luminous blue variable primary star is currently orbited +by a secondary star that is a classical Wolf-Rayet star, as discussed +by Smith et al. (2018). The system began as a hierarchical triple, +and mass transfer led to the initial primary becoming a hydrogen- +deficient Wolf-Rayet star. Mass transfer causes the orbits to become +unstable, which leads to the merger and leaves behind the highly +eccentric binary system we see today. An alternate model for the +eruption relies on the fact that 휂 Car is a binary in a highly eccentric +orbit, and proposes that the periastron events triggered large mass +transfer events that caused the eruptions (Kashi & Soker 2010). A +similar model was used to explain the much less massive eruption +that was seen from the SMC system HD 5980 during its LBV-like +outburst (e.g., Koenigsberger et al. 2021). +While the binary nature of the system was inferred by Damineli +© 2022 The Authors + +2 +Strawn et al. +4830 +4840 +4850 +4860 +4870 +4880 +4890 +4900 +WAVELENGTH (ANGSTROMS) +0 +5 +10 +15 +NORMALIZED FLUX +FECH, 12.0096 +GMOS, 12.0096 +CHIRON, 13.0081 +CHIRON, 14.0103 +-1000 +0 +1000 +2000 +VELOCITY (km s-1) +Figure 1. A comparison of an example Gemini-GMOS spectrum used by +Grant et al. (2020) with the CTIO data from the fiber echelle (FECH) in +2009 and with more recent CHIRON data at the same phase (phases given +in the legend). Note that the pixel sizes are indicated for the spectra, which +is most obvious for the GMOS spectrum. The spectra are offset by orbital +cycle, which highlights the complexities in the echelle spectra compared to +the GMOS data. +(1996b) and Damineli et al. (1997), the orbit of the system has mostly +eluded observers since the discovery of the spectroscopic events by +Damineli (1996a). Davidson (1997) criticized the first orbit pub- +lished by Damineli et al. (1997) and published a higher eccentricity +model using the same data as Damineli et al. (1997). Since these +first attempts to derive the orbital motion of the system, very few +observationally derived models have appeared in the literature, with +most references to the orbit being inferred for modeling purposes. +Recently, Grant et al. (2020) used archival moderate-resolution Gem- +ini/GMOS spectra from 2009 to fit the hydrogen lines using multi- +ple, weighted Gaussians to measure radial velocities corrected to +account for motion from strong stellar winds. They derived a single- +lined spectroscopic orbit based on the upper Balmer lines to be +푇0 = 2454848 (HJD), 푒 = 0.91, 퐾1 = 69 km s−1, and 휔pri = 241◦ +with the period of 2022.7 d that has been widely adopted based +on multi-wavelength observations (e.g., Teodoro et al. 2016). These +are broadly consistent with the smoothed-particle hydrodynamical +(SPH) models used to describe variability across the electromag- +netic spectrum (e.g., Madura et al. 2013) including the X-ray light +curves (e.g., Okazaki et al. 2008), optical He i absorption variability +(Richardson et al. 2016), and the near-UV emission observed with +the Hubble Space Telescope (Madura & Groh 2012). +While the results of Grant et al. (2020) establish the orbital pa- +rameters with greater precision to date, there are potential issues +with the determination of orbital elements from hydrogen lines in +휂 Car’s spectrum, as the strong wind of the primary causes the ef- +fective photospheric radius to be further out from the central star +for lower energy transitions. Indeed, Grant et al. (2020) found better +results with higher-order Balmer lines than with the optically thick +H훼 or H훽. This is a known effect for evolved Wolf-Rayet stars, where +the observed semi-amplitude can change with the ionization poten- +tial of the line measured because lower-energy emission lines tend +to form further out in the wind, where they are more likely to be +perturbed by the companion star as seen in 훾2 Vel (Richardson et al. +2017). This effect causes differences from the true orbital motion +for lower energy transitions, making it difficult to determine accu- +rate orbits (Grant et al. 2020). Grant & Blundell (2022) confirmed +that their methods used for emission-line stars worked for the WR +binaries WR 133 and WR 140 that have combined spectroscopic and +interferometric orbits (Richardson et al. 2021; Thomas et al. 2021). +The primary star in the 휂 Car system is a luminous blue variable +star, with the largest measured value for a mass-loss rate for a mas- +sive star with �푀 = 8.5 × 10−4푀⊙yr−1 and a terminal wind speed +of 푣∞ = 420 km s−1 (Davidson & Humphreys 1997; Groh et al. +2012). Prior to the recent kinematic studies of Grant et al. (2020) +and Grant & Blundell (2022), the best constraints on the compan- +ion star parameters, while indirect, came from the X-ray variability +analyses from RXTE, Swift, and NICER observations of the sys- +tem (Corcoran et al. 2001, 2017; Espinoza-Galeas et al. 2022). These +analyses point to a secondary star with a mass-loss rate on the order +of �푀 ∼ 10−5푀⊙yr−1 and a terminal velocity of 푣∞ ∼ 3000 km s−1 +(Pittard & Corcoran 2002). These values are broadly in agreement +with the suggestion based on the merger models and mass-loss pa- +rameters that the remaining secondary would be a Wolf-Rayet star. +Despite recent work with long-baseline near-infrared interferome- +try by Weigelt et al. (2021), no direct detection of the companion +star has been made to date. From the interferometric data, a mini- +mum primary-secondary flux ratio of ∼50 was derived in the 퐾-band +(Weigelt et al. 2007). Given the extreme luminosity of the LBV pri- +mary, this is consistent with any O or WR star in the Galaxy. +The evolution of the secondary star may well have been signifi- +cantly modified by interactions and mass exchange during formation +of the present-day binary, but if the current secondary star is a clas- +sical H-free Wolf-Rayet star as suggested by Smith et al. (2018) and +Hirai et al. (2021), or a hydrogen-rich WNh star, possibly the best +line to detect it in the optical would be the He ii 휆 4686 line, which +is the dominant line in the optical for the nitrogen-rich WR stars, or +the hydrogen-rich WNh stars. Most of the observations of He ii were +made near periastron, where the He ii excess can be explained by +ionization of He i in the colliding winds in a highly eccentric binary. +Teodoro et al. (2016) showed that the variability could be explained +with the smoothed-particle hydrodynamics models of Madura et al. +(2013). Away from periastron (0.04 < 휙 < 0.96), the He ii line is +typically not observed with moderate resolving power and a nominal +S/N of ∼100. +In this paper, we present our analysis of the spectroscopy collected +with the CTIO 1.5 m telescope and the CHIRON spectrograph, as +well as the data collected with the previous spectrograph on that +telescope with the aim of better constraining the kinematics of the +system. These observations are described in Section 2. In Section +3, we review the variability in the two Balmer lines we can easily +measure (H훼 and H훽). Section 4 describes our techniques of mea- +suring the radial velocity of the H훽 line, and presents observations of +He ii away from periastron in the hope of determining the orbit of the +companion star. We discuss our findings in Section 6, and conclude +this study in Section 7. +2 OBSERVATIONS +We collected high resolution spectra of 휂 Carinae during the peri- +astron passages of 2009, 2014, and 2020. Many additional spectra +were taken in the intermediate phases of the binary orbit as well. +These were collected from the 1.5m telescope at Cerro Tololo Inter- +American Observatory (CTIO 1.5) and both current CHIRON and +the former fiber-fed echelle spectrograph (FECH). The data from the +2009 spectroscopic event spanned from 2008 October 16 to 2010 +March 28, with approximately one spectrum taken every night be- +tween 2008 December 18 to 2009 February 19, which were previ- +MNRAS 000, 1–11 (2022) + +The spectroscopic orbit of 휂 Car +3 +ously used by Richardson et al. (2010, 2015) and cover the spectral +range ∼ 4700 − −7200Å. These spectra with the fiber echelle1 were +collected in late 2009 and 2010, and often had a signal-to-noise ratio +around 80–100 per resolution element with 푅 ∼ 40, 000. In total, we +analyzed 406 spectra of the system. +The 2014–2020 data were collected with the new CHIRON spec- +trograph (Tokovinin et al. 2013), and spanned the time between 2012 +March 2 and 2020 March 16, with high-cadence time-series span- +ning the 2014 and 2020 periastron passages between 2013 Decem- +ber 29 through 2015 April 21 as well as between 2020 January 3 +to 2020 March 16 when the telescope shut down for the COVID-19 +pandemic. The CHIRON spectra cover the spectral range of ∼4500- +8000Å, with some spectral gaps between orders in the red portion +of the spectrum. The data covering the 2014 periastron passage were +previously used by both Richardson et al. (2016) and Teodoro et al. +(2016). These data have a spectral resolution of 푅 ∼ 80, 000 and +typically have a signal-to-noise of 150–200 in the continuum and +were all reduced with the CHIRON pipeline, which is most recently +described by Paredes et al. (2021). In addition to the pipeline reduc- +tions, we perform a blaze correction using fits from an A0V star, +as done by Richardson et al. (2016), allowing orders to be merged +if needed. This process resulted in a flat continuum in regions that +were line-free. +These observations were all fiber-fed with the fiber spanning +2.7′′ on the sky, meaning that the data include the nebular emis- +sion from the Homunculus nebula formed from the eruption of 휂 +Car in the mid-nineteenth century, as well as the Weigelt knots +(Weigelt & Ebersberger 1986) that are thought to have originated +from the second eruption in the 1890s. The CHIRON spectra are +normalized through a comparison with a measured blaze function +from the star HR 4468 (B9.5V), as was done in the analysis of +Richardson et al. (2016). Example spectra are shown in Figure. 1, +with a comparison to a spectrum used by Grant et al. (2020) and +Grant & Blundell (2022). +3 MEASURED VARIABILITY IN THE BALMER LINES, H훼 +AND H훽 +Our observations are unique in providing both the spectral resolution +and signal-to-noise to measure the line strength (equivalent width) +and profile morphology of the emitting gas for the H훼 and H훽 lines +of 휂 Carinae. Here, we detail the observations of the variability of +the hydrogen lines. We estimate errors on equivalent width using the +methods of Vollmann & Eversberg (2006). We note that the analysis +of Richardson et al. (2015) includes many optical wind lines near +the 2009 periastron passage and phases far from periastron. These +line profiles all show minimum line strength near periastron as the +secondary’s high ionizing radiation goes behind the primary star’s +optically thick wind. We use a phase convention in which the low- +ionization state observed by Gaviola (1953) in 1948 is deemed to be +cycle 1, so that the low-ionization state starting in Feb. 2020 marks +the start of cycle 14. We leave the kinematics analysis of the metal +lines for a future analysis in order to confirm the results of Grant et al. +(2020) and Grant & Blundell (2022) here, with plans of using higher +signal-to-noise spectra in a future analysis. +1 http://www.ctio.noao.edu/noao/content/CHIRON +3.1 H훼 +Richardson et al. (2010) examined the variability of the H훼 profile +of 휂 Carinae across the 2009 periastron passage. They found that +the profile’s strength decreased during the periastron passage and +reached a minimum a few days following the X-ray minimum. They +postulated that the changes were caused by the drop in the ionizing +flux from the secondary when the companion moved to the far side. In +addition, they observed an appearance of a P Cygni absorption profile +and an absorption component at −145 km s−1, that also appeared as +the secondary’s ionizing radiation was blocked by the primary star’s +optically thick wind. Richardson et al. (2015) expanded upon this +model to describe the variations of the optical He i profiles while +documenting the variability of the optical wind lines across the 2009 +periastron passage. +We measured the equivalent width of H훼 for all of our spectra in +the range 6500 – 6650Å. These results are shown in Fig. 2, where +we show the measurements both compared to time and to binary +phase, assuming a period of 2022.7 d, and the epoch point given by +Teodoro et al. (2016), which represents the time of the periastron pas- +sage based on a comparison of the He ii observations (Teodoro et al. +2016) to SPH models of the colliding winds. Broadly speaking, the +strength of the line relative to the locally normalized continuum +shows a fast decrease and recovery near each periastron passage. +Richardson et al. (2010) found that the variability is smoother when +considering the photometric flux in the determination of the equiv- +alent widths. We did not make this correction in these data, but do +see the similarities of the events in the context of the raw equivalent +widths. +There is no strong long-term variability in these observations, +and the 2014 and 2020 observations were nearly identical in their +variations. Recently, Damineli et al. (2019, 2021) found that there +are long-term brightness and spectral changes of the system that has +been ongoing for decades and accelerated since the mid-1990s, but +now seems to be stabilizing. The shape of the H훼 variability has +remained similar over these three well-observed periastron passages, +and the line strength has stabilized across the past two cycles, which +could indicate that the system is mostly stable aside from the binary- +induced variability. +Richardson et al. (2010) also documented the timing of the ap- +pearance of the P Cygni absorption component for H훼. In the 2009 +observations we see the absorption occurring at approximately HJD +2454840.7 (휙 ≈ 12.00) and still persisting through the last observa- +tion, 2454881.7 (휙 ≈ 12.02). In 2014 a P Cygni absorption occurs at +2456874.5 (휙 ≈ 13.00) persisting until the object was not observable +at HJD 2456887.5 (휙 ≈ 13.01). In 2020, the absorption is seen at +2458886.8 (휙 ≈ 14.01) and still detected through the last observation +on HJD 2458925.0 (휙 ≈ 14.02). +A narrow absorption component was observed near −145 km s−1 +in the 2009 observations (Richardson et al. 2010) from 2454836.7 +(휙 ≈ 12.00) through the last day of observation, 2454881.7 (휙 ≈ +12.02). In 2014 an absorption in the same location is observed from +2456863.5 (휙 ≈ 13.00) – 2456977.8 (휙 ≈ 13.06). There is no ab- +sorption at this location strong enough to make a definitive detection +in 2020. Pickett et al. (2022) documented the changes in absorp- +tion behavior for the Na D complex at these velocities, showing +that the absorption from these components associated with the Little +Homunculus, formed during the second eruption in the 1890s, are +weakening with time and moving to bluer velocities. +MNRAS 000, 1–11 (2022) + +4 +Strawn et al. +2010 +2012 +2014 +2016 +2018 +2020 +Year +1000 +2000 +3000 +4000 +5000 +HJD-2454000 +200 +300 +400 +500 +600 +-Wλ (ANGSTROMS) +CHIRON +FECH +0.900 +0.925 +0.950 +0.975 +1.000 +1.025 +1.050 +1.075 +1.100 +ORBITAL PHASE ϕ +250 +300 +350 +400 +450 +500 +550 +600 +650 +-Wλ (ANGSTROMS) +11→12 +12→13 +13→14 +Figure 2. Variation in H훼 emission line with respect to time (left) and phase (right); with the data taken between October 2008 and March 2020. Data taken +from the previous echelle spectrograph is indicated by open squares and data from the new CHIRON spectrograph is indicated by solid dots. In the phase plot, +we show the different cycles in different colors to clarify the timing of each data set. Furthermore, the errors are typically the size of the points or smaller. The +phase convention shown in the right panel references the low-ionization spectrum near periastron first observed by Gaviola (1953). +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +Year +2000 +2500 +3000 +3500 +4000 +4500 +5000 +HJD-2454000 +70 +80 +90 +100 +110 +120 +130 +-Wλ (Angstroms) +0.900 +0.925 +0.950 +0.975 +1.000 +1.025 +1.050 +1.075 +1.100 +ORBITAL PHASE ϕ +70 +80 +90 +100 +110 +120 +-Wλ (ANGSTROMS) +12→13 +13→14 +Figure 3. Variation in H훽 emission line with respect to time (left) and phase (right); with the data taken with CHIRON spectrograph as the FECH data were too +noisy to determine equivalent widths. In the phase plot, we show the two recent cycles in different colors to clarify the timing of each data set. Furthermore, the +errors are typically the size of the points or smaller. The phase convention shown in the right panel references the low-ionization spectrum near periastron first +observed by Gaviola (1953). +3.2 H훽 +While some of the H훽 variability was documented for the 2009 +periastron passage of 휂 Car by Richardson et al. (2015), the full +variability and timing of the changes is still not well documented +in the literature. The lack of a more quantitative assessment of the +variability is in part due to the lower signal-to-noise in the H훽 data +from the 2009 event. Similar to the H훼 profile, H훽 experiences a P +Cygni type absorption near −500 km s−1 near periastron. We note the +absorption appears in 2009 at approximately HJD 2454837.7 (휙 ≈ +12.00) and persist through the last observation taken on 2454879.7 +(휙 ≈ 12.01). In 2014, it appears at approximately 2456863.6 (휙 ≈ +13.00) and ends during a seasonal gap in observations beginning at +2456887.5 (휙 ≈ 13.01). In 2020, the P Cygni absorption is observed +between 2458886.8 (휙 ≈ 14.00) and continues through the last day +of observations on 2458925.0 (휙 ≈ 14.02). This transient absorption +was determined to be originating from the downstream bowshock by +Gull et al. (2022). +A +narrow +absorption +component, +previously +observed +by +Richardson et al. (2015), is detected near −145 km s−1 in the 2009 +observations from 2454837.7 (휙 ≈ 12.00) and proceeds through the +end of observations on 2452879.7 (휙 ≈ 12.01). In 2014, this absorp- +tion is observed between 2456864.5 (휙 ≈ 13.00) and also persists +through the last day of observations 2456887.5 (휙 ≈ 13.01). As +with H훼, there is no discernible absorption at −145 km s−1 in 2020 +observations. +Figure 3 shows the time series variation in the H훽 equivalent width +over the last two periastron cycles. We note that the 2009 observa- +tions are not included as they are recorded with the former echelle +spectrograph and have lower signal-to-noise, though the appearance +of the P Cygni absorption remains reliable. As with the H훼 equiva- +lent widths, there is a consistency in the decrease in equivalent width +for the time period corresponding to times close to periastron. +4 LINE KINEMATICS +We measured the bisector velocity of H훽 and the centroid position of +the He ii 휆4686 line. H훽 measurements were taken during the 2009, +2014, and 2020 periastron events and the He ii 4686 measurements +MNRAS 000, 1–11 (2022) + +The spectroscopic orbit of 휂 Car +5 +0 +5 +10 +11.947 +11.998 +12.212 +0 +5 +10 +12.901 +13.007 +13.172 +−750 −500 −250 +0 +250 +500 +750 +0 +5 +10 +13.946 +−750 −500 −250 +0 +250 +500 +750 +14.004 +−750 −500 −250 +0 +250 +500 +750 +14.019 +Radial Velocity (km s−1) +Normali ed Flux +Figure 4. Example polynomial fits to H훽 emission lines from 2009, 2014, and 2020 periastron events. The profiles are shown in black with the portion of the +line wings fit with a polynomial shown in red. The bisector velocity is shown as a vertical line corresponding to the normalized flux at the same level as the +measurements. Near the edges of these ranges, the bisector often appears to curve due to either profile asymmetries or larger errors in the polynomial fits. The +bisector velocities between normalized flux levels of 5 and 6, indicated by the dashed lines, were averaged to obtain a final relative velocity for each day. Further +details are given in Section 4.1. +were taken for 2014 and 2018 and do not include time within 휙 = +0.95 – 1.05 to avoid observations affected by periastron caused by +colliding-wind effects which, to first order, behave with a 퐷−1 trend +for adiabatic and 퐷−2 or steeper for radiative conditions, where 퐷 +is the orbital separation, which is small and quickly changing at +periastron. Teodoro et al. (2016) show the behavior of the He ii 4686 +line near periastron in detail. All measurements are tabulated in +online supplementary data. +4.1 Bisector velocities of H훽 +The process used to find the bisector velocity of H훽 is demon- +strated in Fig. 4. Grant et al. (2020) and Grant & Blundell (2022) +used a method of Gaussian decomposition using many components +to moderate-resolution spectroscopy taken with Gemini-South and +GMOS. Their GMOS spectra of 휂 Car are limited in that the highest +resolving power available is ∼ 4400, whereas our spectroscopy has a +resolving power of 40, 000 from the fiber echelle, and 80, 000 for the +CHIRON data. The profiles become more complex at higher spectral +resolution, making this multiple-Gaussian method more difficult to +implement, likely requiring more than twice as many components +compared to the work of Grant et al. (2020). +In order to create a simpler measurement that has reproducible +results for any spectroscopic data set, we implemented a bisector +technique. We began by fitting two fourth degree polynomials, one +to the red side and another on the blue side of the profile in order to +smooth over any noise inherent in the data. Through this fit, we were +then able to establish the bisecting velocity position at each emission +level with higher precision. Example fits are shown in red in Fig. 4. In +the regions of heights of 4× the continuum up to 10× the continuum, +we calculate the bisecting velocity. This area was chosen based on +the relatively vertical nature of the bisector in this region. We then +created comparisons of all spectra and found that the bisecting line +was nearly always vertical in the region of 5 − 6× the normalized +continuum. We therefore used this region, measuring the velocity at +every 0.1 increment between these values, and adopting an average +measurement as the radial velocity for the spectrum. The choice +of a common emission height with which to measure the bisector +velocities allows us confidence in the results as it would relate to +gas emitting from the same region for all spectra, whether the line is +weak or strong in that particular observation. The resulting velocities +MNRAS 000, 1–11 (2022) + +6 +Strawn et al. +2008 +2010 +2012 +2014 +2016 +2018 +2020 +Year +0 +1000 +2000 +3000 +4000 +5000 +HJD-2454000 +−80 +−60 +−40 +−20 +0 +20 +40 +60 +Velocity (km s−1) +FECH +CHIRON +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +Phase +−80 +−60 +−40 +−20 +0 +20 +40 +60 +Velocity (km s−1) +11→12 +12→13 +13→14 +Figure 5. Radial velocities from H훽 bisector measurements compared to time (left) and orbital phase (right). The orbital fit is described in Section 4.3 and +typical errors are on the order of the size of the points. +−400 +−200 +0 +200 +400 +Velocity (km s−1) +0.98 +0.99 +1.00 +1.01 +1.02 +1.03 +Normalized Flux +ϕ = 13.08 +Figure 6. Gaussian fit to an example He ii emission line with a vertical line +plotted at the fitted peak. This particular spectrum had a signal-to-noise ratio +of 210 per resolution element. +are shown in Fig. 5. We provide this bisector code via GitHub2 for +future use on comparable datasets. +4.2 He ii 휆4686 +The region surrounding, but not blended with, the He ii 휆4686 tran- +sition is complicated by several features including narrow emission +lines from the Weigelt knots (Weigelt & Ebersberger 1986) along +with wind emission from Fe ii and He i lines (for a figure showing +that region of the spectrum, see Teodoro et al. 2016). While these +do not directly overlap with the core of the He ii line, they can +complicate this fitting if not properly avoided. The He ii 휆4686 line +has usually been observed near periastron passage when the line +is dominated by the wind-wind collisions, which has been docu- +mented and modeled by Teodoro et al. (2016). The line was discov- +ered by Steiner & Damineli (2004). Since then, multiple studies have +attempted to explain the formation of the stronger line observed near +periastron (퐿He ii ∼ 300 퐿 ⊙; Martin et al. 2006; Mehner et al. 2011, +2015; Teodoro et al. 2012; Davidson et al. 2015), but the colliding +2 https://github.com/EmilysCode/Radial-Velocity-from-a-Polynomial-Fit- +Bisector.git +wind model best reproduces the emission near periastron. This emis- +sion is strongest for times within ±0.05 in phase from periastron, as +detailed in the recent analysis of Teodoro et al. (2016). +Outside of the phase intervals near periastron, the He ii 휆4686 +line could only be properly observed with high spectral resolution +and high signal-to-noise data (Teodoro et al. 2016). Our data taken +with CHIRON, after the 2014 periastron passage has the necessary +sensitivity to detect this notably weak emission line. We measure the +radial velocity of this line outside of 휙 = ±0.05 of periastron, so that +it minimizes the effects of the colliding winds that peak at periastron. +As shown in Fig. 6, we fit a Gaussian to the He ii emission line +and use the centroid position to determine the radial velocity. Un- +fortunately, the continuum placement for the feature is not reliable +enough to measure equivalent widths with precision, but the line +was nearly constant in equivalent width when considering the errors +of these measurements. Before fitting the 2018 observations near +apastron, we needed to average up to ten observations to improve +the signal-to-noise ratio. The resulting velocities are shown in Fig. 7 +with a resulting total of 19 data points. The averaging of the points +from the 2018 data resulted in a smaller dispersion of the data than +seen in the earlier points. +The He ii line is normally absent in the spectra of luminous blue +variables. The extreme mass-loss rate of 휂 Car does not preclude +this emission line originating in the primary star’s wind, as there +are some combinations of parameters used that can create this weak +emission feature in CMFGEN models. These models and parameters +are very sensitive and depend on the mass-loss rate and stellar radii +used. The He ii can be formed through strong wind collisions at +times close to periastron (e.g., Teodoro et al. 2016). However, this +line moves in opposition to the primary star’s motion, so we consider +this feature as originating from the companion during these phases +far from periastron for the remainder of this analysis. +4.3 Orbital Kinematics and Observed Elements +We began our fit of the kinematics of the primary star with the +BinaryStarSolver software (Milson et al. 2020; Barton & Milson +2020). The resulting orbit is broadly in agreement with the orbit +derived with H훽 velocities by Grant et al. (2020), with the orbital +elements given in Table 1. Our resulting fits are in agreement with +those of Grant et al. (2020) so we did not perform the same correction +for the stellar wind effects as in their analysis. +MNRAS 000, 1–11 (2022) + +The spectroscopic orbit of 휂 Car +7 +Line +푇0 (HJD-2400000) +푒 +퐾 (km s−1) +휔 (degrees) +훾 (km s−1) +Source +Pa훾 +48800 ± 33 +0.63 ± 0.08 +53±6 +286 ±6 +−15 ± 3 +Damineli et al. (1997) +Pa훾, He i 6678 +48829± 8 +0.802 ± 0.033 +65.4 ± 3.5 +286 ± 8 +-12.1 ± 2.7 +Davidson (1997) +Pa훾, Pa훿 +50861 +0.75 +50 +275 +-12 +Damineli et al. (2000) +H훽 +54854.9 +4.5 +−4.1 +0.82 ±0.02 +53.0 +2.1 +−1.9 +254 ±4 +-25.5 ±2.0 +Grant et al. (2020) +All Balmer lines +54848.3 ±0.4 +0.91 ±0.00 +69.0 ±0.9 +241 ±1 +. . . +Grant et al. (2020) +Upper Balmer lines +54848.4 ±0.4 +0.89 ±0.00 +69.9 ±0.8 +246 ±1 +. . . +Grant et al. (2020) +H훽 +56912.2 ±0.3 +0.8100 ±0.0007 +58.13 ±0.08 +251.43 ±0.19 +6.34 ±0.10 +This work(BinaryStarSolver) +H훽 +56927.4 ±0.5 +0.8041 ±0.0008 +54.6±0.2 +260.6 ±0.2 +4.83 ±0.09 +This work (PHOEBE) +He ii +56973.5 ±0.2 +0.937 ±0.001 +129.5±5.0 +80.6 (fixed) +63.1 ±0.4 +This work +Table 1. Orbital elements from previous publications and the results from this work. For the orbits of Grant et al. (2020), Grant & Blundell (2022), and our +work, the period has been held constant at 2022.7 d, while it was fit in the earlier work of Damineli et al. (1997), Davidson (1997), and Damineli et al. (2000) +with periods that agree with 2022.7 d within their errors. Note that our errors from the PHOEBE code may be underestimated, especially for the He ii line (see +text for details). +6500 +7000 +7500 +8000 +8500 +9000 +HJD - 2,450,000 +-50 +0 +50 +100 +150 +200 +RADIAL VELOCITY (km s-1) +2014 +2016 +2018 +2020 +YEAR +Figure 7. Radial velocity as determined using centroid positions in He ii +emission at phases away from periastron during 2014–2018 with CHIRON. +We overplotted the He ii orbit from Table 1, along with the H훽 solution from +our work shifted to the same 훾-velocity as the He ii orbit as a grey dashed +line. +In an attempt to fully assess the errors of the parameters, we used +the PHOEBE code (PHysics Of Eclipsing BinariEs; Prša & Zwitter +2005; Prša et al. 2016) to verify the orbital elements. The latest +version of PHOEBE incorporates the Markov Chain Monte Carlo +package emcee (Foreman-Mackey et al. 2013). Unlike traditional or- +bit fitting routines, PHOEBE fits using the variable of the projected +semi-major axis (푎 sin 푖) rather than the semi-amplitude 퐾, but these +are easily interchangeable using +푎 sin 푖 = (1 − 푒2)1/2 +2휋 +퐾푃. +These orbital elements are also similar to the other published +orbital elements measured with H훽, and the resulting orbit is shown in +Fig. 5. The distribution of the errors from the Monte Carlo simulation, +shown in Fig. 8, is tightly constrained but shows that various orbital +elements have errors that are interdependent with other parameters. +While this represents the best solution to the entire data set, we +explored how the parameters change if we kept only the densest of +the three periastra observed (the 2014 event). Running the PHOEBE +code with the MCMC package on just those data resulted in the +eccentricity being slightly larger (푒 = 0.824), the time of periastron +being later (HJD 2,456,935.31), and the value of 푎 sin 푖 (hence 퐾1) +being slightly larger at 2620.4 푅⊙. These values are outside the +limits given with our MCMC fit of all of the data, so we caution that +the errors in Table 1 are likely underestimated. We include the fit +parameters in the same style as Fig. 8 in the online Fig. A1. +Once the orbital elements for H훽 were fit, we proceeded to run a +simpler model for the He ii emission. For this PHOEBE model, we +keep 휔 constant to that representing the primary star from the upper +Balmer line results from Grant et al. (2020). However, we do allow +the semi-major axis, 훾-velocity, 푒, and time of periastron passage to +vary. The resulting orbit is more eccentric than that of the primary star +when derived using H훽 (and a bit more eccentric than the Grant et al. +(2020) solution) and is shown in Fig. 7. With future observations of +the He ii line at times away from periastron, a combined double-lined +orbit of the system with 휔 being consistent for the two stars will be +possible. +5 DISCUSSION +The optical spectrum of 휂 Car is dominated by emission lines from +the wind of the primary and its ejecta. The dominant emission lines +are the hydrogen Balmer lines, but there are strong lines from He i +and Fe ii in the spectrum as well. The He i lines, when considered +in non-LTE stellar wind models, are a strong function of the adopted +value of the stellar radius. However, if most of the He i emission +comes from the colliding wind interaction region, it forces a larger +stellar core radius value for the primary star, ∼ 120푅⊙ in the pre- +ferred models (see Hillier et al. 2001, for many further details). The +model of Groh et al. (2012) improved previous spherically symmet- +ric models of Hillier et al. (2001) in that the spectrum was modeled +with a cavity carved from the wind of the secondary, which was in- +cluded along with a central occulter or “coronagraph" that extended +∼ 0.033′′ to allow for stronger He i emission, and better agreement +for the P Cygni absorption lines. Given the spectral modeling agree- +ment for the spectroscopically similar star HDE 316285 (Hillier et al. +1998), the strong disagreements for the absorption components and +He i lines led to an interpretation that the He i lines are formed in +the wind-wind collision region of the system (Nielsen et al. 2007). +Indeed, the P Cygni absorption component variability of the optical +He i lines seems to represent the outflowing shocked gas from the +wind-wind collision region (Richardson et al. 2016). These results +all indicate that the best lines in the optical for determination of the +orbit may indeed be the upper hydrogen Balmer lines, even if they +are likely modified by the wind collisions. +All of the measured orbits, including ours, rely on measurements +taken when the line profiles are most variable near periastron. This +MNRAS 000, 1–11 (2022) + +8 +Strawn et al. +2595.0+4.0 +−4.0 R⊙ +4.50 +4.65 +4.80 +4.95 +5.10 +vγ (km +s ) +4.83+0.09 +−0.09 +km +s +0.8020 +0.8035 +0.8050 +0.8065 +ebinary +0.8041+0.0008 +−0.0008 +6 +7 +8 +9 +t0, perpass, binary (d) ++2.45692e6 +2456927.4+0.5 +−0.5 d +2584 +2592 +2600 +2608 +abinarysinibinary (R ) +259.9 +260.2 +260.5 +260.8 +261.1 +ω0, binary (∘) +4.50 +4.65 +4.80 +4.95 +5.10 +vγ (km +s ) +0.8020 +0.8035 +0.8050 +0.8065 +ebinary +6 +7 +8 +9 +t0, perpass, binary (d) ++2.45692e6 +259.9 +260.2 +260.5 +260.8 +261.1 +ω0, binary (∘) +260.6+0.2 +−0.2 +∘ +Figure 8. Results of the Markov chain Monte Carlo fit for the H훽 velocities. Note that 휔0 refers to the value of 휔 for the primary star. +likely causes additional errors in the parameters derived, but we +tried to always sample emission from the same line formation re- +gion by taking bisector velocities at the same height. Furthermore, +our technique produces nearly the same orbital elements as those +from Grant et al. (2020) in the case of H훽. Grant et al. (2020) pro- +ceeded to correct the orbital elements by considering the effects of +the outflowing wind. +These results all show that the system is a long-period and highly +eccentric binary where the primary star is in front of the secondary +at periastron, causing the ionization in our line of sight to drop +during the “spectroscopic events" due to a wind occultation of the +secondary at these times. The results of Grant et al. (2020) show that +the higher-order Balmer lines give different results than that of the +lower-level lines such as H훼 or H훽, which is expected as the higher +level lines form deeper in the wind (e.g. Hillier et al. 2001). As such, +the results of Grant et al. (2020) and Grant & Blundell (2022) should +be considered the best for the primary star at the current time. Similar +differences in the orbital kinematics is sometimes inferred for Wolf- +Rayet stars (e.g., 훾2 Vel; Richardson et al. 2017). +Despite the detection of the He ii 휆4686 emission at times near +apastron by Teodoro et al. (2016), the exact formation channel for this +line remains unclear. The emission lines in colliding wind binaries +often vary as a function of the orbit due to the colliding wind line +excess (e.g., Hill et al. 2000), and the modeling of these variations +has been done in the context of the so-called Lührs model (Lührs +1997). Recently, the excess emission was observed to be a strong +cooling contributor when X-ray cooling becomes less efficient in the +colliding wind binary WR 140 (Pollock et al. 2021). In WR 140, the +MNRAS 000, 1–11 (2022) + +The spectroscopic orbit of 휂 Car +9 +Lührs model was used by Fahed et al. (2011) to explain the variations +in the C III 휆5696 line near periastron. +The Lührs model can explain changes in the radial velocity and +the width of the excess emission. As can be seen in Fig. 6, we +detect the He ii line with our spectra, but the actual characterization +of this line will have large errors in line width due to the limited +signal-to-noise for the detection in the spectroscopy. We used the +models for WR 140 (Fahed et al. 2011) as a starting point, changing +stellar and binary parameters as appropriate to the 휂 Carinae system +to investigate if the He ii velocities in Fig. 7 were from colliding +wind excess emission. For the velocity of the outflow, we can see +that during the periastron passage of 2014, 휂 Car’s outflow reached +velocities faster than the primary star’s wind speed based on the +optical He i lines (Richardson et al. 2016), which are slower than the +excess absorption seen to reach nearly 2000 km s−1 in the meta-stable +He i 휆10830 line (Groh et al. 2010). With these velocities, we expect +to see the observed amplitude of the excess increase between the +times of 2015 and 2018 like we see in Fig. 7, but with amplitudes of +at least 1000 km s−1, much greater than the ∼ 100 km s−1 observed. +Therefore, the analysis of the He ii 휆4686 emission line at times +away from periastron from the CHIRON spectra is an important +observation towards understanding the nature of the companion. We +note that the data indicate a narrower emission line profile then +expected from the parameters inferred for the secondary. However, +the primary star dominates the spectrum, and the motion of this peak +opposite the primary indicate that the He ii excess could be from the +secondary’s wind. In particular, the Lührs models of the kinematics +of the He ii line seem to exclude the possibility that the line is formed +in the colliding winds at times away from periastron. +The models of Smith et al. (2018) suggest that the companion +should be a classical Wolf-Rayet star. The classical hydrogen-free +Wolf-Rayet stars can be split into the WN and WC subtypes. The +WN stars show strong He and N lines, with the He ii 휆4686 typically +being the strongest optical line, whereas the WC subtype exhibits +strong He, C, and O lines with the C IV 휆휆5802,5812 doublet often +being the strongest optical line. There is also the rare WO subtype, +which is similar to the WC subtype but shows more dominant O +lines. The WO stars were recently shown to have higher carbon +and lower helium content than the WC stars, likely representing the +final stages of the WR evolution (Tramper et al. 2015; Aadland et al. +2022). Given the generalized characteristics of WR stars, a WN star +would seem the most likely companion star if the He ii 휆4686 line is +from the companion at times further from periastron. +For contrast, the Carina nebula is also the home to several +hydrogen-rich Wolf-Rayet stars: WR 22, WR 24, and WR 25 +(Rosslowe & Crowther 2015)3. This type of WR star tends to be +considered the higher mass and luminosity extension of the main +sequence. As such, these stars have masses in excess of ∼ 60푀⊙, +with the R145 system in the LMC having masses of the two WNh +stars being 105 and 95 푀⊙ (Shenar et al. 2017). Like the classical +WN stars, these stars have similar nitrogen and helium spectra, along +with stronger emission blended on the Balmer lines which overlap +with Pickering He ii lines. The region surrounding the He ii 휆5411 +line in our 휂 Carinae spectra does not exhibit emission lines at the +same epochs as our observations of He ii 4686, making it difficult to +quantify the companion’s properties without the higher order He ii +lines which would also be notably weaker than He ii 휆4686. +With the assumption that the He ii orbit shown in Table 1 is from +the companion star, and that the semi-amplitude from the higher- +3 http://pacrowther.staff.shef.ac.uk/WRcat/ +order Balmer lines for the primary star (Grant et al. 2020), then the +semi-amplitude ratio shows that the primary star is 2–3 times more +massive than the secondary star. This is also an indicator that the +companion is not likely a WNh star, as that would imply the primary +star could have a mass of in excess of 100 푀⊙. Models of the system, +such as those by Okazaki et al. (2008) and Madura et al. (2013), +typically have the masses of the primary and secondary as 90 and +30 푀⊙ respectively, broadly in agreement with the kinematics of +the orbits presented here. On the other hand, if 휂 Carinae A has a +mass of > 100푀⊙, the secondary would have a mass on the order +of 50–60 푀⊙. This is similar to the nearby WNh star in the Carina +nebula: WR22. The mass of this WNh star in an eclipsing system is +56–58 푀⊙ (Lenoir-Craig et al. 2022). The tidally-induced pulsations +observed by Richardson et al. (2018) were modeled with stars of +masses 100 and 30 푀⊙, and therefore may also support the higher +masses suggested here. +Most models for 휂 Car have a preferred orbital inclination of +130–145◦ (Madura et al. 2012), which agrees with forbidden [Fe iii] +emission observed with Hubble Space Telescope’s Space Telescope +Imaging Spectrograph. This inclination can be used with the mass +function derived from the primary star’s orbit +푓 (푀) = +푚3 +2 sin3 푖 +(푚1 + 푚2)2 = (1.0361 × 10−7)(1 − 푒2)3/2퐾3 +1푃[M⊙] +to constrain the system’s masses with the mass function using the +standard units measured and our measured H훽 orbit using PHOEBE +(Table 1). The mass function is 푓 (푀) = 8.30 ± 0.05 M⊙, and would +indicate a companion star with a mass of at least 60 푀⊙ if we assume +a primary mass of ∼ 90 푀⊙. Given the actual mass functions for the +measured upper Balmer lines and He ii orbits, the minimum masses +required for these measured orbits are 푀 sin3 푖 = 102푀⊙ for the +LBV primary and 푀 sin3 푖 = 55푀⊙ for the secondary, making the +companion star’s identification as a WNh star more likely. These +results are still preliminary and require follow-up observations to +constrain the orbits. +A WNh star can account for the mass of the secondary star in 휂 Car, +but could cause some difficulty for the modeling of the Great Eruption +models of Hirai et al. (2021). In that scenario, the companion star +would be a hydrogen-stripped star, contrary to the hydrogen content +of the WNh stars. Recently modeled WNh systems such as R144 +(Shenar et al. 2021) show that the surface fraction of hydrogen is +about 0.4. This does show some amount of lost hydrogen on the +surface, so the scenario could still be relevant even if the final star +is not a fully stripped classical Wolf-Rayet star, assuming that the +evolution of the secondary star has not been significantly influenced +by mass exchange prior to or during the merger event hypothesized +by both Portegies Zwart & van den Heuvel (2016) and Hirai et al. +(2021). +6 CONCLUSIONS +In this paper, we provide an orbital ephemeris for 휂 Carinae mea- +sured with a bisector method and high resolution ground-based spec- +troscopy of the H훽 emission line, along with an ephemeris for the +He ii 휆4686 emission line at times far from periastron. Our findings +can be be summarized as follows: +• The H훽 emission profile tracks the primary star, and our bisec- +tor method provides similar results as the multiple-Gaussian fitting +method used by Grant et al. (2020). The results show a high ec- +centricity orbit of the system with the primary star in front of the +secondary at periastron. +MNRAS 000, 1–11 (2022) + +10 +Strawn et al. +• The weak He ii 휆4686 emission tracks opposite the kinematics +of the primary star, suggesting it is formed in the secondary star’s +windat timesawayfromperiastron. Thiscouldsupport thehypothesis +of the scenarios presented by Hirai et al. (2021) for a stellar merger +being the cause of the Great Eruption as the secondary could be a +Wolf-Rayet star that has leftover hydrogen on its surface. +• With the assumed inclination of 130–145◦, the masses of the +stars could be around ∼100 푀⊙ for the primary and at least 60 푀⊙ +for the secondary. However, the mass ratio derived by comparing the +two semi-amplitudes is about 1.9. New observations will be needed +to better determine precise masses. +Future studies will be able to better measure the He ii 4686 orbit +and refine its parameters. As shown in Grant et al. (2020), the upper +Balmer lines are more likely to reflect the orbital motion of the +stars, and the upper Paschen lines will also be useful. However, our +work shows that a simpler bisector measurement of higher resolution +spectroscopy results in the same derived orbital elements as that of +Grant et al. (2020). Furthermore, with better signal-to-noise spectra, +we can better determine if the He ii emission near periastron can +be reproduced with a Lührs model or if it is a signature of the +companion. With this information, we will be able to more precisely +measure the kinematics of the two stars and the mass function, and +then we can begin to better understand the current evolutionary status +of the system. +ACKNOWLEDGEMENTS +We thank our referee, Tomer Shenar for many suggestions that +improved this paper. These results are the result of many alloca- +tions of telescope time for the CTIO 1.5-m telescope and echelle +spectrographs. We thank internal SMARTS allocations at Geor- +gia State University, as well as NOIR Lab (formerly NOAO) al- +locations of NOAO-09B-153, NOAO-12A-216, NOAO-12B-194, +NOAO-13B-328, NOAO-15A-0109, NOAO-18A-0295, NOAO-19B- +204, NOIRLab-20A-0054, and NOIRLab-21B-0334. This research +has used data from the CTIO/SMARTS 1.5m telescope, which +is operated as part of the SMARTS Consortium by RECONS +(www.recons.org) members Todd Henry, Hodari James, Wei-Chun +Jao, and Leonardo Paredes. At the telescope, observations were car- +ried out by Roberto Aviles and Rodrigo Hinojosa. C.S.P. and A.L. +were partially supported by the Embry-Riddle Aeronautical Univer- +sity Undergraduate Research Institute. E.S. acknowledges support +from the Arizona Space Grant program. N.D.R., C.S.P., A.L., E.S., +and T.R.G. acknowledge support from the HST GO Programs #15611 +and #15992. AD thanks to FAPESP (2011/51680-6 and 2019/02029- +2) for support. AFJM is grateful for financial aid from NSERC +(Canada). The material is based upon work supported by NASA un- +der award number 80GSFC21M0002. The work of ANC is supported +by NOIRLab, which is managed by the Association of Universities +for Research in Astronomy (AURA) under a cooperative agreement +with the National Science Foundation. +DATA AVAILABILITY +All measurements can be found in Appendix A. Reasonable requests +to use the reduced spectra will be granted by the corresponding +author. +REFERENCES +Aadland E., Massey P., Hillier D. J., Morrell N. I., Neugent K. F., Eldridge +J. J., 2022, arXiv e-prints, p. arXiv:2204.04258 +Barton C., Milson N., 2020, BinaryStarSolver: Orbital elements of binary +stars solver, Astrophysics Source Code Library, record ascl:2012.004 +(ascl:2012.004) +Corcoran M. F., Ishibashi K., Swank J. H., Petre R., 2001, ApJ, 547, 1034 +Corcoran M. F., et al., 2017, ApJ, 838, 45 +Currie D. G., et al., 1996, AJ, 112, 1115 +Damineli A., 1996a, ApJ, 460, L49 +Damineli A., 1996b, ApJ, 460, L49 +Damineli A., Conti P. 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P., Szymkowiak A., 2013, PASP, 125, 1336 +Tramper F., et al., 2015, A&A, 581, A110 +Vollmann K., Eversberg T., 2006, Astronomische Nachrichten, 327, 862 +Weigelt G., Ebersberger J., 1986, A&A, 163, L5 +Weigelt G., et al., 2007, A&A, 464, 87 +Weigelt G., et al., 2021, A&A, 652, A140 +APPENDIX A: APPENDIX +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2022) + +12 +Strawn et al. +2620.4 ++4.6 +−4.7 +R +⊙ +3.60 +3.75 +3.90 +4.05 +v +γ + ( +km +s +) +3.811 ++0.08 +−0.08 +km +s +0.8200 +0.8225 +0.8250 +0.8275 +e +binary +0.824 ++0.0013 +−0.0013 +4.5 +5.0 +5.5 +6.0 +6.5 +t +0, +perpass, +binary + (d) ++2.45693e6 +2456935.31 ++0.28 +−0.29 +d +2608 +2616 +2624 +2632 +a +binary +sin +i +binary + (R +⊙ +) +260.8 +261.2 +261.6 +262.0 +ω +0, +binary + ( +⊙ +) +3.60 +3.75 +3.90 +4.05 +v +γ + ( +km +s +) +0.8200 +0.8225 +0.8250 +0.8275 +e +binary +4.5 +5.0 +5.5 +6.0 +6.5 +t +0, +perpass, +binary + (d) ++2.45693e6 +260.8 +261.2 +261.6 +262.0 +ω +0, +binary + ( +⊙ +) +261.43 ++0.22 +−0.22 +⊙ +Figure A1. Results of the Markov chain Monte Carlo fit for the H훽 velocities for only the data surrounding the 2014 periastron passage. Note that 휔0 refers to +the value of 휔 for the primary star. +MNRAS 000, 1–11 (2022) + diff --git a/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/load_file.txt b/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8afce0286ec9496f2d362b096fd4cada58fcdeb8 --- /dev/null +++ b/-9AyT4oBgHgl3EQfRPbk/content/tmp_files/load_file.txt @@ -0,0 +1,1145 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf,len=1144 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00064v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='SR] 30 Dec 2022 MNRAS 000, 1–11 (2022) Preprint 3 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 The orbital kinematics of 휂 Carinae over three periastra with a possible detection of the elusive secondary’s motion Emily Strawn1, Noel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Richardson1,★ Anthony F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Moffat2, Nour Ibrahim1,3, Alexis Lane1, Connor Pickett1, André-Nicolas Chené4, Michael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Corcoran5,6, Augusto Damineli7, Theodore R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Gull8,9, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' John Hillier10, Patrick Morris11, Herbert Pablo12, Joshua D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Thomas13 Ian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Stevens14, Mairan Teodoro9, Gerd Weigelt15 1 Embry Riddle Aeronautical University, Department of Physics and Astronomy, 3700 Willow Creek Road, Prescott, AZ 86301, United States 2 Département de physique, Université de Montréal, Complexe des Sciences, 1375 Avenue Thérèse-Lavoie-Roux, Montréal (Qc), H2V 0B3, Canada 3 Department of Astronomy, University of Michigan, 1085 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' University, Ann Arbor, MI 48109, USA 4 NSF’s NOIRLab, 670 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A’ohoku Place, Hilo, Hawai’i, 96720, USA 5 CRESST & X-ray Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA 6 The Catholic University of America, 620 Michigan Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Washington, DC 20064, USA 7 Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária, São Paulo, Brasil 8 Exoplanets & Stellar Astrophysics Laboratory, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA 9 Space Telescope Science Institute, 3700 San Martin Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' USA 10 Department of Physics & Astronomy & Pittsburgh Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' & Cosmology Center (PITT PACC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' University of Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 3941 O’Hara Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' PA 15260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' USA 11 California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' IPAC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' M/C 100-22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' USA 12 American Association of Variable Star Observers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 49 Bay State Road,' metadata={'source': 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+page_content=' NY 13699,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' USA 14 School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' University of Birmingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Birmingham B15 2TT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' UK 15 Max Planck Institute for Radio Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Auf dem Hügel 69,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 53121 Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Germany Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The binary 휂 Carinae is the closest example of a very massive star, which may have formed through a merger during its Great Eruption in the mid-nineteenth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We aimed to confirm and improve the kinematics using a spectroscopic data set taken with the CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 m telescope over the time period of 2008–2020, covering three periastron passages of the highly eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We measure line variability of H훼 and H훽, where the radial velocity and orbital kinematics of the primary star were measured from the H훽 emission line using a bisector method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' At phases away from periastron, we observed the He ii 4686 emission moving opposite the primary star, consistent with a possible Wolf-Rayet companion, although with a seemingly narrow emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This could represent the first detection of emission from the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Key words: techniques: spectroscopic — stars: massive — stars: variables: S Doradus — stars: winds, outflows — binaries: spectroscopic — stars: individual: 휂 Carinae 1 INTRODUCTION The binary star system 휂 Carinae is known for being one of the most massive and luminous binaries in our local galaxy (Davidson & Humphreys 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The two stars are locked in a highly eccentric orbit (Damineli 1996a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Envelop- ing these stars is the Homunculus nebula which was formed by a large eruption in the mid-nineteenth century (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The Great Eruption that formed the Homunculus nebula was recently modeled to be the product of a binary merger in a triple sys- tem leading to the current orbit (Portegies Zwart & van den Heuvel 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021), supported by light echo observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2018) and an extended central high-mass torus-like struc- ★ E-mail: noel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='richardson@erau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='edu ture surrounding the central binary (Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In this sce- nario, the luminous blue variable primary star is currently orbited by a secondary star that is a classical Wolf-Rayet star, as discussed by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The system began as a hierarchical triple, and mass transfer led to the initial primary becoming a hydrogen- deficient Wolf-Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Mass transfer causes the orbits to become unstable, which leads to the merger and leaves behind the highly eccentric binary system we see today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' An alternate model for the eruption relies on the fact that 휂 Car is a binary in a highly eccentric orbit, and proposes that the periastron events triggered large mass transfer events that caused the eruptions (Kashi & Soker 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A similar model was used to explain the much less massive eruption that was seen from the SMC system HD 5980 during its LBV-like outburst (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Koenigsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' While the binary nature of the system was inferred by Damineli © 2022 The Authors 2 Strawn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4830 4840 4850 4860 4870 4880 4890 4900 WAVELENGTH (ANGSTROMS) 0 5 10 15 NORMALIZED FLUX FECH, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0096 GMOS, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0096 CHIRON, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0081 CHIRON, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0103 1000 0 1000 2000 VELOCITY (km s-1) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A comparison of an example Gemini-GMOS spectrum used by Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) with the CTIO data from the fiber echelle (FECH) in 2009 and with more recent CHIRON data at the same phase (phases given in the legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Note that the pixel sizes are indicated for the spectra, which is most obvious for the GMOS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The spectra are offset by orbital cycle, which highlights the complexities in the echelle spectra compared to the GMOS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1996b) and Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1997), the orbit of the system has mostly eluded observers since the discovery of the spectroscopic events by Damineli (1996a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Davidson (1997) criticized the first orbit pub- lished by Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1997) and published a higher eccentricity model using the same data as Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Since these first attempts to derive the orbital motion of the system, very few observationally derived models have appeared in the literature, with most references to the orbit being inferred for modeling purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Recently, Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) used archival moderate-resolution Gem- ini/GMOS spectra from 2009 to fit the hydrogen lines using multi- ple, weighted Gaussians to measure radial velocities corrected to account for motion from strong stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' They derived a single- lined spectroscopic orbit based on the upper Balmer lines to be 푇0 = 2454848 (HJD), 푒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='91, 퐾1 = 69 km s−1, and 휔pri = 241◦ with the period of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 d that has been widely adopted based on multi-wavelength observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These are broadly consistent with the smoothed-particle hydrodynamical (SPH) models used to describe variability across the electromag- netic spectrum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Madura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2013) including the X-ray light curves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Okazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2008), optical He i absorption variability (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016), and the near-UV emission observed with the Hubble Space Telescope (Madura & Groh 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' While the results of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) establish the orbital pa- rameters with greater precision to date, there are potential issues with the determination of orbital elements from hydrogen lines in 휂 Car’s spectrum, as the strong wind of the primary causes the ef- fective photospheric radius to be further out from the central star for lower energy transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Indeed, Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) found better results with higher-order Balmer lines than with the optically thick H훼 or H훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This is a known effect for evolved Wolf-Rayet stars, where the observed semi-amplitude can change with the ionization poten- tial of the line measured because lower-energy emission lines tend to form further out in the wind, where they are more likely to be perturbed by the companion star as seen in 훾2 Vel (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This effect causes differences from the true orbital motion for lower energy transitions, making it difficult to determine accu- rate orbits (Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Grant & Blundell (2022) confirmed that their methods used for emission-line stars worked for the WR binaries WR 133 and WR 140 that have combined spectroscopic and interferometric orbits (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The primary star in the 휂 Car system is a luminous blue variable star, with the largest measured value for a mass-loss rate for a mas- sive star with �푀 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 × 10−4푀⊙yr−1 and a terminal wind speed of 푣∞ = 420 km s−1 (Davidson & Humphreys 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Prior to the recent kinematic studies of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) and Grant & Blundell (2022), the best constraints on the compan- ion star parameters, while indirect, came from the X-ray variability analyses from RXTE, Swift, and NICER observations of the sys- tem (Corcoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2001, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Espinoza-Galeas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These analyses point to a secondary star with a mass-loss rate on the order of �푀 ∼ 10−5푀⊙yr−1 and a terminal velocity of 푣∞ ∼ 3000 km s−1 (Pittard & Corcoran 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These values are broadly in agreement with the suggestion based on the merger models and mass-loss pa- rameters that the remaining secondary would be a Wolf-Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Despite recent work with long-baseline near-infrared interferome- try by Weigelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021), no direct detection of the companion star has been made to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' From the interferometric data, a mini- mum primary-secondary flux ratio of ∼50 was derived in the 퐾-band (Weigelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Given the extreme luminosity of the LBV pri- mary, this is consistent with any O or WR star in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The evolution of the secondary star may well have been signifi- cantly modified by interactions and mass exchange during formation of the present-day binary, but if the current secondary star is a clas- sical H-free Wolf-Rayet star as suggested by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2018) and Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021), or a hydrogen-rich WNh star, possibly the best line to detect it in the optical would be the He ii 휆 4686 line, which is the dominant line in the optical for the nitrogen-rich WR stars, or the hydrogen-rich WNh stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Most of the observations of He ii were made near periastron, where the He ii excess can be explained by ionization of He i in the colliding winds in a highly eccentric binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016) showed that the variability could be explained with the smoothed-particle hydrodynamics models of Madura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Away from periastron (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='04 < 휙 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='96), the He ii line is typically not observed with moderate resolving power and a nominal S/N of ∼100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In this paper, we present our analysis of the spectroscopy collected with the CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 m telescope and the CHIRON spectrograph, as well as the data collected with the previous spectrograph on that telescope with the aim of better constraining the kinematics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These observations are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In Section 3, we review the variability in the two Balmer lines we can easily measure (H훼 and H훽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Section 4 describes our techniques of mea- suring the radial velocity of the H훽 line, and presents observations of He ii away from periastron in the hope of determining the orbit of the companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We discuss our findings in Section 6, and conclude this study in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2 OBSERVATIONS We collected high resolution spectra of 휂 Carinae during the peri- astron passages of 2009, 2014, and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Many additional spectra were taken in the intermediate phases of the binary orbit as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These were collected from the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5m telescope at Cerro Tololo Inter- American Observatory (CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5) and both current CHIRON and the former fiber-fed echelle spectrograph (FECH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The data from the 2009 spectroscopic event spanned from 2008 October 16 to 2010 March 28, with approximately one spectrum taken every night be- tween 2008 December 18 to 2009 February 19, which were previ- MNRAS 000, 1–11 (2022) The spectroscopic orbit of 휂 Car 3 ously used by Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2010, 2015) and cover the spectral range ∼ 4700 − −7200Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These spectra with the fiber echelle1 were collected in late 2009 and 2010, and often had a signal-to-noise ratio around 80–100 per resolution element with 푅 ∼ 40, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In total, we analyzed 406 spectra of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The 2014–2020 data were collected with the new CHIRON spec- trograph (Tokovinin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2013), and spanned the time between 2012 March 2 and 2020 March 16, with high-cadence time-series span- ning the 2014 and 2020 periastron passages between 2013 Decem- ber 29 through 2015 April 21 as well as between 2020 January 3 to 2020 March 16 when the telescope shut down for the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The CHIRON spectra cover the spectral range of ∼4500- 8000Å, with some spectral gaps between orders in the red portion of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The data covering the 2014 periastron passage were previously used by both Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016) and Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These data have a spectral resolution of 푅 ∼ 80, 000 and typically have a signal-to-noise of 150–200 in the continuum and were all reduced with the CHIRON pipeline, which is most recently described by Paredes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In addition to the pipeline reduc- tions, we perform a blaze correction using fits from an A0V star, as done by Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016), allowing orders to be merged if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This process resulted in a flat continuum in regions that were line-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These observations were all fiber-fed with the fiber spanning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7′′ on the sky, meaning that the data include the nebular emis- sion from the Homunculus nebula formed from the eruption of 휂 Car in the mid-nineteenth century, as well as the Weigelt knots (Weigelt & Ebersberger 1986) that are thought to have originated from the second eruption in the 1890s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The CHIRON spectra are normalized through a comparison with a measured blaze function from the star HR 4468 (B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5V), as was done in the analysis of Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Example spectra are shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 1, with a comparison to a spectrum used by Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) and Grant & Blundell (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 3 MEASURED VARIABILITY IN THE BALMER LINES, H훼 AND H훽 Our observations are unique in providing both the spectral resolution and signal-to-noise to measure the line strength (equivalent width) and profile morphology of the emitting gas for the H훼 and H훽 lines of 휂 Carinae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Here, we detail the observations of the variability of the hydrogen lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We estimate errors on equivalent width using the methods of Vollmann & Eversberg (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We note that the analysis of Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2015) includes many optical wind lines near the 2009 periastron passage and phases far from periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These line profiles all show minimum line strength near periastron as the secondary’s high ionizing radiation goes behind the primary star’s optically thick wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We use a phase convention in which the low- ionization state observed by Gaviola (1953) in 1948 is deemed to be cycle 1, so that the low-ionization state starting in Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2020 marks the start of cycle 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We leave the kinematics analysis of the metal lines for a future analysis in order to confirm the results of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) and Grant & Blundell (2022) here, with plans of using higher signal-to-noise spectra in a future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='ctio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='noao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='edu/noao/content/CHIRON 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 H훼 Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2010) examined the variability of the H훼 profile of 휂 Carinae across the 2009 periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' They found that the profile’s strength decreased during the periastron passage and reached a minimum a few days following the X-ray minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' They postulated that the changes were caused by the drop in the ionizing flux from the secondary when the companion moved to the far side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In addition, they observed an appearance of a P Cygni absorption profile and an absorption component at −145 km s−1, that also appeared as the secondary’s ionizing radiation was blocked by the primary star’s optically thick wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2015) expanded upon this model to describe the variations of the optical He i profiles while documenting the variability of the optical wind lines across the 2009 periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We measured the equivalent width of H훼 for all of our spectra in the range 6500 – 6650Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2, where we show the measurements both compared to time and to binary phase, assuming a period of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 d, and the epoch point given by Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016), which represents the time of the periastron pas- sage based on a comparison of the He ii observations (Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016) to SPH models of the colliding winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Broadly speaking, the strength of the line relative to the locally normalized continuum shows a fast decrease and recovery near each periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2010) found that the variability is smoother when considering the photometric flux in the determination of the equiv- alent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We did not make this correction in these data, but do see the similarities of the events in the context of the raw equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' There is no strong long-term variability in these observations, and the 2014 and 2020 observations were nearly identical in their variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Recently, Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2019, 2021) found that there are long-term brightness and spectral changes of the system that has been ongoing for decades and accelerated since the mid-1990s, but now seems to be stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The shape of the H훼 variability has remained similar over these three well-observed periastron passages, and the line strength has stabilized across the past two cycles, which could indicate that the system is mostly stable aside from the binary- induced variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2010) also documented the timing of the ap- pearance of the P Cygni absorption component for H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In the 2009 observations we see the absorption occurring at approximately HJD 2454840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and still persisting through the last observa- tion, 2454881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2014 a P Cygni absorption occurs at 2456874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) persisting until the object was not observable at HJD 2456887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2020, the absorption is seen at 2458886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 (휙 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01) and still detected through the last observation on HJD 2458925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 (휙 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A narrow absorption component was observed near −145 km s−1 in the 2009 observations (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2010) from 2454836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) through the last day of observation, 2454881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2014 an absorption in the same location is observed from 2456863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) – 2456977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' There is no ab- sorption at this location strong enough to make a definitive detection in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Pickett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2022) documented the changes in absorp- tion behavior for the Na D complex at these velocities, showing that the absorption from these components associated with the Little Homunculus, formed during the second eruption in the 1890s, are weakening with time and moving to bluer velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 4 Strawn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2010 2012 2014 2016 2018 2020 Year 1000 2000 3000 4000 5000 HJD-2454000 200 300 400 500 600 Wλ (ANGSTROMS) CHIRON FECH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='050 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='075 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='100 ORBITAL PHASE ϕ 250 300 350 400 450 500 550 600 650 Wλ (ANGSTROMS) 11→12 12→13 13→14 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Variation in H훼 emission line with respect to time (left) and phase (right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' with the data taken between October 2008 and March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Data taken from the previous echelle spectrograph is indicated by open squares and data from the new CHIRON spectrograph is indicated by solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In the phase plot, we show the different cycles in different colors to clarify the timing of each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Furthermore, the errors are typically the size of the points or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The phase convention shown in the right panel references the low-ionization spectrum near periastron first observed by Gaviola (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2012 2013 2014 2015 2016 2017 2018 2019 2020 Year 2000 2500 3000 3500 4000 4500 5000 HJD-2454000 70 80 90 100 110 120 130 Wλ (Angstroms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='050 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='075 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='100 ORBITAL PHASE ϕ 70 80 90 100 110 120 Wλ (ANGSTROMS) 12→13 13→14 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Variation in H훽 emission line with respect to time (left) and phase (right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' with the data taken with CHIRON spectrograph as the FECH data were too noisy to determine equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In the phase plot, we show the two recent cycles in different colors to clarify the timing of each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Furthermore, the errors are typically the size of the points or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The phase convention shown in the right panel references the low-ionization spectrum near periastron first observed by Gaviola (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 H훽 While some of the H훽 variability was documented for the 2009 periastron passage of 휂 Car by Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2015), the full variability and timing of the changes is still not well documented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The lack of a more quantitative assessment of the variability is in part due to the lower signal-to-noise in the H훽 data from the 2009 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Similar to the H훼 profile, H훽 experiences a P Cygni type absorption near −500 km s−1 near periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We note the absorption appears in 2009 at approximately HJD 2454837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and persist through the last observation taken on 2454879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2014, it appears at approximately 2456863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='6 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and ends during a seasonal gap in observations beginning at 2456887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2020, the P Cygni absorption is observed between 2458886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 (휙 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and continues through the last day of observations on 2458925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 (휙 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This transient absorption was determined to be originating from the downstream bowshock by Gull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A narrow absorption component, previously observed by Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2015), is detected near −145 km s−1 in the 2009 observations from 2454837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and proceeds through the end of observations on 2452879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 (휙 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In 2014, this absorp- tion is observed between 2456864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00) and also persists through the last day of observations 2456887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 (휙 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As with H훼, there is no discernible absorption at −145 km s−1 in 2020 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Figure 3 shows the time series variation in the H훽 equivalent width over the last two periastron cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We note that the 2009 observa- tions are not included as they are recorded with the former echelle spectrograph and have lower signal-to-noise, though the appearance of the P Cygni absorption remains reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As with the H훼 equiva- lent widths, there is a consistency in the decrease in equivalent width for the time period corresponding to times close to periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4 LINE KINEMATICS We measured the bisector velocity of H훽 and the centroid position of the He ii 휆4686 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' H훽 measurements were taken during the 2009, 2014, and 2020 periastron events and the He ii 4686 measurements MNRAS 000, 1–11 (2022) The spectroscopic orbit of 휂 Car 5 0 5 10 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='947 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='998 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='212 0 5 10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='901 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='007 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='172 −750 −500 −250 0 250 500 750 0 5 10 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='946 −750 −500 −250 0 250 500 750 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='004 −750 −500 −250 0 250 500 750 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='019 Radial Velocity (km s−1) Normali ed Flux Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Example polynomial fits to H훽 emission lines from 2009, 2014, and 2020 periastron events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The profiles are shown in black with the portion of the line wings fit with a polynomial shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The bisector velocity is shown as a vertical line corresponding to the normalized flux at the same level as the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Near the edges of these ranges, the bisector often appears to curve due to either profile asymmetries or larger errors in the polynomial fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The bisector velocities between normalized flux levels of 5 and 6, indicated by the dashed lines, were averaged to obtain a final relative velocity for each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Further details are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' were taken for 2014 and 2018 and do not include time within 휙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='95 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='05 to avoid observations affected by periastron caused by colliding-wind effects which, to first order, behave with a 퐷−1 trend for adiabatic and 퐷−2 or steeper for radiative conditions, where 퐷 is the orbital separation, which is small and quickly changing at periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016) show the behavior of the He ii 4686 line near periastron in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' All measurements are tabulated in online supplementary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 Bisector velocities of H훽 The process used to find the bisector velocity of H훽 is demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) and Grant & Blundell (2022) used a method of Gaussian decomposition using many components to moderate-resolution spectroscopy taken with Gemini-South and GMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Their GMOS spectra of 휂 Car are limited in that the highest resolving power available is ∼ 4400, whereas our spectroscopy has a resolving power of 40, 000 from the fiber echelle, and 80, 000 for the CHIRON data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The profiles become more complex at higher spectral resolution, making this multiple-Gaussian method more difficult to implement, likely requiring more than twice as many components compared to the work of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In order to create a simpler measurement that has reproducible results for any spectroscopic data set, we implemented a bisector technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We began by fitting two fourth degree polynomials, one to the red side and another on the blue side of the profile in order to smooth over any noise inherent in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Through this fit, we were then able to establish the bisecting velocity position at each emission level with higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Example fits are shown in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In the regions of heights of 4× the continuum up to 10× the continuum, we calculate the bisecting velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This area was chosen based on the relatively vertical nature of the bisector in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We then created comparisons of all spectra and found that the bisecting line was nearly always vertical in the region of 5 − 6× the normalized continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We therefore used this region, measuring the velocity at every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 increment between these values, and adopting an average measurement as the radial velocity for the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The choice of a common emission height with which to measure the bisector velocities allows us confidence in the results as it would relate to gas emitting from the same region for all spectra, whether the line is weak or strong in that particular observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The resulting velocities MNRAS 000, 1–11 (2022) 6 Strawn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2008 2010 2012 2014 2016 2018 2020 Year 0 1000 2000 3000 4000 5000 HJD-2454000 −80 −60 −40 −20 0 20 40 60 Velocity (km s−1) FECH CHIRON −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 Phase −80 −60 −40 −20 0 20 40 60 Velocity (km s−1) 11→12 12→13 13→14 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Radial velocities from H훽 bisector measurements compared to time (left) and orbital phase (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The orbital fit is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 and typical errors are on the order of the size of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' −400 −200 0 200 400 Velocity (km s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='03 Normalized Flux ϕ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='08 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Gaussian fit to an example He ii emission line with a vertical line plotted at the fitted peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This particular spectrum had a signal-to-noise ratio of 210 per resolution element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We provide this bisector code via GitHub2 for future use on comparable datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 He ii 휆4686 The region surrounding, but not blended with, the He ii 휆4686 tran- sition is complicated by several features including narrow emission lines from the Weigelt knots (Weigelt & Ebersberger 1986) along with wind emission from Fe ii and He i lines (for a figure showing that region of the spectrum, see Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' While these do not directly overlap with the core of the He ii line, they can complicate this fitting if not properly avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The He ii 휆4686 line has usually been observed near periastron passage when the line is dominated by the wind-wind collisions, which has been docu- mented and modeled by Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The line was discov- ered by Steiner & Damineli (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Since then, multiple studies have attempted to explain the formation of the stronger line observed near periastron (퐿He ii ∼ 300 퐿 ⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Mehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2011, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Davidson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2015), but the colliding 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='com/EmilysCode/Radial-Velocity-from-a-Polynomial-Fit- Bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='git wind model best reproduces the emission near periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This emis- sion is strongest for times within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='05 in phase from periastron, as detailed in the recent analysis of Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Outside of the phase intervals near periastron, the He ii 휆4686 line could only be properly observed with high spectral resolution and high signal-to-noise data (Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Our data taken with CHIRON, after the 2014 periastron passage has the necessary sensitivity to detect this notably weak emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We measure the radial velocity of this line outside of 휙 = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='05 of periastron, so that it minimizes the effects of the colliding winds that peak at periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 6, we fit a Gaussian to the He ii emission line and use the centroid position to determine the radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Un- fortunately, the continuum placement for the feature is not reliable enough to measure equivalent widths with precision, but the line was nearly constant in equivalent width when considering the errors of these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Before fitting the 2018 observations near apastron, we needed to average up to ten observations to improve the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The resulting velocities are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 7 with a resulting total of 19 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The averaging of the points from the 2018 data resulted in a smaller dispersion of the data than seen in the earlier points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The He ii line is normally absent in the spectra of luminous blue variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The extreme mass-loss rate of 휂 Car does not preclude this emission line originating in the primary star’s wind, as there are some combinations of parameters used that can create this weak emission feature in CMFGEN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These models and parameters are very sensitive and depend on the mass-loss rate and stellar radii used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The He ii can be formed through strong wind collisions at times close to periastron (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, this line moves in opposition to the primary star’s motion, so we consider this feature as originating from the companion during these phases far from periastron for the remainder of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 Orbital Kinematics and Observed Elements We began our fit of the kinematics of the primary star with the BinaryStarSolver software (Milson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Barton & Milson 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The resulting orbit is broadly in agreement with the orbit derived with H훽 velocities by Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020), with the orbital elements given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Our resulting fits are in agreement with those of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) so we did not perform the same correction for the stellar wind effects as in their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) The spectroscopic orbit of 휂 Car 7 Line 푇0 (HJD-2400000) 푒 퐾 (km s−1) 휔 (degrees) 훾 (km s−1) Source Pa훾 48800 ± 33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='08 53±6 286 ±6 −15 ± 3 Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1997) Pa훾, He i 6678 48829± 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='802 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='033 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 286 ± 8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 Davidson (1997) Pa훾, Pa훿 50861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='75 50 275 12 Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2000) H훽 54854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='82 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='02 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 254 ±4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) All Balmer lines 54848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='91 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 241 ±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) Upper Balmer lines 54848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='89 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='00 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 246 ±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) H훽 56912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8100 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0007 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='13 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='08 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='43 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='34 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='10 This work(BinaryStarSolver) H훽 56927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8041 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0008 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='6 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='83 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='09 This work (PHOEBE) He ii 56973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='937 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='001 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='6 (fixed) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 This work Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Orbital elements from previous publications and the results from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' For the orbits of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020), Grant & Blundell (2022), and our work, the period has been held constant at 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 d, while it was fit in the earlier work of Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (1997), Davidson (1997), and Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2000) with periods that agree with 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='7 d within their errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Note that our errors from the PHOEBE code may be underestimated, especially for the He ii line (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 6500 7000 7500 8000 8500 9000 HJD - 2,450,000 50 0 50 100 150 200 RADIAL VELOCITY (km s-1) 2014 2016 2018 2020 YEAR Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Radial velocity as determined using centroid positions in He ii emission at phases away from periastron during 2014–2018 with CHIRON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We overplotted the He ii orbit from Table 1, along with the H훽 solution from our work shifted to the same 훾-velocity as the He ii orbit as a grey dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In an attempt to fully assess the errors of the parameters, we used the PHOEBE code (PHysics Of Eclipsing BinariEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Prša & Zwitter 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Prša et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016) to verify the orbital elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The latest version of PHOEBE incorporates the Markov Chain Monte Carlo package emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Unlike traditional or- bit fitting routines, PHOEBE fits using the variable of the projected semi-major axis (푎 sin 푖) rather than the semi-amplitude 퐾, but these are easily interchangeable using 푎 sin 푖 = (1 − 푒2)1/2 2휋 퐾푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These orbital elements are also similar to the other published orbital elements measured with H훽, and the resulting orbit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The distribution of the errors from the Monte Carlo simulation, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 8, is tightly constrained but shows that various orbital elements have errors that are interdependent with other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' While this represents the best solution to the entire data set, we explored how the parameters change if we kept only the densest of the three periastra observed (the 2014 event).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Running the PHOEBE code with the MCMC package on just those data resulted in the eccentricity being slightly larger (푒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='824), the time of periastron being later (HJD 2,456,935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='31), and the value of 푎 sin 푖 (hence 퐾1) being slightly larger at 2620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4 푅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These values are outside the limits given with our MCMC fit of all of the data, so we caution that the errors in Table 1 are likely underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We include the fit parameters in the same style as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 8 in the online Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Once the orbital elements for H훽 were fit, we proceeded to run a simpler model for the He ii emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' For this PHOEBE model, we keep 휔 constant to that representing the primary star from the upper Balmer line results from Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, we do allow the semi-major axis, 훾-velocity, 푒, and time of periastron passage to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The resulting orbit is more eccentric than that of the primary star when derived using H훽 (and a bit more eccentric than the Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) solution) and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' With future observations of the He ii line at times away from periastron, a combined double-lined orbit of the system with 휔 being consistent for the two stars will be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 5 DISCUSSION The optical spectrum of 휂 Car is dominated by emission lines from the wind of the primary and its ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The dominant emission lines are the hydrogen Balmer lines, but there are strong lines from He i and Fe ii in the spectrum as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The He i lines, when considered in non-LTE stellar wind models, are a strong function of the adopted value of the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, if most of the He i emission comes from the colliding wind interaction region, it forces a larger stellar core radius value for the primary star, ∼ 120푅⊙ in the pre- ferred models (see Hillier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2001, for many further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The model of Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2012) improved previous spherically symmet- ric models of Hillier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2001) in that the spectrum was modeled with a cavity carved from the wind of the secondary, which was in- cluded along with a central occulter or “coronagraph" that extended ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='033′′ to allow for stronger He i emission, and better agreement for the P Cygni absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Given the spectral modeling agree- ment for the spectroscopically similar star HDE 316285 (Hillier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 1998), the strong disagreements for the absorption components and He i lines led to an interpretation that the He i lines are formed in the wind-wind collision region of the system (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Indeed, the P Cygni absorption component variability of the optical He i lines seems to represent the outflowing shocked gas from the wind-wind collision region (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These results all indicate that the best lines in the optical for determination of the orbit may indeed be the upper hydrogen Balmer lines, even if they are likely modified by the wind collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' All of the measured orbits, including ours, rely on measurements taken when the line profiles are most variable near periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This MNRAS 000, 1–11 (2022) 8 Strawn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0 R⊙ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='10 vγ (km s ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='83+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='09 km s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8065 ebinary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8041+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0008 6 7 8 9 t0, perpass, binary (d) +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='45692e6 2456927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 d 2584 2592 2600 2608 abinarysinibinary (R ) 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 ω0, binary (∘) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='10 vγ (km s ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8065 ebinary 6 7 8 9 t0, perpass, binary (d) +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='45692e6 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='8 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='1 ω0, binary (∘) 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='2 ∘ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Results of the Markov chain Monte Carlo fit for the H훽 velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Note that 휔0 refers to the value of 휔 for the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' likely causes additional errors in the parameters derived, but we tried to always sample emission from the same line formation re- gion by taking bisector velocities at the same height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Furthermore, our technique produces nearly the same orbital elements as those from Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) in the case of H훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) pro- ceeded to correct the orbital elements by considering the effects of the outflowing wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These results all show that the system is a long-period and highly eccentric binary where the primary star is in front of the secondary at periastron, causing the ionization in our line of sight to drop during the “spectroscopic events" due to a wind occultation of the secondary at these times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The results of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) show that the higher-order Balmer lines give different results than that of the lower-level lines such as H훼 or H훽, which is expected as the higher level lines form deeper in the wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Hillier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As such, the results of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020) and Grant & Blundell (2022) should be considered the best for the primary star at the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Similar differences in the orbital kinematics is sometimes inferred for Wolf- Rayet stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', 훾2 Vel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Despite the detection of the He ii 휆4686 emission at times near apastron by Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2016), the exact formation channel for this line remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The emission lines in colliding wind binaries often vary as a function of the orbit due to the colliding wind line excess (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2000), and the modeling of these variations has been done in the context of the so-called Lührs model (Lührs 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Recently, the excess emission was observed to be a strong cooling contributor when X-ray cooling becomes less efficient in the colliding wind binary WR 140 (Pollock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In WR 140, the MNRAS 000, 1–11 (2022) The spectroscopic orbit of 휂 Car 9 Lührs model was used by Fahed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2011) to explain the variations in the C III 휆5696 line near periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The Lührs model can explain changes in the radial velocity and the width of the excess emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 6, we detect the He ii line with our spectra, but the actual characterization of this line will have large errors in line width due to the limited signal-to-noise for the detection in the spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We used the models for WR 140 (Fahed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2011) as a starting point, changing stellar and binary parameters as appropriate to the 휂 Carinae system to investigate if the He ii velocities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 7 were from colliding wind excess emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' For the velocity of the outflow, we can see that during the periastron passage of 2014, 휂 Car’s outflow reached velocities faster than the primary star’s wind speed based on the optical He i lines (Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2016), which are slower than the excess absorption seen to reach nearly 2000 km s−1 in the meta-stable He i 휆10830 line (Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' With these velocities, we expect to see the observed amplitude of the excess increase between the times of 2015 and 2018 like we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 7, but with amplitudes of at least 1000 km s−1, much greater than the ∼ 100 km s−1 observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Therefore, the analysis of the He ii 휆4686 emission line at times away from periastron from the CHIRON spectra is an important observation towards understanding the nature of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We note that the data indicate a narrower emission line profile then expected from the parameters inferred for the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, the primary star dominates the spectrum, and the motion of this peak opposite the primary indicate that the He ii excess could be from the secondary’s wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In particular, the Lührs models of the kinematics of the He ii line seem to exclude the possibility that the line is formed in the colliding winds at times away from periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The models of Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2018) suggest that the companion should be a classical Wolf-Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The classical hydrogen-free Wolf-Rayet stars can be split into the WN and WC subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The WN stars show strong He and N lines, with the He ii 휆4686 typically being the strongest optical line, whereas the WC subtype exhibits strong He, C, and O lines with the C IV 휆휆5802,5812 doublet often being the strongest optical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' There is also the rare WO subtype, which is similar to the WC subtype but shows more dominant O lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The WO stars were recently shown to have higher carbon and lower helium content than the WC stars, likely representing the final stages of the WR evolution (Tramper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Aadland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Given the generalized characteristics of WR stars, a WN star would seem the most likely companion star if the He ii 휆4686 line is from the companion at times further from periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' For contrast, the Carina nebula is also the home to several hydrogen-rich Wolf-Rayet stars: WR 22, WR 24, and WR 25 (Rosslowe & Crowther 2015)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This type of WR star tends to be considered the higher mass and luminosity extension of the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As such, these stars have masses in excess of ∼ 60푀⊙, with the R145 system in the LMC having masses of the two WNh stars being 105 and 95 푀⊙ (Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Like the classical WN stars, these stars have similar nitrogen and helium spectra, along with stronger emission blended on the Balmer lines which overlap with Pickering He ii lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The region surrounding the He ii 휆5411 line in our 휂 Carinae spectra does not exhibit emission lines at the same epochs as our observations of He ii 4686, making it difficult to quantify the companion’s properties without the higher order He ii lines which would also be notably weaker than He ii 휆4686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' With the assumption that the He ii orbit shown in Table 1 is from the companion star, and that the semi-amplitude from the higher- 3 http://pacrowther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='shef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='uk/WRcat/ order Balmer lines for the primary star (Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2020), then the semi-amplitude ratio shows that the primary star is 2–3 times more massive than the secondary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This is also an indicator that the companion is not likely a WNh star, as that would imply the primary star could have a mass of in excess of 100 푀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Models of the system, such as those by Okazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2008) and Madura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2013), typically have the masses of the primary and secondary as 90 and 30 푀⊙ respectively, broadly in agreement with the kinematics of the orbits presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' On the other hand, if 휂 Carinae A has a mass of > 100푀⊙, the secondary would have a mass on the order of 50–60 푀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This is similar to the nearby WNh star in the Carina nebula: WR22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The mass of this WNh star in an eclipsing system is 56–58 푀⊙ (Lenoir-Craig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The tidally-induced pulsations observed by Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2018) were modeled with stars of masses 100 and 30 푀⊙, and therefore may also support the higher masses suggested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Most models for 휂 Car have a preferred orbital inclination of 130–145◦ (Madura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2012), which agrees with forbidden [Fe iii] emission observed with Hubble Space Telescope’s Space Telescope Imaging Spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This inclination can be used with the mass function derived from the primary star’s orbit 푓 (푀) = 푚3 2 sin3 푖 (푚1 + 푚2)2 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='0361 × 10−7)(1 − 푒2)3/2퐾3 1푃[M⊙] to constrain the system’s masses with the mass function using the standard units measured and our measured H훽 orbit using PHOEBE (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The mass function is 푓 (푀) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='05 M⊙, and would indicate a companion star with a mass of at least 60 푀⊙ if we assume a primary mass of ∼ 90 푀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Given the actual mass functions for the measured upper Balmer lines and He ii orbits, the minimum masses required for these measured orbits are 푀 sin3 푖 = 102푀⊙ for the LBV primary and 푀 sin3 푖 = 55푀⊙ for the secondary, making the companion star’s identification as a WNh star more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These results are still preliminary and require follow-up observations to constrain the orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' A WNh star can account for the mass of the secondary star in 휂 Car, but could cause some difficulty for the modeling of the Great Eruption models of Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' In that scenario, the companion star would be a hydrogen-stripped star, contrary to the hydrogen content of the WNh stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Recently modeled WNh systems such as R144 (Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 2021) show that the surface fraction of hydrogen is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This does show some amount of lost hydrogen on the surface, so the scenario could still be relevant even if the final star is not a fully stripped classical Wolf-Rayet star, assuming that the evolution of the secondary star has not been significantly influenced by mass exchange prior to or during the merger event hypothesized by both Portegies Zwart & van den Heuvel (2016) and Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' 6 CONCLUSIONS In this paper, we provide an orbital ephemeris for 휂 Carinae mea- sured with a bisector method and high resolution ground-based spec- troscopy of the H훽 emission line, along with an ephemeris for the He ii 휆4686 emission line at times far from periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Our findings can be be summarized as follows: The H훽 emission profile tracks the primary star, and our bisec- tor method provides similar results as the multiple-Gaussian fitting method used by Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The results show a high ec- centricity orbit of the system with the primary star in front of the secondary at periastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 10 Strawn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The weak He ii 휆4686 emission tracks opposite the kinematics of the primary star, suggesting it is formed in the secondary star’s windat timesawayfromperiastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Thiscouldsupport thehypothesis of the scenarios presented by Hirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2021) for a stellar merger being the cause of the Great Eruption as the secondary could be a Wolf-Rayet star that has leftover hydrogen on its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' With the assumed inclination of 130–145◦, the masses of the stars could be around ∼100 푀⊙ for the primary and at least 60 푀⊙ for the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, the mass ratio derived by comparing the two semi-amplitudes is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' New observations will be needed to better determine precise masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Future studies will be able to better measure the He ii 4686 orbit and refine its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' As shown in Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020), the upper Balmer lines are more likely to reflect the orbital motion of the stars, and the upper Paschen lines will also be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' However, our work shows that a simpler bisector measurement of higher resolution spectroscopy results in the same derived orbital elements as that of Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' Furthermore, with better signal-to-noise spectra, we can better determine if the He ii emission near periastron can be reproduced with a Lührs model or if it is a signature of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' With this information, we will be able to more precisely measure the kinematics of the two stars and the mass function, and then we can begin to better understand the current evolutionary status of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank our referee, Tomer Shenar for many suggestions that improved this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' These results are the result of many alloca- tions of telescope time for the CTIO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5-m telescope and echelle spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' We thank internal SMARTS allocations at Geor- gia State University, as well as NOIR Lab (formerly NOAO) al- locations of NOAO-09B-153, NOAO-12A-216, NOAO-12B-194, NOAO-13B-328, NOAO-15A-0109, NOAO-18A-0295, NOAO-19B- 204, NOIRLab-20A-0054, and NOIRLab-21B-0334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' This research has used data from the CTIO/SMARTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='5m telescope, which is operated as part of the SMARTS Consortium by RECONS (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='org) members Todd Henry, Hodari James, Wei-Chun Jao, and Leonardo Paredes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' At the telescope, observations were car- ried out by Roberto Aviles and Rodrigo Hinojosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' were partially supported by the Embry-Riddle Aeronautical Univer- sity Undergraduate Research Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' acknowledges support from the Arizona Space Grant program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' acknowledge support from the HST GO Programs #15611 and #15992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' AD thanks to FAPESP (2011/51680-6 and 2019/02029- 2) for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' AFJM is grateful for financial aid from NSERC (Canada).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The material is based upon work supported by NASA un- der award number 80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' The work of ANC is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfRPbk/content/2301.00064v1.pdf'} +page_content=' DATA AVAILABILITY All measurements can be found in Appendix A.' metadata={'source': 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authors; +Email: kasirga@unam.bilkent.edu.tr; abdulsalam@unam.bilkent.edu.tr + + +2 + +ABSTRACT +The need for novel multifunctional nanomaterials capable of meeting new demands in the realm +of nanotechnology coupled with versatility of chemical vapor deposition (CVD) technique (in +large-area growth of crystals), encourages innovative methods for synthesis of untried two- +dimensional (2D) crystals. While there exist reports on both top-down and bottom-up synthesis +methodologies of different Cu2S-based nanostructures, CVD-based synthesis of 2D crystals of +copper(II) sulfide (CuS) has not been investigated. This work represents details of CVD method +in systematic growth of highly crystalline 2D CuS sheets as thin as ~ 6 nm with lateral sizes +exceeding 60 μm, at a relatively low temperature of 560 °C in ambient pressure. Samples were +characterized via X-ray diffraction, Raman, atomic force microscopy, and high-resolution +transmission electron microscopy. SAED revealed a 6-fold symmetric structure and identical +atomic ratio of copper:sulphur was corroborated from the energy-dispersive X-ray spectroscopy. +The as-prepared 2D CuS sheets were successfully utilized in second harmonic generation (SHG) +and their strong response was found to be highly polarization angle-sensitive as well. The CVD- +synthesized 2D CuS crystals in this study are considered to be of great significance in a diverse +range of future applications, as in energy storage, next-generation solar cells, nonlinear +optoelectronic-related devices, and even bioelectronics pursuits. +KEYWORDS: 2D materials, CVD method, Covellite, Nonlinear optics, Second harmonic +generation. + +3 + +ToC + +560C +2D lattice of +Arflow~ +Copper (l) Sulfide +S +CuCl +2W +w +SecondHarmonic +Generation yia CuS4 + +Two-dimensional (2D) materials could make the way for myriad of unprecedented functional +devices such as high on/off ratio field-effect transistors (at room temperature),1 electric nose,2 +mode-locked laser,3 and broad-band photodetectors.4-5 However, top-down methods of +chemical exfoliation,6 solvothermal,7 or supercritical8 lead to defective structures that deviate +from the required parameters regarding fabrication of highly efficient nanodevices sought in +different areas of optoelectronics, bioelectronics,9 and spintronics.10-13 Among bottom-up +approaches for synthesis of highly-ordered crystalline 2D structure14-16, chemical vapor +deposition (CVD) provides high-quality and high-yield products.17-18 Engineering its +corresponding parameters such as temperature, substrate, gas flow, dwell time, fast/natural +cooling, can herald novel 2D structures, pursued in realization of innovative devices.19-21. +Copper(II) sulfide (covellite, CuS), being highly conductive, chemically stable, and +having ultralow thermal conductivity, is widely used in solar cells, batteries, and photothermal +treatments.22-27 CuS nanoparticles have been reported to be prepared via wet-chemistry, as a +promising candidate for a lithium-ion battery.28 In another report, their photocatalysis property +was investigated and attributed to the large specific surface area.29 In addition, light harvesting +and charge separation activities can be significantly enhanced by nanosheets of ZnIn2S4/CuS,30 +without necessity of co-catalyst thanks to the strong interactions between assembled p-n +heterostructures. Some nanoflakes of CuCrS2 showing switchable ferroelectric polarization +have been also reported to be synthesized recently.31 While important phenomenon of +superconductivity has been theoretically predicted from 2D lattice of CuS,32 the +physicochemical properties of 2D copper-based chalcogenides have scarcely been studied and +there exists no report on CVD growth of 2D CuS yet. +Highly crystalline 2D CuS synthesized in this work, are produced using a single-step +CVD technique at a relatively low temperature of 560 °C. The as-grown 2D CuS sheets having +nanometer thickness, were characterized by atomic force microscopy (AFM), X-ray diffraction + +5 + +(XRD), Raman, high-resolution transmission electron microscopy (HRTEM), and energy- +dispersive X-ray spectroscopy (EDX). A 6-fold symmetric structure was revealed via selected +area diffraction (SAED). As an application of 2D CuS, a single sheet of it was utilized in second +harmonic generation (SHG) with the nonlinear susceptibility of up to 1.4 × 10−11 m/V. In +addition, the nonlinear optical characteristic of 2D CuS crystals was utilized in broad-spectrum +wavelength and polarization-resolved SHG. + +Copper(I) chloride (CuCl) powder was opted as the Cu source for growth of 2D CuS +lattices, due to its suitable chemical property and the relatively low melting point temperature. +An asymmetric tiny crucible was chosen to this end, filled with scant amount of CuCl powder, +and put in middle of tubular CVD furnace as can be seen in the schematic setup in the supporting +information (Figure S1). During the synthesis process, the optimum growth temperature was +found to be about 560 °C, way lower than the other CVD synthesis33 of copper-based +chalcogenides. Additional information regarding the CVD growth process is provided in the +experimental section. Figure 1a shows a typical optical image of CuS crystals grown on a mica. +The lateral length of the grown 2D CuS crystals can be up to 70 µm (Figure 1b). Mica is used +as a substrate because of its atomic-level smooth and inert surface, which has been widely +reported as a favorable substrate for 2D material synthesis.34 AFM height trace map given in +Figure 1c confirms that the surface of 2D CuS is very smooth and the thickness was found to +be about 14 nm according to AFM measurements. + +6 + + +Figure 1: Typical OM image of the as-grown 2D CuS crystals (a and b), the scale bars are 10 +and 20 μm, respectively. AFM image of the CuS crystal (c), and its height profile of the in the +inset. XRD pattern of 2D CuS crystals on SiO₂/Si substrate (d). EBSD inverse pole figure +(IPF) map along the c-axis of 2D CuS crystal on SiO₂/Si substrate (e), the length of the scale +bar corresponds to 5 μm. Color coded map type of IPF (f). + +XRD pattern of as-grown (Figure S2) and transferred CuS crystals on the SiO2/Si +substrate depicted in Figure 1d clearly identifies the hexagonal phase of CuS (PDF No. 06- +0464). The strong characteristic peaks of (002), (006), and (008) show that 2D CuS crystals +preferentially grow in the basal plane (00l). The plane interspacing can be estimated using +Bragg’s relation, 𝑛𝜆 = 2𝑑(ℎ𝑘𝑙)𝑠𝑖𝑛(𝜃) where n is an integer corresponds to the other of +diffraction peak and λ is wavelength of X-ray. The mean dimensions of the crystallite +perpendicular to the (00l) plane (L002) can be determined by using Scherrer equation, +𝐿(ℎ𝑘𝑙) = 𝐾𝜆/𝛽𝑐𝑜𝑠(𝜃) + +a) +b) +c) +30nm +Position(μm) +d) +(006) +f) +Intensity (a.u.) +1010 +Cus crystal +PDF06-0464 +(002) +IS +(008) +0001 +2110 +10 +20 +30 +40 +50 +60 +20 (deg.)7 + +where K is a constant (0.89) and β is the integral full widths at half maximum (in radians, in +our case 0.07 regarding 2θ peak at 11°). The average number of layers can be estimated by +simply dividing L(002) over d(002). Therefore, the average number of layers was found ~ 16, +implying the presence of multi-layer sheets in the structure, consistent with AFM investigations. +In addition, we utilized electron backscatter diffraction (EBSD), a powerful method for +identifying the microstructural characterization of materials, to determine the crystallographic +orientation of the 2D CuS crystals.35-36 Figure 1e-f displays a uniform color contrast of the +EBSD inverse pole figure (IPF) map within the hexagonal domains along the basal plane of +CuS ([00l] direction), implying a single-crystalline nature and ordered in-plane orientation +throughout the hexagonal CuS crystal, which is consistent with the XRD results. As illustrated +in Figure S3, the CuS structure belongs to the space group P63/mmc (hexagonal symmetry) +with Z = 6 per unit cell. Cu atoms exist in two types of environments: CuS3 (triangular planes) +and CuS4 (rectangular planes) (tetrahedra). The unit cell can be assumed as plates connected by +S-S bonds, and through triangular planes, vortices merge the tetrahedral units. According to +previous research, the Cu(1)-S(1) bonds (∼2.19 Å ) which occur in triangle units, have a length +much shorter than the Cu-S bonds seen in most other copper sulfides (∼2.33 Å).37 Because of +this, it is rather conceivable that the Cu(1)-S(1) bond will have a stronger bond. It has also been +reported that Cu(1) ions in the [Cu(1)-S(1)3] triangles exhibit significantly high thermal motion +along the c-axis.38 +Raman spectroscopy with a 532 nm excitation laser was used to investigate the intrinsic +properties and identify the fingerprint of the 2D CuS crystal structure. As shown in Figure 2a, +the Raman spectrum of the 2D CuS crystal shows four distinct Raman peaks at 90.1, 130, 279, +and 471 cm-1, representing E2g, A1, Eg +1, and A1 modes, respectively. Among these peaks, the +strong characteristic peak at 471.0 cm-1 can be attributed to the stretching mode of the S-S bond, +corresponding to the S2 groups of the recognized crystal structure of 2D CuS lattice.39-40 + +8 + + +Figure 2: Raman spectrum of the 2D CuS lattice on mica substrate (a). Spatially resolved +Raman mapping images (b) of the 2D CuS characteristic peaks E2g +3 (P1), A1g +2 (P2), E2g +2 (P3), +and A1g +1 (P4), scale bars: 5 μm. Temperature-dependent spectra (c) of 2D CuS crystal (80-300 +K, step: 20 K). Raman peak positions of 2D CuS (P1-4) as a function of the measured +temperature (d). + +Temperature-dependent Raman spectroscopy is a classical method to study the atomic +bonding and thermal expansion of 2D materials.41-42 The spatially resolved Raman mapping +images (Figure 2b) of the four characteristic peaks (60, 138, 267, and 471 cm–1) exhibit +uniformity throughout the 2D crystalline sheet of CuS. Figure 2c shows the typical +temperature-dependent Raman spectra for the grown 2D CuS crystal at temperatures ranging + +a) +b) +P2 +Intensity (a.u.) +P4 +A2 +2c +100 +200 +300 +400 +500 +600 +Raman shift (cm-1) +c) +d +P1 +P4 +476.1 +P4 +P2 +P3 +Fit +300 K +473.8 +471.5 +Slope=-0.02535 +Raman shift (cm' +Intensity (a.u.) +267.5 +P3 +Fit +265.0 +262.5 +Slope=-0.02563 +140.0 +P2 +Fit +137.2 +134.4 +Slope=-0.02677 +P1 +60.48 +Fit +59.85 +80 K +59.22 +Slope=-0.00662 +60 +120 +100 +200300400500 +180 +240 +300 +600700 +800 +Raman shift (cm-1) +Temperature (K)9 + +from 80 to 300 K. It can be clearly seen that the positions of the peaks exhibit a slight "redshift" +with increasing temperature, which is mainly due to anharmonic vibrations of the lattice +induced by the thermal expansion of the lattice at elevated temperatures.43 The correlation +between them can be described by a linear equation: 𝜔(𝑇) = 𝜔0 + 𝜒𝑇, where 𝜔0, T, and χ +are the Raman peak position at 0 K, the Kelvin temperature, and the first-order temperature +coefficient, respectively. As shown in previous reports,44-45 the first-order temperature +coefficient of 2D materials is related to the van der Waals interaction between the neighboring +layers and is usually used to explain the temperature dependence of the Raman peak shift. +Notably, the derived χ-values for P1, P2, P3, and P4 of CuS crystals are - 0.00662, -0.02677, - +0.02563, and - 0.02535 cm-1K-1, respectively (Figure 2d), which is larger than that of ordinary +layered materials.46-47 +Further analyses, such as high-resolution transmission electron microscopy (HR-TEM), +SAED, and EDX, were carried out to investigate the crystal structure and atomic composition +of the 2D CuS crystal. Figure 3a shows Fast Fourier Transform-filtered HR-TEM image and it +can be clearly seen that the atoms are arranged hexagonally. The interplanar spacing of the two +planes crossing at an angle of 120° is 0.35 nm, corresponding to planes (100) and (010), +respectively. The corresponding SAED image shows a 6-fold symmetric structure with an [001] +axis presented in Figure 3b, and the EDX spectrum of the 2D CuS crystal is shown in Figure +3c. The detected peaks suggest that the crystal is made entirely of Cu and S components, which +is supported by X-ray photoelectron spectroscopy (XPS) results which is shown in Figure 3d- +e. + +10 + + +Figure 3: Structural and chemical compositional characterization of 2D CuS crystals. +High-resolution TEM image. Scale bar: 5 nm (a). The SAED patterns, scale bar: 5 nm-1 (b), +EDX spectrum, inset shows atomic ratios of the chemical composition (c), XPS spectra +deconvoluted peaks of Cu2p (d), and S2p (e) core levels. + +New materials that have nonlinear optical response, can be of beneficial application in +different areas ranging from photon generation, imaging, and photon manipulation in ultrafast + +0.35 nm +0.35 nm +(010) +(100) +b) +Intensity (a.u.) +Cu +s +cu +52 % +48 % +（100 +S +Cu +（010) +cu +0 +2 +4 +6 +8 +10 +Energy (keV) +d) +Cu,2p3/2 +e +Intensity (a.u.) +S 2p3/2 +Intensity (a.u.), +S 2p1/2 +Cu 2p1/2 +096 +950 +940 +930 +920 +170 +168 +166 +164 +162 +Binding energy (eV) +Binding energy (eV)11 + +lasers, optical modulators, and pulse characterization.48-53 Stacking faults existing in the +synthesized CuS crystals (in this study), such as an interlayer slip, dislocation, and undulation +of the atomic layers, can induce multi-oriented domains in the crystal and deemed responsible +for the observed nonlinear optical behavior.54 The SHG is a very useful technique, where the +incident laser (ω) generates an (2ω) response, as shown in Figure 4a. The SHG response of a +CuS crystal (14.5 nm) under various incident laser wavelengths from the edge of visible light +to near-infrared (760 to 1020 nm) is presented in Figure 4b, which shows a wide spectrum +response with distinct wavelength selectivity. Moreover, the SHG mappings display a uniform +response throughout the entire 2D CuS lattice (Figure 4b inset). +Evolution of SHG intensity with changing incident laser power was also further +systematically investigated. With increasing the incident laser power from 0.7 to 1.6 mW under +800 nm laser excitation, the intensity of the SHG signal at 400 nm exhibits significant +enhancement (Figure 4c). The relationship between SHG intensity and laser power was fitted +linearly in the log-log coordinate, as displayed in Figure 4d. Interestingly, the slope of 2.05 is +close to the theoretical value of 2 calculated from the electric dipole theory.41 + + + + + +12 + + +Figure 4: SHG characterization of 2D CuS crystal. Basic mechanism of nonlinear optical +effects (a), The SHG spectra of 2D CuS crystal under various excitation wavelengths (760 - +1020 nm). Inset is the SHG mapping of 2D CuS crystal under 800 nm laser excitation, scale +bar corresponds to 3 μm (b), The SHG spectra of the 2D CuS crystal with different incident +powers (c), The SHG intensities as a function of incident power (d), Polarization angle- +dependent SHG intensity under parallel (e), and perpendicular (f) polarization configurations +(The excitation laser is 800 nm with a power of 1.2 mW). + + +a) +b) +(a.u.) +2=760-1020 nm +5k +3 +2w +m +2w +4k +SHG intensity +3k +3 +2k +SHG +1k +Mica substrate +380400420440460480500 +520 +Wavelength (nm) +6k +0.7 mW +1.3 mW +Data +(a.u. +5k +0.8 mW +1.4 mW +Linear fit +0.9 mW +1.5 mW +a +4k +Intensity +1.0 mW +1.6 mW +3k +1.2 mw +Intensi +2k +Slope = 2.05 +1k +385 +390 +395 +400 +405 +410 +415 +0.6 +0.9 +1.2 +1.5 +1.8 +Wavelength (nm) +Laser power (mw) +e) +f) +06 +xX +06 +120 +60 +120 +60 +XY +Fit +Fit +150 +30 +150 +30 +180 +0 +180 +0 +210 +330 +210 +330 +240 +300 +240 +300 +270 +27013 + +The nonlinear susceptibility of our newly synthesized CuS crystal was estimated to be +𝜒𝐶𝑢𝑆 +(2) = 1.4 × 10−11 m/V. Before assessing polarization, we rotated the sample to a position +where the highest SHG response could be generated by setting the initial azimuthal angle to 0°. +In parallel (XX) and perpendicular (XY) directions, the typical 6-fold symmetry pattern fitted +proportionally with sin2 3𝜃 and cos2 3𝜃 can be detected, as presented in Figure 4e-f. It implied +broken inversion symmetry property that is characteristic of hexagonal-symmetric structures +similar to other SHG sensitive materials (Table 1). This feature imparts 2D CuS crystals with +promising properties of interest in the field of applied nonlinear optics. In addition, utilizing +Piezoresponse force microscopy (PFM), we explored the unexpected SHG response in CuS, +and interestingly, we detected switchable hysteretic behavior in the dual-pass remnant +hysteresis measurement on numerous CuS crystals (Figure S4). The findings of the PFM +support the SHG response. +Table 1: Properties of the synthesized 2D CuS sheets and comparison with other +nanomaterials used in SHG (C: centrosymmetric, and N: noncentrosymmetric). +2D +Material +Synthesis +method +Sample +Thickness +(nm) +C +N +𝝌(𝟐) +Refs. +CuS +CVD +14.5 +* + +1.4 × 10–11 +This +work +MoS2 + +Exfoliation +0.8 + +* +1 × 10–7 +[55] +GaSe + +CVD +0.83 + +* +0.7 × 10-9 +[56] +SnP2S6 + +Exfoliation +8 +* + +4 × 10-9 +[57] +WS2 + +CVD +0.65 + +* +4.5 × 10-9 +[58] +RhI3 + +Exfoliation +12 +* + +- +[59] + + +14 + +To summarize, in a single-step CVD technique (at a growth temperature of less than +600 °C), highly crystalline 2D lattice CuS was synthesized for the first time. The as-grown 2D +CuS sheets with nanoscale thickness were thoroughly characterized (phase and orientation of +its lattice were verified). A sheet of 2D CuS crystal was utilized in SHG, with a nonlinear +susceptibility of up to 1.4 × 10-11 m/V and the underlying mechanism was discussed. The +nanoscale 2D sheets of CuS are therefore expected to have a wide range of optoelectronic +applications. + +Materials and Characterization +CVD growth: 2D CuS crystals were grown in a tubular furnace with a single temperature zone +and atmospheric pressure CVD conditions. A quartz boat containing a CuCl powder (97%, +Sigma Aldrich) was placed in the middle of the temperature zone. S powder (99.5%, Sigma +Aldrich) was inserted at the upstream end of the tube, and the temperature was maintained at +200. Substrates, e.g., cleaved fluorphlogopite mica, were positioned 8 cm apart from the +furnace's center in the downstream position. The tube was pumped and cleaned with 500 sccm +Ar flow to drain air prior to heating. Then, the furnace was heated to 560 °C at a rate of 30 +⁰C/min using steady 50 sccm Ar as the carrier gas, and it was held at that temperature for 30 +minutes. After the procedure was concluded, the furnace was allowed to cool naturally. +Characterizations: 2D CuS crystal morphologies were examined using an OM (BX51, +OLYMPUS) and an AFM (Bruker Dimension Icon). The crystalline structure, orientation, and +composition were investigated using XRD (λ: 1.54 Å, D2 phaser, Bruker), XPS (AXIS-ULTRA +DLD-600W, Kratos), EBSD (FEI Quanta650), and TEM (Tecnai G30 F30, FEI). Raman +spectra were acquired using a confocal Raman system (Alpha 300R, WITec) equipped with a +532 nm laser. + +15 + +SHG measurements: SHG measurements were performed in an (alpha300RS+, WITec) +Raman system with a reflection mode under normal incidence excitation using a femtosecond +laser as the excitation source. A mode-locked Ti: sapphire laser with a pulse duration of 140 fs +and repetition rate of 80 MHZ generated the output laser with a continually varying wavelength +ranging from 340 nm to 1600 nm, which was then filtered into an optical parametric oscillator +(Chameleon Compact OPO-Vis). A dichroic beam splitter was used to reflect the laser beam +into the 100x objective lens with a spot size of roughly 1.8 μm and communicate the reflected +SHG signal. The reflected SHG signal was then filtered with a short pass (SP) filter before +being sent to the spectrometer and CCD. The collected polarized SHG signal was sent through +a linear polarized analyzer for SHG polarization measurement by rotating the sample with a +step of 10° relative to fixed light polarization (Figure S5). All experiments were carried out in +a natural setting. +ASSOCIATED CONTENT +AUTHOR CONTRIBUTION +A.A.S: Synthesis, characterizations, conceptualization, data curation, writing-original draft, co- +corresponding. R.R: Writing, data curation and editing. A.P: Characterization and editing. +T.S.K: Supervision, conceptualization, funding, editing, corresponding. All authors have +agreed on the final version of the manuscript. +CONFLICT OF INTEREST +No competing financial interest are declared. +ACKNOWLEDGMENTS +The authors acknowledge funding from the Scientific and Technological Research Council of +Turkey (TUBITAK) under grant number 120N885. + +16 + +REFERENCES +1. +Sarkar, D.; Xie, X.; Liu, W.; Cao, W.; Kang, J.; Gong, Y.; Kraemer, S.; Ajayan, P. M.; +Banerjee, K. A subthermionic tunnel field-effect transistor with an atomically thin channel. +Nature 2015, 526 (7571), 91-5. +2. +Gaggiotti, S.; Scroccarello, A.; Della Pelle, F.; Ferraro, G.; Del Carlo, M.; Mascini, +M.; Cichelli, A.; Compagnone, D. An electronic nose based on 2D group VI transition metal +dichalcogenides/organic compounds sensor array. 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OM mages of CuS crystals grown on various substrates of mica, Si/SiO2, and Si +(a-c), scale bars are 10, 20, and 20 μm, respectively. Corresponding AFM images of CuS +crystals with their height profiles 40, 28, and 6 nm respectively (d-f). Raman spectra of CuS +crystals grown on mica, SiO2, and Si, respectively (g-i). + +a) +b) +c) +d) +e +f) +16.7 nm +48.1nm +12.5nm +-8.3 nm +-12.8nm +-3.7nm +HeightSensor +2.0um +Height Sensor +3.0um +HeightSensor +4.0um +g) +h) +D +Intensity (a.u.) +Al1g +3 +(n +Intensity (a. +(a. +Intensity +E2 +E2g +A1g +E2 +100200300400500 +600 +100200300400500 +600 +100200300400500 +600 +Raman shift (cm-1) +Raman shift (cm-1) +Raman shift (cm-1)S4 + + +Figure S3: Side view and top view of CuS crystal structure. + + +Topview +CuS4 +Cu +CuS3 +S +CuS2 +S-S +CuS4 +CuS3 +CuS4S5 + + +Figure S4: On-field hysteresis loops for (a) PFM amplitude and (b) PFM phase on CuS +crystal. + + +a) +b) +100 +100 +Amplitude (pm) +80 +50 +Phase (°) +60 +40 +-50 +20 +100 +0 +-150 +8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +-8 +9- +-2 +0 +2 +4 +6 +8 +Bias (V) +Bias (V)S6 + +Calculation of Second-order nonlinear susceptibility +Using similar estimation of second order susceptibility χ(2) as reported by several studies +in the literature, the χ(2) value of thin CuS crystal could be calculated: +𝐼2𝜔 = [𝜒(2)] +2𝐼𝜔 +28𝜖0𝑐3. +1 +𝑛2𝜔𝑛𝜔 +2 . +𝜔2𝑑2 +8𝜖0𝑐3, +where 𝐼𝜔 and 𝐼2𝜔 are the intensity of excitation laser and SHG signal, respectively; 𝜒(2) is +the second-order susceptibility; 𝜖0 is the vacuum dielectric constant; 𝑐 is the speed of light +in vacuum; 𝑛𝜔 ≈ 2.6 and 𝑛2𝜔 ≈ 2.6 are respectively the refractive index of CuS at +frequency ω of excitation laser and at frequency 2ω of SHG field1; 𝑑 = 14.3 nm is the +thickness. However, it is challenging to obtain 𝐼2𝜔 as it involves many experimental +parameters including the optical absorptions of the optical setup, the detector efficiencies +and laser frequency and duration. For this reason, typically the susceptibility is referenced +with respect to the monolayer MoS2 (𝜒𝑀𝑜𝑆2 +(2) += 4.05 × 10−10 m/V) using the following +relation2: +𝜒𝐶𝑢𝑆 +(2) = √ +𝐼2𝜔−𝐶𝑢𝑆 +𝐼2𝜔−𝑀𝑜𝑆2 +. +𝑑𝑀𝑜𝑆2 +𝑑𝐶𝑢𝑆 +. √ 𝑛2𝜔−𝐶𝑢𝑆 +𝑛2𝜔−𝑀𝑜𝑆2 +𝑛𝜔−𝐶𝑢𝑆 +2 +𝑛𝜔−𝑀𝑜𝑆2 +2 +. 𝜒𝑀𝑜𝑆2 +(2) +Our result was attained after giving due consideration to the efficacy of signal collection +and detection as 𝜒(2) = 1.4 × 10−11 m/V for 800 nm. This calculation was only an +approximation of the order of magnitude because the value of χ(2) depends on many +accurate experimental parameters. + +S7 + + +Figure S5: Setup for measuring second harmonic generation. + +Pinhole +Mirror +Spectrometer +Polarizer +SP Filter +Camera +Mirror +Dichroic beamsplitter +Beam +expander +Ti-Sapphire +OPO +Laser +Mirror +N +Len +100x +SampleS8 + + +References +(1) Aziz, S. B.; Abdulwahid, R. T.; Rsaul, H. A.; Ahmed, H. M., In situ synthesis of +CuS nanoparticle with a distinguishable SPR peak in NIR region. J. Mater. Sci. Mater. +2016, 27 (5), 4163-4171. +(2) Shi, J.; Yu, P.; Liu, F.; He, P.; Wang, R.; Qin, L.; Zhou, J.; Li, X.; Zhou, J.; Sui, +X.; et al., 3R MoS2 with broken inversion symmetry: a promising ultrathin nonlinear +optical device. Adv. Mater. 2017, 29 (30). + + + diff --git a/0NAzT4oBgHgl3EQfDPod/content/tmp_files/load_file.txt b/0NAzT4oBgHgl3EQfDPod/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6fee88758a287cd272371dc6b3e0b3a13454b9b --- /dev/null +++ b/0NAzT4oBgHgl3EQfDPod/content/tmp_files/load_file.txt @@ -0,0 +1,1478 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf,len=1477 +page_content='1 Low-Temperature Chemical Vapor Deposition of Copper(II) Sulfide Crystals and its Nonlinear Optical Response Abdulsalam Aji Suleimana*, Reza Rahighia, Amir Parsia, and Talip Serkan Kasirgaa,b* aInstitute of Materials Science and Nanotechnology, Bilkent University UNAM, Ankara 06800, Turkey bDepartment of Physics, Bilkent University, Ankara 06800, Turkey Corresponding authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Email: kasirga@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' abdulsalam@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='tr 2 ABSTRACT The need for novel multifunctional nanomaterials capable of meeting new demands in the realm of nanotechnology coupled with versatility of chemical vapor deposition (CVD) technique (in large-area growth of crystals), encourages innovative methods for synthesis of untried two- dimensional (2D) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' While there exist reports on both top-down and bottom-up synthesis methodologies of different Cu2S-based nanostructures, CVD-based synthesis of 2D crystals of copper(II) sulfide (CuS) has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' This work represents details of CVD method in systematic growth of highly crystalline 2D CuS sheets as thin as ~ 6 nm with lateral sizes exceeding 60 μm, at a relatively low temperature of 560 °C in ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Samples were characterized via X-ray diffraction, Raman, atomic force microscopy, and high-resolution transmission electron microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' SAED revealed a 6-fold symmetric structure and identical atomic ratio of copper:sulphur was corroborated from the energy-dispersive X-ray spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The as-prepared 2D CuS sheets were successfully utilized in second harmonic generation (SHG) and their strong response was found to be highly polarization angle-sensitive as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The CVD- synthesized 2D CuS crystals in this study are considered to be of great significance in a diverse range of future applications, as in energy storage, next-generation solar cells, nonlinear optoelectronic-related devices, and even bioelectronics pursuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' KEYWORDS: 2D materials, CVD method, Covellite, Nonlinear optics, Second harmonic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 3 ToC 560C 2D lattice of Arflow~ Copper (l) Sulfide S CuCl 2W w SecondHarmonic Generation yia CuS4 Two-dimensional (2D) materials could make the way for myriad of unprecedented functional devices such as high on/off ratio field-effect transistors (at room temperature),1 electric nose,2 mode-locked laser,3 and broad-band photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4-5 However, top-down methods of chemical exfoliation,6 solvothermal,7 or supercritical8 lead to defective structures that deviate from the required parameters regarding fabrication of highly efficient nanodevices sought in different areas of optoelectronics, bioelectronics,9 and spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='10-13 Among bottom-up approaches for synthesis of highly-ordered crystalline 2D structure14-16, chemical vapor deposition (CVD) provides high-quality and high-yield products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='17-18 Engineering its corresponding parameters such as temperature, substrate, gas flow, dwell time, fast/natural cooling, can herald novel 2D structures, pursued in realization of innovative devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='19-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Copper(II) sulfide (covellite, CuS), being highly conductive, chemically stable, and having ultralow thermal conductivity, is widely used in solar cells, batteries, and photothermal treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='22-27 CuS nanoparticles have been reported to be prepared via wet-chemistry, as a promising candidate for a lithium-ion battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='28 In another report, their photocatalysis property was investigated and attributed to the large specific surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='29 In addition, light harvesting and charge separation activities can be significantly enhanced by nanosheets of ZnIn2S4/CuS,30 without necessity of co-catalyst thanks to the strong interactions between assembled p-n heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Some nanoflakes of CuCrS2 showing switchable ferroelectric polarization have been also reported to be synthesized recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='31 While important phenomenon of superconductivity has been theoretically predicted from 2D lattice of CuS,32 the physicochemical properties of 2D copper-based chalcogenides have scarcely been studied and there exists no report on CVD growth of 2D CuS yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Highly crystalline 2D CuS synthesized in this work, are produced using a single-step CVD technique at a relatively low temperature of 560 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The as-grown 2D CuS sheets having nanometer thickness, were characterized by atomic force microscopy (AFM), X-ray diffraction 5 (XRD), Raman, high-resolution transmission electron microscopy (HRTEM), and energy- dispersive X-ray spectroscopy (EDX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A 6-fold symmetric structure was revealed via selected area diffraction (SAED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' As an application of 2D CuS, a single sheet of it was utilized in second harmonic generation (SHG) with the nonlinear susceptibility of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 × 10−11 m/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' In addition, the nonlinear optical characteristic of 2D CuS crystals was utilized in broad-spectrum wavelength and polarization-resolved SHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Copper(I) chloride (CuCl) powder was opted as the Cu source for growth of 2D CuS lattices, due to its suitable chemical property and the relatively low melting point temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' An asymmetric tiny crucible was chosen to this end, filled with scant amount of CuCl powder, and put in middle of tubular CVD furnace as can be seen in the schematic setup in the supporting information (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' During the synthesis process, the optimum growth temperature was found to be about 560 °C, way lower than the other CVD synthesis33 of copper-based chalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Additional information regarding the CVD growth process is provided in the experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Figure 1a shows a typical optical image of CuS crystals grown on a mica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The lateral length of the grown 2D CuS crystals can be up to 70 µm (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Mica is used as a substrate because of its atomic-level smooth and inert surface, which has been widely reported as a favorable substrate for 2D material synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='34 AFM height trace map given in Figure 1c confirms that the surface of 2D CuS is very smooth and the thickness was found to be about 14 nm according to AFM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 6 Figure 1: Typical OM image of the as-grown 2D CuS crystals (a and b), the scale bars are 10 and 20 μm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' AFM image of the CuS crystal (c), and its height profile of the in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' XRD pattern of 2D CuS crystals on SiO₂/Si substrate (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' EBSD inverse pole figure (IPF) map along the c-axis of 2D CuS crystal on SiO₂/Si substrate (e), the length of the scale bar corresponds to 5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Color coded map type of IPF (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' XRD pattern of as-grown (Figure S2) and transferred CuS crystals on the SiO2/Si substrate depicted in Figure 1d clearly identifies the hexagonal phase of CuS (PDF No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 06- 0464).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The strong characteristic peaks of (002), (006), and (008) show that 2D CuS crystals preferentially grow in the basal plane (00l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The plane interspacing can be estimated using Bragg’s relation, 𝑛𝜆 = 2𝑑(ℎ𝑘𝑙)𝑠𝑖𝑛(𝜃) where n is an integer corresponds to the other of diffraction peak and λ is wavelength of X-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The mean dimensions of the crystallite perpendicular to the (00l) plane (L002) can be determined by using Scherrer equation, 𝐿(ℎ𝑘𝑙) = 𝐾𝜆/𝛽𝑐𝑜𝑠(𝜃) a) b) c) 30nm Position(μm) d) (006) f) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') 1010 Cus crystal PDF06 0464 (002) IS (008) 0001 2110 10 20 30 40 50 60 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' )7 where K is a constant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='89) and β is the integral full widths at half maximum (in radians, in our case 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='07 regarding 2θ peak at 11°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The average number of layers can be estimated by simply dividing L(002) over d(002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Therefore, the average number of layers was found ~ 16, implying the presence of multi-layer sheets in the structure, consistent with AFM investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' In addition, we utilized electron backscatter diffraction (EBSD), a powerful method for identifying the microstructural characterization of materials, to determine the crystallographic orientation of the 2D CuS crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='35-36 Figure 1e-f displays a uniform color contrast of the EBSD inverse pole figure (IPF) map within the hexagonal domains along the basal plane of CuS ([00l] direction), implying a single-crystalline nature and ordered in-plane orientation throughout the hexagonal CuS crystal, which is consistent with the XRD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' As illustrated in Figure S3, the CuS structure belongs to the space group P63/mmc (hexagonal symmetry) with Z = 6 per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Cu atoms exist in two types of environments: CuS3 (triangular planes) and CuS4 (rectangular planes) (tetrahedra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The unit cell can be assumed as plates connected by S-S bonds, and through triangular planes, vortices merge the tetrahedral units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' According to previous research, the Cu(1)-S(1) bonds (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='19 Å ) which occur in triangle units, have a length much shorter than the Cu-S bonds seen in most other copper sulfides (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='33 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='37 Because of this, it is rather conceivable that the Cu(1)-S(1) bond will have a stronger bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' It has also been reported that Cu(1) ions in the [Cu(1)-S(1)3] triangles exhibit significantly high thermal motion along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='38 Raman spectroscopy with a 532 nm excitation laser was used to investigate the intrinsic properties and identify the fingerprint of the 2D CuS crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' As shown in Figure 2a, the Raman spectrum of the 2D CuS crystal shows four distinct Raman peaks at 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='1, 130, 279, and 471 cm-1, representing E2g, A1, Eg 1, and A1 modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Among these peaks, the strong characteristic peak at 471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0 cm-1 can be attributed to the stretching mode of the S-S bond, corresponding to the S2 groups of the recognized crystal structure of 2D CuS lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='39-40 8 Figure 2: Raman spectrum of the 2D CuS lattice on mica substrate (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Spatially resolved Raman mapping images (b) of the 2D CuS characteristic peaks E2g 3 (P1), A1g 2 (P2), E2g 2 (P3), and A1g 1 (P4), scale bars: 5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Temperature-dependent spectra (c) of 2D CuS crystal (80-300 K, step: 20 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Raman peak positions of 2D CuS (P1-4) as a function of the measured temperature (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Temperature-dependent Raman spectroscopy is a classical method to study the atomic bonding and thermal expansion of 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='41-42 The spatially resolved Raman mapping images (Figure 2b) of the four characteristic peaks (60, 138, 267, and 471 cm–1) exhibit uniformity throughout the 2D crystalline sheet of CuS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Figure 2c shows the typical temperature-dependent Raman spectra for the grown 2D CuS crystal at temperatures ranging a) b) P2 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') P4 A2 2c 100 200 300 400 500 600 Raman shift (cm 1) c) d P1 P4 476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='1 P4 P2 P3 Fit 300 K 473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8 471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 Slope= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content="02535 Raman shift (cm' Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 P3 Fit 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 Slope= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='02563 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0 P2 Fit 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='2 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 Slope= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='02677 P1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='48 Fit 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='85 80 K 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='22 Slope= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='00662 60 120 100 200300400500 180 240 300 600700 800 Raman shift (cm 1) Temperature (K)9 from 80 to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' It can be clearly seen that the positions of the peaks exhibit a slight "redshift" with increasing temperature, which is mainly due to anharmonic vibrations of the lattice induced by the thermal expansion of the lattice at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='43 The correlation between them can be described by a linear equation: 𝜔(𝑇) = 𝜔0 + 𝜒𝑇, where 𝜔0, T, and χ are the Raman peak position at 0 K, the Kelvin temperature, and the first-order temperature coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' As shown in previous reports,44-45 the first-order temperature coefficient of 2D materials is related to the van der Waals interaction between the neighboring layers and is usually used to explain the temperature dependence of the Raman peak shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Notably, the derived χ-values for P1, P2, P3, and P4 of CuS crystals are - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='00662, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='02677, - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='02563, and - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='02535 cm-1K-1, respectively (Figure 2d), which is larger than that of ordinary layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='46-47 Further analyses, such as high-resolution transmission electron microscopy (HR-TEM), SAED, and EDX, were carried out to investigate the crystal structure and atomic composition of the 2D CuS crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Figure 3a shows Fast Fourier Transform-filtered HR-TEM image and it can be clearly seen that the atoms are arranged hexagonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The interplanar spacing of the two planes crossing at an angle of 120° is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='35 nm, corresponding to planes (100) and (010), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The corresponding SAED image shows a 6-fold symmetric structure with an [001] axis presented in Figure 3b, and the EDX spectrum of the 2D CuS crystal is shown in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The detected peaks suggest that the crystal is made entirely of Cu and S components, which is supported by X-ray photoelectron spectroscopy (XPS) results which is shown in Figure 3d- e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 10 Figure 3: Structural and chemical compositional characterization of 2D CuS crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' High-resolution TEM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Scale bar: 5 nm (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The SAED patterns, scale bar: 5 nm-1 (b), EDX spectrum, inset shows atomic ratios of the chemical composition (c), XPS spectra deconvoluted peaks of Cu2p (d), and S2p (e) core levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' New materials that have nonlinear optical response, can be of beneficial application in different areas ranging from photon generation, imaging, and photon manipulation in ultrafast 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='35 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='35 nm (010) (100) b) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') Cu s cu 52 % 48 % （100 S Cu （010) cu 0 2 4 6 8 10 Energy (keV) d) Cu,2p3/2 e Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') S 2p3/2 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' ), S 2p1/2 Cu 2p1/2 096 950 940 930 920 170 168 166 164 162 Binding energy (eV) Binding energy (eV)11 lasers, optical modulators, and pulse characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='48-53 Stacking faults existing in the synthesized CuS crystals (in this study), such as an interlayer slip, dislocation, and undulation of the atomic layers, can induce multi-oriented domains in the crystal and deemed responsible for the observed nonlinear optical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='54 The SHG is a very useful technique, where the incident laser (ω) generates an (2ω) response, as shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The SHG response of a CuS crystal (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 nm) under various incident laser wavelengths from the edge of visible light to near-infrared (760 to 1020 nm) is presented in Figure 4b, which shows a wide spectrum response with distinct wavelength selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Moreover, the SHG mappings display a uniform response throughout the entire 2D CuS lattice (Figure 4b inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Evolution of SHG intensity with changing incident laser power was also further systematically investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' With increasing the incident laser power from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='6 mW under 800 nm laser excitation, the intensity of the SHG signal at 400 nm exhibits significant enhancement (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The relationship between SHG intensity and laser power was fitted linearly in the log-log coordinate, as displayed in Figure 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Interestingly, the slope of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='05 is close to the theoretical value of 2 calculated from the electric dipole theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='41 12 Figure 4: SHG characterization of 2D CuS crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Basic mechanism of nonlinear optical effects (a), The SHG spectra of 2D CuS crystal under various excitation wavelengths (760 - 1020 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Inset is the SHG mapping of 2D CuS crystal under 800 nm laser excitation, scale bar corresponds to 3 μm (b), The SHG spectra of the 2D CuS crystal with different incident powers (c), The SHG intensities as a function of incident power (d), Polarization angle- dependent SHG intensity under parallel (e), and perpendicular (f) polarization configurations (The excitation laser is 800 nm with a power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='2 mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' a) b) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') 2=760 1020 nm 5k 3 2w m 2w 4k SHG intensity 3k 3 2k SHG 1k Mica substrate 380400420440460480500 520 Wavelength (nm) 6k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='7 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='3 mW Data (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 mW Linear fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='9 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 mW a 4k Intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='6 mW 3k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='2 mw Intensi 2k Slope = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='05 1k 385 390 395 400 405 410 415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8 Wavelength (nm) Laser power (mw) e) f) 06 xX 06 120 60 120 60 XY Fit Fit 150 30 150 30 180 0 180 0 210 330 210 330 240 300 240 300 270 27013 The nonlinear susceptibility of our newly synthesized CuS crystal was estimated to be 𝜒𝐶𝑢𝑆 (2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 × 10−11 m/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Before assessing polarization, we rotated the sample to a position where the highest SHG response could be generated by setting the initial azimuthal angle to 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' In parallel (XX) and perpendicular (XY) directions, the typical 6-fold symmetry pattern fitted proportionally with sin2 3𝜃 and cos2 3𝜃 can be detected, as presented in Figure 4e-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' It implied broken inversion symmetry property that is characteristic of hexagonal-symmetric structures similar to other SHG sensitive materials (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' This feature imparts 2D CuS crystals with promising properties of interest in the field of applied nonlinear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' In addition, utilizing Piezoresponse force microscopy (PFM), we explored the unexpected SHG response in CuS, and interestingly, we detected switchable hysteretic behavior in the dual-pass remnant hysteresis measurement on numerous CuS crystals (Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The findings of the PFM support the SHG response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Table 1: Properties of the synthesized 2D CuS sheets and comparison with other nanomaterials used in SHG (C: centrosymmetric, and N: noncentrosymmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 2D Material Synthesis method Sample Thickness (nm) C N 𝝌(𝟐) Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' CuS CVD 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 × 10–11 This work MoS2 Exfoliation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8 1 × 10–7 [55] GaSe CVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='7 × 10 9 [56] SnP2S6 Exfoliation 8 4 × 10 9 [57] WS2 CVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5 × 10 9 [58] RhI3 Exfoliation 12 [59] 14 To summarize, in a single-step CVD technique (at a growth temperature of less than 600 °C), highly crystalline 2D lattice CuS was synthesized for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The as-grown 2D CuS sheets with nanoscale thickness were thoroughly characterized (phase and orientation of its lattice were verified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A sheet of 2D CuS crystal was utilized in SHG, with a nonlinear susceptibility of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 × 10-11 m/V and the underlying mechanism was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The nanoscale 2D sheets of CuS are therefore expected to have a wide range of optoelectronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Materials and Characterization CVD growth: 2D CuS crystals were grown in a tubular furnace with a single temperature zone and atmospheric pressure CVD conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A quartz boat containing a CuCl powder (97%, Sigma Aldrich) was placed in the middle of the temperature zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' S powder (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5%, Sigma Aldrich) was inserted at the upstream end of the tube, and the temperature was maintained at 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Substrates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=", cleaved fluorphlogopite mica, were positioned 8 cm apart from the furnace's center in the downstream position." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The tube was pumped and cleaned with 500 sccm Ar flow to drain air prior to heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Then, the furnace was heated to 560 °C at a rate of 30 ⁰C/min using steady 50 sccm Ar as the carrier gas, and it was held at that temperature for 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' After the procedure was concluded, the furnace was allowed to cool naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Characterizations: 2D CuS crystal morphologies were examined using an OM (BX51, OLYMPUS) and an AFM (Bruker Dimension Icon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The crystalline structure, orientation, and composition were investigated using XRD (λ: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='54 Å, D2 phaser, Bruker), XPS (AXIS-ULTRA DLD-600W, Kratos), EBSD (FEI Quanta650), and TEM (Tecnai G30 F30, FEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Raman spectra were acquired using a confocal Raman system (Alpha 300R, WITec) equipped with a 532 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 15 SHG measurements: SHG measurements were performed in an (alpha300RS+, WITec) Raman system with a reflection mode under normal incidence excitation using a femtosecond laser as the excitation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A mode-locked Ti: sapphire laser with a pulse duration of 140 fs and repetition rate of 80 MHZ generated the output laser with a continually varying wavelength ranging from 340 nm to 1600 nm, which was then filtered into an optical parametric oscillator (Chameleon Compact OPO-Vis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A dichroic beam splitter was used to reflect the laser beam into the 100x objective lens with a spot size of roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8 μm and communicate the reflected SHG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The reflected SHG signal was then filtered with a short pass (SP) filter before being sent to the spectrometer and CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' The collected polarized SHG signal was sent through a linear polarized analyzer for SHG polarization measurement by rotating the sample with a step of 10° relative to fixed light polarization (Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' All experiments were carried out in a natural setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' ASSOCIATED CONTENT AUTHOR CONTRIBUTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='S: Synthesis, characterizations, conceptualization, data curation, writing-original draft, co- corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='R: Writing, data curation and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='P: Characterization and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='K: Supervision, conceptualization, funding, editing, corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' All authors have agreed on the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' CONFLICT OF INTEREST No competing financial interest are declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors acknowledge funding from the Scientific and Technological Research Council of Turkey (TUBITAK) under grant number 120N885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 16 REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Sarkar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Xie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Zhai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Inversion symmetry broken 2D SnP2S6 with strong nonlinear optical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Nano Research 2021, 15 (3), 2391-2398.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Mehta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Elias, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Perea-Lopez, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Terrones, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Crespi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Extraordinary Second Harmonic Generation in tungsten disulfide monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Sci Rep 2014, 4, 5530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Nie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Huang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Zhai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Honeycomb RhI3 Flakes with High Environmental Stability for Optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Adv Mater 2020, e2001979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' S1 SUPPORTING INFORMATION Low-Temperature Chemical Vapor Deposition of Copper(II) Sulfide Crystals and its Nonlinear Optical Response Abdulsalam Aji Suleimana*, Reza Rahighia, Amir Parsia, and Talip Serkan Kasirgaa,b* aInstitute of Materials Science and Nanotechnology, Bilkent University UNAM, Ankara 06800, Turkey bDepartment of Physics, Bilkent University, Ankara 06800, Turkey Corresponding authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Email: kasirga@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' abdulsalam@unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='tr S2 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Schematic image of the CVD setup (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' CVD growth temperature curve (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' a) 560°C Arflow 2D CuS sheets S Cucl d) T/°C Growth time T1 rate °C/min 30 1 0 t1 t2 t3 t/minS3 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' OM mages of CuS crystals grown on various substrates of mica, Si/SiO2, and Si (a-c), scale bars are 10, 20, and 20 μm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Corresponding AFM images of CuS crystals with their height profiles 40, 28, and 6 nm respectively (d-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Raman spectra of CuS crystals grown on mica, SiO2, and Si, respectively (g-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' a) b) c) d) e f) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='7 nm 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='1nm 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='5nm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='3 nm 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='8nm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='7nm HeightSensor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0um Height Sensor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0um HeightSensor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='0um g) h) D Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=') Al1g 3 (n Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Intensity E2 E2g A1g E2 100200300400500 600 100200300400500 600 100200300400500 600 Raman shift (cm 1) Raman shift (cm 1) Raman shift (cm 1)S4 Figure S3: Side view and top view of CuS crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Topview CuS4 Cu CuS3 S CuS2 S S CuS4 CuS3 CuS4S5 Figure S4: On-field hysteresis loops for (a) PFM amplitude and (b) PFM phase on CuS crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' a) b) 100 100 Amplitude (pm) 80 50 Phase (°) 60 40 50 20 100 0 150 8 6 4 2 0 2 4 6 8 8 9 2 0 2 4 6 8 Bias (V) Bias (V)S6 Calculation of Second-order nonlinear susceptibility Using similar estimation of second order susceptibility χ(2) as reported by several studies in the literature, the χ(2) value of thin CuS crystal could be calculated: 𝐼2𝜔 = [𝜒(2)] 2𝐼𝜔 28𝜖0𝑐3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 1 𝑛2𝜔𝑛𝜔 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝜔2𝑑2 8𝜖0𝑐3, where 𝐼𝜔 and 𝐼2𝜔 are the intensity of excitation laser and SHG signal, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝜒(2) is the second-order susceptibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝜖0 is the vacuum dielectric constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝑐 is the speed of light in vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝑛𝜔 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='6 and 𝑛2𝜔 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='6 are respectively the refractive index of CuS at frequency ω of excitation laser and at frequency 2ω of SHG field1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝑑 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='3 nm is the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' However, it is challenging to obtain 𝐼2𝜔 as it involves many experimental parameters including the optical absorptions of the optical setup, the detector efficiencies and laser frequency and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' For this reason, typically the susceptibility is referenced with respect to the monolayer MoS2 (𝜒𝑀𝑜𝑆2 (2) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='05 × 10−10 m/V) using the following relation2: 𝜒𝐶𝑢𝑆 (2) = √ 𝐼2𝜔−𝐶𝑢𝑆 𝐼2𝜔−𝑀𝑜𝑆2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝑑𝑀𝑜𝑆2 𝑑𝐶𝑢𝑆 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' √ 𝑛2𝜔−𝐶𝑢𝑆 𝑛2𝜔−𝑀𝑜𝑆2 𝑛𝜔−𝐶𝑢𝑆 2 𝑛𝜔−𝑀𝑜𝑆2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' 𝜒𝑀𝑜𝑆2 (2) Our result was attained after giving due consideration to the efficacy of signal collection and detection as 𝜒(2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content='4 × 10−11 m/V for 800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' This calculation was only an approximation of the order of magnitude because the value of χ(2) depends on many accurate experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' S7 Figure S5: Setup for measuring second harmonic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Pinhole Mirror Spectrometer Polarizer SP Filter Camera Mirror Dichroic beamsplitter Beam expander Ti Sapphire OPO Laser Mirror N Len 100x SampleS8 References (1) Aziz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Abdulwahid, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Rsaul, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' Ahmed, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=', In situ synthesis of CuS nanoparticle with a distinguishable SPR peak in NIR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfDPod/content/2301.00971v1.pdf'} +page_content=' J.' metadata={'source': 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CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Abstract. The hypoplactic monoid was introduced by Krob and Thibon +through a presentation and through quasi-ribbon tableaux and an insertion +algorithm. Just as Kashiwara crystals enriched the structure of the plactic +monoid and allowed its generalization, the first and third authors of this paper +introduced a construction of the hypoplactic monoid by identifying vertices +in a quasi-crystal graph derived from the crystal graph associated to the gen- +eral linear Lie algebra. Although this construction is based on Kashiwara’s +work, it cannot be extended to other crystal graphs, since the analogous quasi- +Kashiwara operators on words do not admit a recursive definition. This paper +addresses these issues. +A general notion of quasi-crystal is introduced, fol- +lowed by a study of its properties and relation with crystals. A combinatorial +study of quasi-crystals is then made by associating a quasi-crystal graph to +each quasi-crystal, which for the class of seminormal quasi-crystals results in a +one-to-one correspondence. To model the binary operation of the hypoplactic +monoid by quasi-crystals, a notion of quasi-tensor product of quasi-crystals +is introduced, along with a combinatorial way of computing it similar to the +signature rule for the tensor product of crystals. This framework allows the +generalization of the classical hypoplactic monoid to a family of hypoplactic +monoids associated to the various simple Lie algebras. The quasi-crystal struc- +ture is then used to establish algebraic properties of the hypoplactic monoid +associated to the symplectic Lie algebra. +Contents +1. +Introduction +2 +2. +Preliminaries +4 +3. +Quasi-crystals and homomorphisms +5 +4. +Quasi-crystal graphs +11 +5. +Quasi-tensor product of quasi-crystals +17 +5.1. +Definition and results +17 +5.2. +The signature rule +24 +6. +Quasi-crystal monoids +25 +6.1. +Quasi-crystal monoids and homomorphisms +25 +6.2. +The free quasi-crystal monoid +29 +6.3. +Congruences and quotients +33 +7. +The hypoplactic congruence +36 +2020 Mathematics Subject Classification. Primary 05E16; Secondary 05E10, 20M05, 20M10. +Key words and phrases. Quasi-crystal, hypoplactic monoid, crystal, plactic monoid, Kashiwara +operator, weight labelled graph. +The second author was funded by national funds through the FCT – Fundação para a Ciência +e a Tecnologia, I.P., under grant reference SFRH/BD/121819/2016. +For all three authors, this work was funded by national funds through the FCT – Fun- +dação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 +and UIDP/00297/2020 (Center for Mathematics and Applications), and under the scope of the +SemiComb project PTDC/MAT-PUR/31174/2017. +1 + +2 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +8. +Crystallizing the classical hypoplactic monoid +39 +9. +The hypoplactic monoid of type Cn +41 +9.1. +The definition of hypo(Cn) +41 +9.2. +Highest-weight words +44 +9.3. +Isolated words +46 +9.4. +Relations +48 +9.5. +Identities +53 +9.6. +Presentations +57 +9.7. +From hypo(An−1) to hypo(Cn) +60 +9.8. +From hypo(Cn−1) to hypo(Cn) +62 +References +64 +1. Introduction +The plactic monoid, formally introduced by Lascoux and Schützenberger [LS81], +is an algebraic object of great interest, with connections to several fields such as +representation theory, combinatorics [Ful97], symmetric functions, and Schubert +polynomials [LS85, LS89]. It was also used to give a first rigorous proof of the +Littlewood–Richardson rule [LR34]. This led Schützenberger [Sch97] to consider +it one of the most fundamental monoids in algebra. +There are numerous ways +of obtaining the plactic monoid; we highlight three of them. First, it originally +emerged from Young tableaux and the Schensted insertion algorithm [Sch61]. Sec- +ond, it also has a presentation by the so-called Knuth relations [Knu70]. Third, it +can be obtained by identifying words in the same position of isomorphic connected +components of a certain crystal graph. +Kashiwara [Kas90, Kas91, Kas94] introduced crystal bases for modules of quan- +tized universal enveloping algebras, discovered independently by Drinfel’d [Dri85] +and Jimbo [Jim85], and showed that the plactic monoid arises from the crystal ba- +sis associated with the vector representation of the quantized universal enveloping +general linear Lie algebra. This result allowed a deeper study of the plactic monoid +and its generalization, because the underlying construction still results in a monoid +for crystal bases associated with other quantized universal enveloping algebras. +Thus, Kashiwara and Nakashima [KN94] studied crystal graphs for the Cartan +types An, Bn, Cn and Dn, leading to a notion of Kashiwara–Nakashima tableaux. +Based on this, Lecouvey [Lec02, Lec03] presented comprehensive descriptions of +the plactic monoids for the Cartan types Bn, Cn, and Dn, which later appeared +in a survey [Lec07]. In recent works, Cain, Gray and Malheiro [CGM15a, CGM19] +presented rewriting systems and biautomatic structures for these monoids. In an +independent work and by an alternative approach, Hage [Hag15] described a finite +convergent presentation of the plactic monoid for type Cn. +The hypoplactic monoid was introduced by Krob and Thibon [KT97] from +representation-theoretical interpretations of quasi-symmetric functions and non- +commutative symmetric functions. It emerged from a noncommutative realization +of quasi-symmetric functions analogous to the realization of symmetric functions +by the plactic monoid presented by Lascoux and Schützenberger [LS81]. This led +to a construction of the hypoplactic monoid through quasi-ribbon tableaux and an +insertion algorithm, and to a presentation consisting of the Knuth relations and +the quartic relations. A detailed study of the hypoplactic monoid was done by +Novelli [Nov00]. A comparative study with other monoids was done by Cain, Gray +and Malheiro in [CGM15b], where a rewriting system and a biautomatic structure +for the hypoplactic monoid is presented. Recently, following the work in [Rib22], + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +3 +a complete description of the identities satisfied by the hypoplactic monoid was +presented by Cain, Malheiro and Ribeiro [CMR22]. +A first notion of quasi-crystal graph was introduced by Krob and Thibon [KT99] +to encode the full structure of the modules that give rise to the hypoplactic monoid. +The vertex set of such a graph is formed by the quasi-ribbon words over the alphabet +{12, . . ., n}, which also form a complete set of representatives for the hypoplactic +congruence. Therefore, these quasi-crystal graphs do not allow a construction of +the hypoplactic monoid analogous to the construction of the plactic monoid from +crystal graphs, because they do not have isomorphic connected components. +To overcome the limitations of the first notion of quasi-crystal graph, Cain and +Malheiro [CM17] described a new quasi-crystal graph, derived from the crystal +graph for type An, that allows a construction of the hypoplactic monoid by identi- +fying words in the same position of isomorphic connected components, and induces +the definition of an analogue of Kashiwara operators on words over the alpha- +bet {1, 2, . . ., n}, called quasi-Kashiwara operators. However, this construction is +purely combinatorial and does not have an algebraic foundation. It cannot be used +to construct a monoid starting with a crystal graph of another type [Gui22, Re- +mark 6.17]. It is therefore natural to ask whether quasi-Kashiwara operators on +words can be defined recursively. +The main goal of this paper is to establish a general theory of quasi-crystals +that allows a generalization of the hypoplactic monoid. It addresses the problems +discussed above, while showing that the construction in [CM17] can be placed in +the context of this new theory. It follows the work in [Gui22] and presents a more +consolidated theory with new and improved results. +This paper is structured as follows. Section 2 introduces notation and discusses +preliminaries relating to monoids, root systems, and graphs. Section 3 states the +definitions of quasi-crystals and homomorphisms between quasi-crystals, which give +rise to a category, and is devoted to making an algebraic study of them. Section 4 +presents the notion of the quasi-crystal graph associated to a quasi-crystal, leading +to a combinatorial study of quasi-crystals, and describes a one-to-one correspon- +dence between the class of seminormal quasi-crystals and a class of weighted labelled +graphs. Section 5 states the definition of the quasi-tensor product of quasi-crystals +and describes a practical method to compute it. Section 6 states the definition of +the quasi-crystal monoid and is devoted to making an algebraic study of it, concern- +ing homomorphisms, congruences and free objects. It is shown that a free quasi- +crystal monoid satisfies a universal property which defines it up to isomorphism, +and that congruences on a quasi-crystal monoid form a lattice. Homomorphism +theorems for quasi-crystal monoids are also proven. Section 7 shows that identi- +fying elements in isomorphic connected components of a free quasi-crystal monoid +gives rise to a congruence, called the hypoplactic congruence, which leads to the +definition of hypoplactic monoid associated to a quasi-crystal. It is shown that the +central elements of a hypoplactic monoid correspond to the isolated elements of the +free quasi-crystal monoid, and the idempotents correspond to isolated elements of +weight zero, leading to the conclusion that the idempotents of a hypoplactic monoid +commute. Section 8 proves that the hypoplactic monoid associated to the standard +quasi-crystal of type An is isomorphic to the classical hypoplactic monoid of rank +n, indicating that this approach results in a genuine generalization of the classical +hypoplactic monoid, by showing that the construction in [CM17] can be placed in +the context of the developed framework. Section 9 is devoted to the study of the +hypoplactic monoid associated to the standard quasi-crystal of type Cn. Highest- +weight and isolated words are characterized, allowing an identification of central +and idempotent elements of this monoid. Relations satisfied by the hypoplactic + +4 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +monoid of type Cn are then studied, in particular, it is investigated whether this +monoid satisfies the Knuth relations. it is shown that the hypoplactic monoid of +type Cn satisfies a non-trivial identity if and only if n = 2, in contrast to the classi- +cal hypoplactic monoid, which satisfies a non-trivial identity independently of rank. +It is proven that the hypoplactic monoid of type Cn is not finitely presented, for +any n ≥ 2. Finally, embeddings of the hypoplactic monoids of types An−1 and +Cn−1 into the hypoplactic monoid of type Cn are presented, and it is shown that +the ‘obvious’ approach to defining such embeddings does not work. +2. Preliminaries +We assume some familiarity with the basic concepts related with monoids and +graphs, so we will not make a proper introduction to them. For background on +monoids see [How95], on presentations see [Hig92], and on graphs see [Bol98]. +We will introduce crystals as a subclass of quasi-crystals, and so, we will not need +to present a complete introduction to crystals. We refer to [Kas95] for an intro- +duction to crystals as they originally emerged in connection to quantized universal +envelopping algebras (also called quantum groups) or [HK02] for a comprehensive +background on this approch, to [BS17] for a study of crystals detached from their +origin, and to [CGM19] for the relations between crystals and plactic monoids for +the infinite Cartan types. +In this section, we give the essential background on root systems, as these alge- +braic structures will be used throughout this paper. Root systems are commonly +found in representation theory, in particular, they arise on the study of Lie groups +and Lie algebras, but we will detach them from this context, as our aim is to con- +struct an algebraic structure for defining crystals and quasi-crystals. Thus, we only +introduce the necessary notions needed for this purpose. For further context see +for example [FH91, Bou02, EW06, Bum13]. +Let V be a Euclidean space, that is, a real vector space with an inner product +⟨ · , · ⟩. For α ∈ V other than 0, denote by rα the reflection in the hyperplane +orthogonal to α, which is given by +rα(v) = v − +� +v, α∨� +α, +where +α∨ = +2 +⟨α, α⟩α, +for each v ∈ V . Note that rα is bijective, as rα +� +rα(v) +� += v, for all v ∈ V . Also, rα +preserves the inner product, as +� +rα(u), rα(v) +� += ⟨u, v⟩ for any u, v ∈ V . +A root system in V is a subset Φ of V satisfying the following conditions: +(RS1) Φ is nonempty, finite, and 0 /∈ Φ; +(RS2) rα(β) ∈ Φ, for all α, β ∈ Φ; +(RS3) +� +α, β∨� +∈ Z, for all α, β ∈ Φ, +(RS4) if α ∈ Φ and kα ∈ Φ, then k = ±1. +The elements of Φ are called roots, and the elements α∨, with α ∈ Φ, are called +coroots. Note that the definition of root system may differ in the literature, as +some authors omit some of the conditions above and use them to characterize root +systems. +For instance, some authors say that a root system is crystallographic +when (RS3) is satisfied, or that it is reduced when (RS4) is satisfied. On the other +hand, some authors require Φ to span V , we say that a root system is semisimple +when this happens. +Together with a root system, we always fix an index set I and simple roots +(αi)i∈I, that is, a collection of roots satisfying the following conditions: +(SR1) {αi | i ∈ I} is a linearly independent subset of V ; and +(SR2) every root β ∈ Φ can be expressed as β = � +i∈I kiαi, where all ki are either +nonnegative or nonpositive integers. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +5 +For each i ∈ I, the reflection rαi is called a simple reflection and is denoted by si. +We also fix a weight lattice Λ, that is, a Z-submodule of V satisfying the following +conditions: +(WL1) Λ spans V ; +(WL2) Φ ⊆ Λ; +(WL3) +� +λ, α∨� +∈ Z, for any λ ∈ Λ and α ∈ Φ. +The elements of Λ are called weights and are compared using the following partial +order +λ ≥ µ ⇐⇒ λ − µ = +� +i∈I +kiαi, for some ki ∈ R≥0, i ∈ I. +(2.1) +Finally, we draw attention to the root systems associated to Cartan types An and +Cn, which will be the only non-arbitrary root systems considered in the subsequent +sections. Let n ≥ 2. Consider V to be the real vector space Rn with the usual +inner product, and denote by ei ∈ Rn the n-tuple with 1 in the i-th position, and +0 elsewhere, i = 1, 2, . . . , n. The root system associated to Cartan type An based +on the general linear Lie algebra gln consists of Φ = {ei − ej | i ̸= j}, the index set +for the simple roots is I = {1, 2, . . ., n − 1}, the simple roots are αi = ei − ei+1, +i = 1, 2, . . . , n − 1, and the weight lattice is Λ = Zn. +The root system associated to Cartan type Cn based on the symplectic Lie al- +gebra sp2n consists of Φ = {±ei ± ej | i < j} ∪ {±2ei | i = 1, 2, . . . , n}, the index +set for the simple roots is I = {1, 2, . . ., n}, the simple roots are αi = ei − ei+1, +i = 1, 2, . . . , n − 1, and αn = 2en, and the weight lattice is Λ = Zn. For more +examples of root systems see [BS17, Examples 2.4 to 2.10]. +3. Quasi-crystals and homomorphisms +In this section we introduce the notion of quasi-crystals associated to a root +system. We then study some basic properties satisfied by quasi-crystals, some of +which correspond to generalizations of properties verified by crystals. Finally, we +introduce the notion of quasi-crystal homomorphisms and study their properties. +Although we rely on root systems (Section 2) to define quasi-crystals, we only +make use of properties that are also satisfied by other algebraic structures com- +monly used to define crystals. Thus, all subsequent definitions and results can be +reinterpreted using the algebraic data in [Kas95] or a Cartan datum as in [HK02]. +Consider Z ∪ {−∞, +∞} to be the usual set of integers where we add a minimal +element −∞ and a maximal element +∞, that is, −∞ < m < +∞ for all m ∈ Z. +Also, set m + (−∞) = (−∞) + m = −∞ and m + (+∞) = (+∞) + m = +∞, for +all m ∈ Z. +Definition 3.1. Let Φ be a root system with weight lattice Λ and index set I for +the simple roots (αi)i∈I. A quasi-crystal Q of type Φ consists of a set Q together +with maps wt : Q → Λ, ¨ei, ¨fi : Q → Q ⊔ {⊥} and ¨εi, ¨ϕi : Q → Z ∪ {−∞, +∞}, for +each i ∈ I, satisfying the following conditions: +(1) ¨ϕi(x) = ¨εi(x) + +� +wt(x), α∨ +i +� +; +(2) if ¨ei(x) ∈ Q, then wt +� +¨ei(x) +� += wt(x) + αi, ¨εi +� +¨ei(x) +� += ¨εi(x) − 1, and +¨ϕi +� +¨ei(x) +� += ¨ϕi(x) + 1; +(3) if ¨fi(x) ∈ Q, then wt +� ¨fi(x) +� += wt(x) − αi, ¨εi +� ¨fi(x) +� += ¨εi(x) + 1, and +¨ϕi +� ¨fi(x) +� += ¨ϕi(x) − 1; +(4) ¨ei(x) = y if and only if x = ¨fi(y); +(5) if ¨εi(x) = −∞ then ¨ei(x) = ¨fi(x) = ⊥; +(6) if ¨εi(x) = +∞ then ¨ei(x) = ¨fi(x) = ⊥; + +6 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +for x, y ∈ Q and i ∈ I. The set Q is called the underlying set of Q, and the maps +wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) form the quasi-crystal structure of Q. Also, the map +wt is called the weight map, where wt(x) is said to be the weight of x ∈ Q, and +the maps ¨ei and ¨fi (i ∈ I) are called the raising and lowering quasi-Kashiwara +operators, respectively. +In this definition, ⊥ is an auxilary symbol. In the definition of crystals, 0 is +oftenly used instead of ⊥, but since some well-known crystals have 0 as an element, +we have adopted this notation for quasi-crystals to avoid ambiguity. For x ∈ Q, +by ¨ei(x) = ⊥ (or ¨fi(x) = ⊥) we mean that ¨ei (resp., ¨fi) is undefined on x. On +the other hand, we say that ¨ei (or ¨fi) is defined on x whenever ¨ei(x) ∈ Q (resp., +¨fi(x) ∈ Q). So, alternatively one can consider the quasi-Kashiwara operators ¨ei +and ¨fi (i ∈ I) to be partial maps from Q to Q. When this point of view is more +suitable to describe quasi-Kashiwara operators, we will make use of it. +In comparison with the definition of crystal [BS17, Definition 2.12], we have +that ¨εi and ¨ϕi (i ∈ I) can also take the value +∞. This leads to the addition of +condition (6), as conditions (1) to (5) coincide in both definitions. Thus, we can +take the following as the definition of crystal. +Remark 3.2. A crystal is a quasi-crystal B where ¨εi(x) ̸= +∞ and ¨ϕi(x) ̸= +∞, +for all x ∈ B and i ∈ I. +From condition (1) of Definition 3.1, we get that ¨εi(x) = ±∞ if and only if +¨ϕi(x) = ±∞. And if so, ¨εi(x) = ¨ϕi(x). Thus, conditions (5) and (6) could have been +stated replacing ¨εi by ¨ϕi. Moreover, we could have only stated one of conditions (2) +and (3) as justified by the following result. +Proposition 3.3. Let Φ be a root system with weight lattice Λ and index set I for +the simple roots (αi)i∈I. Consider a set Q and maps wt : Q → Λ, ¨ei, ¨fi : Q → +Q⊔{⊥} and ¨εi, ¨ϕi : Q → Z∪{−∞, +∞}, for each i ∈ I, satisfying Definition 3.1(4). +Then Definition 3.1(2) holds if and only if Definition 3.1(3) holds. +Proof. Assume Definition 3.1(2) holds. Let x ∈ Q and i ∈ I such that ¨fi(x) ∈ Q. +By Definition 3.1(4), we have that ¨ei +� ¨fi(x) +� += x, and so, +wt(x) = wt +� +¨ei( ¨fi(x)) +� += wt +� ¨fi(x) +� +− αi, +¨εi(x) = ¨εi +� +¨ei( ¨fi(x)) +� += ¨εi +� ¨fi(x) +� +− 1, +and +¨ϕi(x) = ¨ϕi +� +¨ei( ¨fi(x)) +� += ¨ϕi +� ¨fi(x) +� ++ 1, +by Definition 3.1(2). Hence, Definition 3.1(3) holds. +The converse implication is analogous. +□ +In the same way, by conditions (1) and (4) of Definition 3.1 we have that a quasi- +crystal is determined by a set Q and the weight map wt together with either ¨ei or +¨fi, and either ¨εi or ¨ϕi, for each i ∈ I. However, for a purpose of clarity, we usually +give explicit definitions for each map when defining a quasi-crystal. +Example 3.4. (1) Consider the root system of type An. +By Remark 3.2, the +standard crystal of type An gives rise to the quasi-crystal An defined as follows. +The underlying set is the ordered set An = {1 < 2 < · · · < n}. For x ∈ An, the +weight of x is wt(x) = ex. For i = 1, 2, . . . , n − 1, the quasi-Kashiwara operators +¨ei and ¨fi are only defined on i + 1 and i, respectively, where ¨ei(i + 1) = i and +¨fi(i) = i + 1. Finally, ¨εi(x) = δx,i+1 and ¨ϕi(x) = δx,i, where δk,l = 1 if k = l, and +δk,l = 0 whenever k ̸= l. We call An the standard quasi-crystal of type An. +(2) Consider the root system of type Cn. By Remark 3.2, the standard crystal +of type Cn gives rise to the quasi-crystal Cn defined as follows. The underlying set + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +7 +is Cn = {1 < 2 < · · · < n < n < n − 1 < · · · < 1}. For x ∈ {1, 2, . . ., n}, the weight +of x is wt(x) = ex, and the weight of x is wt(x) = −ex. For i = 1, 2, . . ., n − 1, +the quasi-Kashiwara operators ¨ei and ¨fi are only defined on the following cases: +¨ei(i + 1) = i, ¨ei(i) = i + 1, ¨fi(i) = i + 1, and ¨fi(i + 1) = i. The quasi-Kashiwara +operators ¨en and ¨fn are only defined in n and n, respectively, where ¨en(n) = n +and ¨fn(n) = n. Finally, for y ∈ Cn, ¨εi(y) = δy,i+1 + δy,i, ¨εn(y) = δy,n, ¨ϕi(y) = +δy,i + δy,i+1, and ¨ϕn(y) = δy,n. We call Cn the standard quasi-crystal of type Cn. +(3) Consider the root system of type A3. We have a quasi-crystal A2 +3 of type A3 +whose underlying set is A2 +3 = A3 × A3 and whose quasi-crystal structure is given +as follows. +x +wt(x) +¨e1(x) +¨e2(x) +¨f1(x) +¨f2(x) +¨ε1(x) +¨ε2(x) +¨ϕ1(x) +¨ϕ2(x) +(1, 1) +2e1 +⊥ +⊥ +(2, 1) +⊥ +0 +0 +2 +0 +(1, 2) +e1 + e2 +⊥ +⊥ +⊥ +(1, 3) ++∞ +0 ++∞ +1 +(1, 3) +e1 + e3 +⊥ +(1, 2) +(2, 3) +⊥ +0 +1 +1 +0 +(2, 1) +e1 + e2 +(1, 1) +⊥ +(2, 2) +(3, 1) +1 +0 +1 +1 +(2, 2) +2e2 +(2, 1) +⊥ +⊥ +(3, 2) +2 +0 +0 +2 +(2, 3) +e2 + e3 +(1, 3) +⊥ +⊥ +⊥ +1 ++∞ +0 ++∞ +(3, 1) +e1 + e3 +⊥ +(2, 1) +(3, 2) +⊥ +0 +1 +1 +0 +(3, 2) +e2 + e3 +(3, 1) +(2, 2) +⊥ +(3, 3) +1 +1 +0 +1 +(3, 3) +2e3 +⊥ +(3, 2) +⊥ +⊥ +0 +2 +0 +0 +(4) Consider the root system of type A2. We have a quasi-crystal Q of type A2 +consisting of a set Q = {a, b} and maps defined as follows. +x +wt(x) +¨e(x) +¨f(x) +¨ε(x) +¨ϕ(x) +a +e1 +⊥ +⊥ +0 +1 +b +e2 +⊥ +⊥ +1 +0 +Since the root system of type A2 has exactly one simple root, we omit the subscript +index in the maps, for instance ¨e instead of ¨e1. +In the previous example we only introduce quasi-crystals that will be relevant +below. As crystals are quasi-crystals (Remark 3.2), more examples can be found +in [BS17, Examples 2.21 to 2.25], where the standard crystals for types Bn and Dn +are included. +Recall the partial order defined on a weight lattice Λ described in (2.1). The +following result justifies the terminology of raising and lowering used to characterize +the quasi-Kashiwara operators ¨ei and ¨fi (i ∈ I). +Proposition 3.5. Let Q be a quasi-crystal, and let x ∈ Q and i ∈ I. If ¨ei(x) ∈ Q, +then wt +� +¨ei(x) +� +> wt(x). If ¨fi(x) ∈ Q, then wt(x) > wt +� ¨fi(x) +� +. +Proof. If ¨ei(x) ∈ Q, then +wt +� +˜ei(x) +� +− wt(x) = wt(x) + αi − wt(x) = αi, +by Definition 3.1(2), and so, wt +� +¨ei(x) +� +≥ wt(x). Since αi is a root, we have that +αi ̸= 0, which implies that wt +� +¨ei(x) +� +̸= wt(x). Hence, wt +� +¨ei(x) +� +> wt(x). +If ¨fi(x) ∈ Q, then x = ¨ei +� ¨fi(x) +� +, by Definition 3.1(4). As proved above, we have +that wt(x) = wt +� +¨ei( ¨fi(x)) +� +> wt +� ¨fi(x) +� +. +□ +From the previous result, we have that, like the Kashiwara operators in crystals, +the raising quasi-Kashiwara operators ¨ei (i ∈ I) increase the weight of elements, +whenever defined, and the lowering quasi-Kashiwara operators ¨fi (i ∈ I) decrease +the weight of elements, whenever they are defined. Thus, the notions of highest- +and lowest-weight elements from crystals can be generalized in a natural way. +Definition 3.6. Let x ∈ Q be an element of a quasi-crystal Q. + +8 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +(1) x is said to be of highest weight if ¨ei(x) = ⊥, for all i ∈ I. +(2) x is said to be of lowest weight if ¨fi(x) = ⊥, for all i ∈ I. +Similar to crystals, notice that a quasi-crystal may have a highest-weight element +whose weight is less than or equal to the weight of an element that is not of highest +weight. For instance, consider the quasi-crystal A2 +3 described in Example 3.4(3), +take x = (1, 1), y = (1, 2) and z = (2, 1), then x and y are of highest weight, z is not +of highest weight, wt(x) > wt(y) and wt(y) = wt(z). Moreover, if a quasi-crystal +Q has an element x ∈ Q such that ¨εi(x) ∈ {−∞, +∞}, for all i ∈ I, then we can +change the weight wt(x) of x to any weight in Λ, and the resulting structure is still +a quasi-crystal. However, if we extend this definition to the weights as follows, we +get some more natural results. +Definition 3.7. Let Q be a quasi-crystal, and let λ ∈ Λ be a weight. +(1) λ is called a highest weight in Q if there exists a highest-weight element +x ∈ Q such that λ = wt(x). +(2) λ is called a lowest weight in Q if there exists a lowest-weight element x ∈ Q +such that λ = wt(x). +Proposition 3.8. Let Q be a quasi-crystal, and let λ be a weight in wt(Q) = +{wt(x) | x ∈ Q}. +(1) If λ is maximal among weights in wt(Q), then λ is a highest weight, and +any element x ∈ Q such that wt(x) = λ is of highest weight. +(2) If λ is minimal among weights in wt(Q), then λ is a lowest weight, and any +element x ∈ Q such that wt(x) = λ is of lowest weight. +Proof. (1) Let x ∈ Q be such that wt(x) = λ. If x is not of highest weight, then +¨ei(x) ∈ Q, for some i ∈ I, and by Proposition 3.5, wt +� +¨ei(x) +� +> wt(x). Therefore, λ +is not maximal among weights in wt(Q). +(2) Let x ∈ Q be such that wt(x) = λ. +If x is not of lowest weight, then +¨fi(x) ∈ Q, for some i ∈ I, and by Proposition 3.5, wt(x) > wt +� ¨fi(x) +� +. Hence, λ is +not minimal among weights in wt(Q). +□ +Since the quasi-Kashiwara operators of a quasi-crystal Q can be regarded as +partial maps from Q to Q, we can compose them in a natural way. As usual, for +i ∈ I, set ¨e0 +i and ¨f 0 +i to be the identity map on Q, and recursively, define ¨ek+1 +i += ¨ei¨ek +i +and ¨f k+1 +i += ¨fi ¨f k +i , for k ≥ 0. +Definition 3.9. A quasi-crystal Q is said to be seminormal if for any x ∈ Q and +i ∈ I, +¨εi(x) = max +� +k ∈ Z≥0 +�� ¨ek +i (x) ∈ Q +� +and +¨ϕi(x) = max +� +k ∈ Z≥0 +�� ¨f k +i (x) ∈ Q +� +, +whenever ¨εi(x) ̸= +∞. +The quasi-crystals described in items (1) to (3) of Example 3.4 are seminormal. +On the other hand, the quasi-crystal described in item (4) is not seminormal. +As pointed out in Remark 3.2, a crystal B satisfies ¨εi(x) ̸= +∞, for all x ∈ B +and i ∈ I. If B is seminormal, then the equalities in Definition 3.9 are verified for +any x ∈ B and i ∈ I, and so, B is seminormal as a crystal [BS17, formula (2.6)]. +Thus, the seminormal property for quasi-crystals generalize the one for crystals in +the following sense. +Remark 3.10. For a crystal B, we have that B is seminormal as a crystal if and only +if it is seminormal as a quasi-crystal. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +9 +We have just seen that the seminormal property for quasi-crystals is consistent +with the corresponding property for crystals. The exception when ¨εi(x) takes the +value +∞ is crucial. Without this exception, in the case ¨εi(x) = +∞ we would have +that max +� +k ∈ Z≥0 +�� ¨ek +i (x) ∈ Q +� += 0, by Definition 3.1(6), and hence the class of +seminormal quasi-crystals would coincide with the class of seminormal crystals, and +we would not have a proper generalization of the seminormal property as intended. +Thus this exception is vital. However, it has deep implications, as some common +results for seminormal crystals are not satisfied by seminormal quasi-crystals. For +example, we can no longer guarantee the weight of a highest-weight element to be +dominant. Instead we have the following result. +Proposition 3.11. Let x ∈ Q be an element of a seminormal quasi-crystal Q. +(1) If x is of highest weight and +� +wt(x), α∨ +i +� +< 0, for some i ∈ I, then ¨εi(x) = +¨ϕi(x) = +∞. +(2) If x is of lowest weight and +� +wt(x), α∨ +i +� +> 0, for some i ∈ I, then ¨εi(x) = +¨ϕi(x) = +∞. +Proof. (1) Suppose that ¨εi(x) ̸= +∞ (or equivalently, ¨ϕi(x) ̸= +∞), for some i ∈ I. +As Q is seminormal, we have that ¨εi(x), ¨ϕi(x) ∈ Z≥0 and by Definition 3.1(1), +� +wt(x), α∨ +i +� += ¨ϕi(x) − ¨εi(x). +Thus, if +� +wt(x), α∨ +i +� +< 0, then ¨εi(x) > 0, which implies that ¨ei(x) ∈ Q, because Q +is seminormal. Hence, x is not of highest weight. +(2) As above, if ¨εi(x) ̸= +∞ and +� +wt(x), α∨ +i +� +> 0, then ¨ϕi(x) ∈ Z>0, which +implies that ¨fi(x) ∈ Q. And therefore, x is not of lowest weight. +□ +Now, we introduce the definition of a homomorphism between quasi-crystals, +which is analogous to the one for crystals. +Definition 3.12. Let Q and Q′ be quasi-crystals of the same type. A quasi-crystal +homomorphism ψ from Q to Q′, denoted by ψ : Q → Q′, is a map ψ : Q ⊔ {⊥} → +Q′ ⊔ {⊥} that satisfies the following conditions: +(1) ψ(⊥) = ⊥; +(2) if ψ(x) ∈ Q′, then wt +� +ψ(x) +� += wt(x), ¨εi +� +ψ(x) +� += ¨εi(x), and ¨ϕi +� +ψ(x) +� += +¨ϕi(x); +(3) if ¨ei(x) ∈ Q and ψ(x), ψ +� +¨ei(x) +� +∈ Q′, then ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� +; +(4) if ¨fi(x) ∈ Q and ψ(x), ψ +� ¨fi(x) +� +∈ Q′, then ψ +� ¨fi(x) +� += ¨fi +� +ψ(x) +� +; +for x ∈ Q and i ∈ I. +A quasi-crystal isomorphism ψ between Q and Q′ is a bijection ψ : Q ⊔ {⊥} → +Q′⊔{⊥} such that ψ : Q → Q′ and ψ−1 : Q′ → Q are quasi-crystal homomorphisms. +We say that Q and Q′ are isomorphic if there exists a quasi-crystal isomorphism +between Q and Q′. +Due to condition (1), when defining a quasi-crystal homomorphism ψ, we omit +the explicit mention to ψ(⊥) = ⊥. Moreover, as ⊥ is an auxilary symbol which +stands for undefinition, alternatively a quasi-crystal homomorphism ψ : Q → Q′ +can be regarded as a partial map ψ from Q to Q′ satisfying conditions (2) to (4). +Thus, when defining a quasi-crystal homomorphism, we usually only give the images +for the elements x ∈ Q such that ψ(x) ∈ Q′. For the sake of simplicity, by saying +that a map ψ : Q → Q′ is a quasi-crystal homomorphism from Q to Q′, we mean +that the map ψ′ : Q ⊔ {⊥} → Q′ ⊔ {⊥}, given by ψ′(⊥) = ⊥ and ψ′(x) = ψ(x), for +each x ∈ Q, is a quasi-crystal homomorphism from Q to Q′. +The notion of crystal homomorphism can be placed in the context of quasi- +crystals in the following way. + +10 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Remark 3.13. A crystal homomorphism is a quasi-crystal homomorphism between +two crystals. +At this point we defined quasi-crystals and homomorphisms between them. It is +immediate from Definition 3.12 that given a quasi-crystal Q, the identity map on +Q is a quasi-crystal homomorphism from Q to Q. The following result follows by +a straightforward application of the definitions. +Proposition 3.14. Let Q1, Q2 and Q3 be quasi-crystals of the same type, and let +ψ1 : Q1 → Q2 and ψ2 : Q2 → Q3 be quasi-crystal homomorphisms. Then, ψ2 ◦ ψ1 +is a quasi-crystal homomorphism from Q1 to Q3. +Thus we obtain a category whose objects are quasi-crystals of the same type and +morphisms are quasi-crystal homomorphisms. +We say that a quasi-crystal homomorphism ψ : Q → Q′ is injective, surjective +or bijective if the map ψ : Q ⊔ {⊥} → Q′ ⊔ {⊥} is injective, surjective or bijective, +respectively. As the following example shows, a bijective quasi-crystal homomor- +phosm is not necessarily a quasi-crystal isomorphism. +Example 3.15. Let A2 and Q be the quasi-crystals of type A2 described respec- +tively in items (1) and (4) of Example 3.4. Define a map ψ : Q → A2 by ψ(a) = 1 +and ψ(b) = 2. Then, ψ is a quasi-crystal homomorphism from Q to A2. But ψ is +not a quasi-crystal isomorphism as ψ−1 does not verify conditions (3) and (4) of +Definition 3.12. +By Remarks 3.2 and 3.13, we have that ψ is a bijective crystal homomorphism +that is not a crystal isomorphism. Also, notice that Q is not seminormal. So, +in the following results we present an alternative characterization of quasi-crystal +isomorphisms for seminormal quasi-crystals. +Lemma 3.16. Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → Q′ +be a bijective quasi-crystal homomorphism. The following conditions are equivalent +(1) ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� +for all x ∈ Q and i ∈ I; +(2) ψ +� ¨fi(x) +� += ¨fi +� +ψ(x) +� +for all x ∈ Q and i ∈ I. +Proof. Suppose that ψ satisfies (1). Let x ∈ Q and i ∈ I. Since ψ is bijective and +ψ(⊥) = ⊥ by Definition 3.12(1), then ψ(y) ∈ Q′ for all y ∈ Q. Thus, if ¨fi(x) ∈ Q, +then ψ +� ¨fi(x) +� +∈ Q′ which implies ψ +� ¨fi(x) +� += ¨fi +� +ψ(x) +� +by Definition 3.12(4). If +¨fi +� +ψ(x) +� +∈ Q′, or equivalently, ψ−1� ¨fi(ψ(x)) +� +∈ Q, then +ψ +� +¨ei +� +ψ−1� ¨fi(ψ(x)) +��� += ¨ei +� +ψψ−1� ¨fi(ψ(x)) +�� += ¨ei ¨fi +� +ψ(x) +� += ψ(x), +as we assumed that ψ satisfies (1), and so, x = ¨ei +� +ψ−1� ¨fi(ψ(x)) +�� +which implies +ψ +� ¨fi(x) +� += ψ +� ¨fi¨ei +� +ψ−1� ¨fi(ψ(x)) +��� += ψψ−1� ¨fi(ψ(x)) +� += ¨fi +� +ψ(x) +� +. +Hence, ψ satisfies (2). +The fact that (2) implies (1) follows analogously. +□ +Theorem 3.17. Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → +Q′ be a quasi-crystal homomorphism. Then, ψ is a quasi-crystal isomorphism if +and only if ψ is bijective and satisfies (1) or (2) of Lemma 3.16. +Proof. Suppose that ψ is a quasi-crystal isomorphism. +By Definition 3.12, ψ is +bijective. Let x ∈ Q and i ∈ I. If ¨ei(x) ∈ Q, we also have that ψ(x), ψ +� +¨ei(x) +� +∈ Q′ +as ψ is bijective and ψ(⊥) = ⊥, and so, ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� +by Definition 3.12(3). +Similarly, since ψ−1 is also a quasi-crystal isomorphism, if ¨ei +� +ψ(x) +� +∈ Q′, then +ψ−1� +¨ei(ψ(x)) +� += ¨ei +� +ψ−1ψ(x) +� += ¨ei(x), + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +11 +which implies ¨ei +� +ψ(x) +� += ψ +� +¨ei(x) +� +. Hence, ψ satisfies Lemma 3.16(1). +Conversely, by Lemma 3.16, we can assume that ψ is bijective and satisfies con- +ditions (1) and (2) of that lemma. Clearly, ψ−1(⊥) = ⊥. Let x′ ∈ Q′ and i ∈ I. +Since ψ is a quasi-crystal homomorphism, by Definition 3.12(2) we have that +wt +� +ψ−1(x′) +� += wt +� +ψ +� +ψ−1(x′) +�� += wt(x′). +Similarly, we get ¨εi +� +ψ−1(x′) +� += ¨εi(x′) and ¨ϕi +� +ψ−1(x′) +� += ¨ϕi(x′). Since ψ satisfies +Lemma 3.16(1), then +¨ei +� +ψ−1(x′) +� += ψ−1ψ +� +¨ei +� +ψ−1(x′) +�� += ψ−1� +¨ei +� +ψψ−1(x′) +�� += ψ−1� +¨ei(x′) +� +. +And since ψ satisfies Lemma 3.16(2), then +¨fi +� +ψ−1(x′) +� += ψ−1ψ +� ¨fi +� +ψ−1(x′) +�� += ψ−1� ¨fi +� +ψψ−1(x′) +�� += ψ−1� ¨fi(x′) +� +. +Hence, ψ−1 is a quasi-crystal homomorphism from Q′ to Q, and therefore, ψ is a +quasi-crystal isomorphism between Q and Q′. +□ +Corollary 3.18. Let Q and Q′ be seminormal quasi-crystals of the same type, +and let ψ : Q → Q′ be a bijective quasi-crystal homomorphism. +Then, ψ is a +quasi-crystal isomorphism. +Proof. Let x ∈ Q and i ∈ I. As ψ is bijective, we get that ψ(x) ∈ Q′. Since +Q and Q′ are seminormal and ¨εi +� +ψ(x) +� += ¨εi(x), we have that ¨ei(x) ∈ Q if and +only if ¨ei +� +ψ(x) +� +∈ Q′. +So, if ¨ei(x) ∈ Q, then ψ +� +¨ei(x) +� +∈ Q, as ψ is bijective, +and ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� +by Definition 3.12(3). Otherwise, ¨ei(x) = ⊥ = ¨ei +� +ψ(x) +� +, +which implies that ψ +� +¨ei(x) +� += ⊥ = ¨ei +� +ψ(x) +� +, by Definition 3.12(1). +Hence, ψ +satisfies Lemma 3.16(1), and by Theorem 3.17, ψ is a quasi-crystal isomorphism. +□ +4. Quasi-crystal graphs +In this section we present a combinatorial approach to quasi-crystals, which +results in a generalization of the notion of crystal graph. In this framework we +are able to characterize some substructures of quasi-crystals, generalizing similar +structures described for crystals based on crystal graphs. +Finally, as a crystal +graph of a seminormal crystal completely determines its crystal structure, we show +a similar connection between quasi-crystal graphs and seminormal quasi-crystals. +Definition 4.1. Let Λ be a weight lattice. A weight map on a graph Γ with vertex +set X is a map wt : X → Λ. For a vertex x ∈ X of Γ, wt(x) is called the weight of +x. In this case, we say the graph Γ is Λ-weighted. +Definition 4.2. Let Φ be a root system with weight lattice Λ and index set I for +the simple roots (αi)i∈I. The quasi-crystal graph ΓQ of a quasi-crystal Q of type Φ +is a Λ-weighted I-labelled directed graph with vertex set Q and an edge x +i +−−−→ y +from x ∈ Q to y ∈ Q labelled by i ∈ I whenever ¨fi(x) = y, and a loop on x ∈ Q +labelled by i ∈ I whenever ¨εi(x) = +∞. For x ∈ Q, let ΓQ(x) denote the connected +component of ΓQ containing the vertex x. +In comparison with crystal graphs, by requiring quasi-crystal graphs to be Λ- +weighted, we accommodate the weight map wt of a quasi-crystal directly in the +definition of its quasi-crystal graph. Also, we have that a quasi-crystal graph may +not be simple. Moreover, a quasi-crystal graph is simple if the maps ¨εi (i ∈ I) do +not take the value +∞, and from Remark 3.2, we observe the following. +Remark 4.3. For a quasi-crystal Q, the quasi-crystal graph ΓQ is simple if and only +if Q is a crystal. And if so, the quasi-crystal graph of Q coincides with its crystal +graph. + +12 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Example 4.4. (1) The quasi-crystal graph ΓAn of the standard quasi-crystal An +of type An, described in Example 3.4(1), is the following. +1 +1 +−−−→ 2 +2 +−−−→ 3 +3 +−−−→ · · · +n−1 +−−−→ n +(2) The quasi-crystal graph ΓCn of the standard quasi-crystal Cn of type Cn, +described in Example 3.4(2), is the following. +1 +1 +−−−→ 2 +2 +−−−→ · · · +n−1 +−−−→ n +n +−−−→ n +n−1 +−−−→ n − 1 +n−2 +−−−→ · · · +1 +−−−→ 1 +(3) The quasi-crystal graph ΓA2 +3 of the quasi-crystal A2 +3 of type A3, described in +Example 3.4(3), is the following. +(1,1) +(2,1) +(3,1) +(1,2) +(2,2) +(3,2) +(1,3) +(2,3) +(3,3) +1 +1 +2 +1 +1 +2 +2 +2 +1 +2 +Note that (1) and (2) of the previous example are crystal graphs (Remark 4.3). +For the crystal graphs associated to the standard crystals of type Bn and Dn, see +for example [CGM19, § 3.3]. +Let x +i +−−−→ y be an edge of a quasi-crystal graph ΓQ of a quasi-crystal Q. +If x ̸= y, then y = ¨fi(x) by Definition 4.2, or equivalently, x = ¨ei(y) due to +Definition 3.1(4). Otherwise, we have that x = y, that is, x has a loop labelled by +i, and so, ¨εi(x) = +∞ by Definition 4.2, which implies that ¨ei and ¨fi are undefined +on x, by Definition 3.1(6). In either case, we have that if x +i +−−−→ y′ is an edge of +ΓQ, then y = y′, and similarly, if x′ +i +−−−→ y is an edge of ΓQ, then x = x′. Hence, +for i ∈ I, a vertex of ΓQ is the start of at most one edge, and is the end of at most +one edge labelled by i. +We will show that quasi-crystal graphs provide a combinatorial framework to +study quasi-crystals, analogous to the tools that crystal graphs provide for crystals. +For instance, Proposition 3.5 is equivalent to state that for a quasi-crystal Q, if +x +i +−−−→ y is an edge of ΓQ with x ̸= y, then wt(x) > wt(y). Also, Definition 3.6 is +equivalent to stating that an element x ∈ Q is of highest (or lowest) weight if the +only edges of ΓQ ending (resp., starting) at x are loops. +From Remark 4.3, the combinatorial framework formed by quasi-crystal graphs +is a genuine generalization of the framework formed by crystal graphs. This allows +a natural generalization of structures such as connected components. +Definition 4.5. Let Q be a quasi-crystal. A connected component of Q is a subset +Q′ of Q that satisfies the following conditions: +(1) for each x, y ∈ Q′ there exist g1, . . . , gm ∈ {¨ei, ¨fi | i ∈ I} such that +g1 · · · gm(x) = y; +(2) ¨ei(x), ¨fi(x) ∈ Q′ ⊔ {⊥} for all x ∈ Q′ and i ∈ I. +We also use the term connected component to refer to the quasi-crystal Q′ consisting +of Q′ together with the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) of Q restricted to Q′. For +each x ∈ Q, the connected component of Q containing x is denoted by Q(x), and +the associated quasi-crystal is denoted by Q(x). +As a justification for this terminology, we check that connected components of +a quasi-crystal Q and the vertex sets of connected components of the quasi-crystal +graph ΓQ identify the same subsets of Q. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +13 +Proposition 4.6. Let Q be a quasi-crystal, and let Q′ ⊆ Q. Then, Q′ is a con- +nected component of Q if and only if Q′ is the vertex set of a connected component +of ΓQ. +Proof. Assume that Q′ is a connected component of Q. Let x, y ∈ Q′. Then, by +Definition 4.5(1), there exist g1, . . . , gm ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gm(x) = y. +Set x0 = x, and xk+1 = gm−k(xk), for k = 0, 1, . . ., m − 1. Note that xk+1 = +gm−k · · · gm(x) ∈ Q′, by Definition 4.5(2). In particular, xm = g1 · · · gm(x) = y. +If gm−k = ¨fi, for some i ∈ I, then xk +i +−−−→ xk+1 is an edge of ΓQ. Otherwise, +gm−k = ¨ei, for some i ∈ I, and so, xk+1 +i +−−−→ xk is an edge of ΓQ. For z ∈ Q, if +x +i +−−−→ z or z +i +−−−→ x is an edge of ΓQ, then z = ¨fi(x) or z = ¨ei(x), respectively, +which implies that z ∈ Q′, by Definition 4.5(2). Therefore, the subgraph of ΓQ +induced by Q′ is a connected component. +Conversely, assume that Q′ is a vertex set of a connected component of ΓQ. Let +x, y ∈ Q′. Then there exist x0, . . . , xm ∈ Q′ such that x = x0, y = xm, and for +k = 0, . . . , m − 1, xk +i +−−−→ xk+1 or xk+1 +i +−−−→ xk is an edge of ΓQ, for some i ∈ I. +If xk +i +−−−→ xk+1 is an edge of ΓQ, for some i ∈ I, set gm−k = ¨fi. +Otherwise, +xk+1 +i +−−−→ xk is an edge of ΓQ, for some i ∈ I, and so, set gm−k = ¨ei. In any case +we have that xk+1 = gm−k(xk), which implies that g1 · · · gm(x) = y. For i ∈ I, +if ¨fi(x) ∈ Q, then x +i +−−−→ ¨fi(x) is an edge of ΓQ, and so, ¨fi(x) ∈ Q′. Similarly, +if ¨ei(x) ∈ Q, then ¨ei(x) +i +−−−→ x is an edge of ΓQ, which implies that ¨ei(x) ∈ Q′. +Therefore, Q′ is a connected component of Q. +□ +Given Λ-weighted I-labelled directed graphs Γ1 and Γ2 with vertex sets X1 and +X2, respectively, a homomorphism ψ from Γ1 to Γ2 is a map ψ : X1 → X2 such that +wt +� +ψ(x) +� += wt(x), for all x ∈ X1, and ψ(x) +i +−−−→ ψ(y) is an edge of Γ2, whenever +x +i +−−−→ y is an edge of Γ1. If ψ is also bijective and ψ−1 is a homomorphism from +Γ2 to Γ1, then ψ is said to be an isomorphism between Γ1 and Γ2. +We also have the following relation between quasi-crystal homomorphisms and +graph homomorphisms. +Lemma 4.7. Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → +Q′ be a quasi-crystal homomorphism such that ψ(Q) ⊆ Q′. Then, ψ is a graph +homomorphism from ΓQ to ΓQ′. +Proof. Let x, y ∈ Q and i ∈ I. By Definition 3.12(2), we have that wt +� +ψ(x) +� += +wt(x). Suppose that x +i +−−−→ y is an edge of ΓQ. If x = y, that is, x has a loop +labelled by i, then ¨εi(x) = +∞, and by Definition 3.12(2), ¨εi +� +ψ(x) +� += +∞, which +implies that ψ(x) has a loop labelled by i in ΓQ′. +Otherwise, x ̸= y, we have +by Definition 4.2 that ¨fi(x) = y, and since ψ(x), ψ(y) ∈ ψ(Q) ⊆ Q′, we get by +Definition 3.12(4) that ¨fi +� +ψ(x) +� += ψ(y), which implies that ψ(x) +i +−−−→ ψ(y) is an +edge of ΓQ′. Therefore, ψ is a graph homomorphism. +□ +Notice that the converse of the previous result does not hold, as ψ may be a +graph homomorphism from ΓQ to ΓQ′ and not be a quasi-crystal homomorphism +from Q to Q′. +Example 4.8. Consider the root system of type A2. Take Q consisting of the set +Q = {x}, where wt(x) = 0, ¨e(x) = ¨f(x) = ⊥ and ¨ε(x) = ¨ϕ(x) = 0, and take + +14 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Q′ consisting of the set Q′ = {x′}, where wt(x′) = 0, ¨e(x′) = ¨f(x′) = ⊥ and +¨ε(x′) = ¨ϕ(x′) = +∞. The quasi-crystal graphs of Q and Q′ are respectively +x +and +x′ +1 . +The map ψ : Q → Q′, defined by ψ(x) = x′, is a graph homomorphism, but not a +quasi-crystal homomorphism, because ¨ε +� +ψ(x) +� += +∞ ̸= 0 = ¨ε(x). +A quasi-crystal isomorphism ψ between quasi-crystals Q and Q′ satisfies the +property ψ(Q) = Q′. Thus, it is immediate from Lemma 4.7 that ψ and ψ−1 are +graph homomorphisms, which implies that ψ is a graph isomorphism between ΓQ +and ΓQ′. This leads to the following result. +Proposition 4.9. Let Q and Q′ be two quasi-crystals of the same type, and let +ψ : Q → Q′ be a quasi-crystal isomorphism. Then, Q0 is a connected component +of Q if and only if ψ(Q0) is a connected component of Q′. Furthermore, for each +x ∈ Q, the restriction of ψ to Q(x) is a quasi-crystal isomorphism between Q(x) +and Q′� +ψ(x) +� +. +Proof. Suppose Q0 is a connected component of Q. Let x′, y′ ∈ ψ(Q0). Set x = +ψ−1(x′) and y = ψ−1(y′). As x, y ∈ Q0, there exist g1, . . . , gm ∈ {¨ei, ¨fi | i ∈ I} +such that g1 · · · gm(x) = y. By Lemma 3.16 and Theorem 3.17, we have that +g1 · · · gm(x′) = g1 · · · gm +� +ψ(x) +� += ψ +� +g1 · · · gm(x) +� += ψ(y) = y′, +which implies that ψ(Q0) satisfies Definition 4.5(1). By the same results, for i ∈ I, +we have that ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� += ¨ei(x′) and ψ +� ¨fi(x) +� += ¨fi +� +ψ(x) +� += ¨fi(x′), and +since ¨ei(x), ¨fi(x) ∈ Q0 ⊔ {⊥}, we get that ¨ei(x′), ¨fi(x′) ∈ ψ(Q0) ⊔ {⊥}. Therefore, +ψ(Q0) is a connected component of Q′. +Since ψ−1 is a quasi-crystal isomorphism between Q′ and Q, by the previous +implication, if ψ(Q0) is a connected component of Q′, then ψ−1� +ψ(Q0) +� += Q0 is a +connected component of Q. +Finally, let z ∈ Q. Since Q(z) is a connected component of Q, then ψ +� +Q(z) +� +is a +connected component of Q′. As ψ(z) ∈ ψ +� +Q(z) +� +, we get that ψ +� +Q(z) +� += Q′� +ψ(z) +� +. +The restriction of ψ to Q(z) is a bijective quasi-crystal homomorphism from Q(z) +to Q′� +ψ(z) +� +, because ψ is a quasi-crystal homomorphism from Q to Q′. Similarly, +the restriction of ψ−1 to Q′� +ψ(z) +� +is a bijective quasi-crystal homomorphism from +Q′� +ψ(z) +� +to Q(z). And therefore, the restriction of ψ to Q(z) is a quasi-crystal +isomorphism between Q(z) and Q′� +ψ(z) +� +. +□ +For a quasi-crystal Q, it is immediate that the quasi-Kashiwara operators ¨ei and +¨fi (i ∈ I) are completely determined by the quasi-crystal graph ΓQ, because given +x, y ∈ Q with x ̸= y, we have that x +i +−−−→ y is an edge of ΓQ if and only if ¨fi(x) = y +and ¨ei(y) = x. Now, we show that if Q is seminormal, then also the maps ¨εi and +¨ϕi (i ∈ I) are completely determined by ΓQ. +Lemma 4.10. Let Q be a quasi-crystal, and let i ∈ I. Given an i-labelled walk +x0 +i +−−−→ x1 +i +−−−→ · · · +i +−−−→ xm +on ΓQ, then either +(1) x0 = x1 = · · · = xm; or +(2) xk = ¨f k +i (x0), for k = 0, 1, . . . , m, and thus, x0, x1, . . . , xm form the unique +i-labelled path on ΓQ starting at x0 and ending at xm. +Proof. If x0 = x1, then x0 has an i-labelled loop, and so, ¨εi(x0) = +∞. Thus, +¨fi(x1) = ¨fi(x0) = ⊥, which implies that x1 = x2. And recursively, we obtain that +x0 = x1 = · · · = xm. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +15 +Otherwise, we have that x0 ̸= x1, which implies that ¨fi(x0) = x1 (or equivalently, +¨ei(x1) = x0), because x0 +i +−−−→ x1 is an edge of ΓQ. Since ¨ei is defined on x1, then +¨εi(x1) ̸= +∞, and so, x1 does not have an i-labelled loop. +Hence, x1 ̸= x2, +and so, ¨fi(x1) = x2, because x1 +i +−−−→ x2 is an edge of ΓQ. Recursively, we get +that ¨fi(xk) = xk+1, for k = 0, 1, . . . , m − 1. +By Proposition 3.5, we have that +wt(x0) > wt(x1) > · · · > wt(xm), which implies that x0, x1, . . . , xm are pairwise +distinct, and thus, form an i-labelled path on ΓQ. It is the unique i-labelled path on +ΓQ starting at x0 and ending at xm, because in a quasi-crystal graph every vertex +is the start of at most an edge and is the end of at most an edge labelled by i. +□ +Proposition 4.11. Let Q be a seminormal quasi-crystal. For x ∈ Q and i ∈ I, we +have that +(1) ¨ϕi(x) is the supremum among nonnegative integers m ∈ Z≥0 such that there +exists an i-labelled walk on ΓQ starting at x with length m; +(2) ¨εi(x) is the supremum among nonnegative integers m ∈ Z≥0 such that there +exists an i-labelled walk on ΓQ ending at x with length m. +Proof. (1) Let z ∈ Z≥0 ∪ {+∞} be the supremum among nonnegative integers +m ∈ Z≥0 such that there exists an i-labelled walk on ΓQ starting on x with length +m. If ¨ϕi(x) = +∞, then x has an i-labelled loop on ΓQ. And so, for any m ∈ Z≥0, +the sequence x0, . . . , xm, where x0 = · · · = xm = x, is an i-labelled walk starting +on x with length m. Hence, z = +∞ = ¨ϕi(x). +Otherwise, we have that +¨ϕi(x) = max +� +k ∈ Z≥0 +�� ¨f k +i (x) ∈ Q +� +, +because Q is seminormal. Since +x +i +−−−→ ¨fi(x) +i +−−−→ · · · +i +−−−→ ¨f ¨ϕi(x) +i +(x) +is an i-labelled path on ΓQ starting on x, we have that ¨ϕi(x) ≤ z. Since x has no +i-labelled loops, if +x0 +i +−−−→ x1 +i +−−−→ · · · +i +−−−→ xm +is a walk on ΓQ such that x0 = x, then xk = ¨f k +i (x), for k = 0, 1, . . . , m, by +Lemma 4.10. And since ¨f ¨ϕi(x)+1 +i +(x) = ⊥, we get that m ≤ ¨ϕi(x). Hence, z ≤ ¨ϕi(x), +and therefore, ¨ϕi(x) = z. +(2) Analogously to (1), we have that if ¨εi(x) = +∞, then the lengths of i-labelled +walks on ΓQ ending on x are unbounded. And otherwise, +¨e¨εi(x) +i +(x) +i +−−−→ ¨e¨εi(x)−1 +i +(x) +i +−−−→ · · · +i +−−−→ x +is the longest i-labelled walk on ΓQ ending on x. +□ +We have shown how the maps ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) of a seminormal quasi- +crystal can be described by an I-labelled directed graph. +And so a seminor- +mal quasi-crystal can be completely described by a Λ-weighted I-labelled directed +graph. From this correspondence between seminormal quasi-crystals and weighted +I-labelled directed graphs, we can identify a subclass of graphs which leads to a +purely combinatorial description of seminormal quasi-crystals. +Remark 4.12. By translating Definitions 3.1 and 3.9 for weighted labelled directed +graphs we obtain a subclass of graphs, whose elements are call seminormal quasi- +crystal graphs. Consider a root system Φ with weight lattice Λ and index set I for +the simple roots (αi)i∈I. A Λ-weighted I-labelled directed graph Γ is a seminormal + +16 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +quasi-crystal graph if for any vertices x and y, and any i ∈ I, the following conditions +are satisfied: +(1) x is the start of at most one edge labelled by i, and is the end of at most +one edge labelled by i; +(2) any i-labelled path of Γ is finite; +(3) if x +i +−−−→ y is an edge of Γ with x ̸= y, then wt(y) = wt(x) − αi; +(4) ¨ϕi(x) = ¨εi(x)+ +� +wt(x), α∨ +i +� +, where ¨ϕi(x) is the supremum among nonnega- +tive integers k ∈ Z≥0 such that there exists an i-labelled walk on Γ starting +on x with length k, and ¨εi(x) is the supremum among nonnegative integers +l ∈ Z≥0 such that there exists an i-labelled walk on Γ ending on x with +length l. +Note that if Γ is a seminormal quasi-crystal with vertex set Q, we can define partial +maps ¨ei and ¨fi (i ∈ I) on Q, by setting ¨ei(y) = x and ¨fi(x) = y, whenever x +i +−−−→ y +is an edge of Γ with x ̸= y. Then, we get a seminormal quasi-crystal Q, and Γ +coincides with ΓQ. +Due to this relation between seminormal quasi-crystals and weighted labelled +directed graphs we obtain the following result. +Theorem 4.13. Let Q and Q′ be seminormal quasi-crystals of the same type, and +let ψ : Q → Q′. Then, ψ is a quasi-crystal isomorphism between Q and Q′ if and +only if ψ is a graph isomorphism between the weighted labelled directed graphs ΓQ +and ΓQ′. +Proof. If ψ is a quasi-crystal isomorphism between Q and Q′, then ψ and ψ−1 are +graph homomorphisms, by Lemma 4.7. Hence, ψ is a graph isomorphism between +ΓQ and ΓQ′. +Conversely, suppose that ψ is a graph isomorphism between ΓQ and ΓQ′. Let +x ∈ Q and i ∈ I. By definition, ψ is weight-preserving, that is, wt +� +ψ(x) +� += wt(x). +If ¨fi(x) ∈ Q, then x +i +−−−→ ¨fi(x) is an edge of ΓQ, which implies that ψ(x) +i +−−−→ +ψ +� ¨fi(x) +� +is an edge of ΓQ′. Since x ̸= ¨fi(x) and ψ is bijective, then ψ(x) ̸= ψ +� ¨fi(x) +� +, +which implies that ¨fi +� +ψ(x) +� += ψ +� ¨fi(x) +� +. Analogously, if ¨ei(x) ∈ Q, then ψ +� +¨ei(x) +� += +¨ei +� +ψ(x) +� +. Since ψ is a graph isomorphism, we have that x0, x1, . . . , xm ∈ Q form +an i-labelled walk on ΓQ if and only if ψ(x0), ψ(x1), . . . , ψ(xm) form an i-labelled +walk on ΓQ′. Thus, by Proposition 4.11, ¨εi +� +ψ(x) +� += ¨εi(x) and ¨ϕi +� +ψ(x) +� += ¨ϕi(x). +Therefore, ψ is a bijective quasi-crystal homomorphism from Q to Q′, and by +Corollary 3.18, ψ is a quasi-crystal isomorphism between Q and Q′. +□ +In contrast to the relation between graph homomorphisms and crystal homo- +morphisms of seminormal crystals associated to semisimple root systems, we can- +not replace the word isomorphism by homomorphism in the previous result. In +Example 4.8, we have two seminormal quasi-crystals Q and Q′, and a map ψ which +is a graph homomorphism from ΓQ to ΓQ′, but not a quasi-crystal homomorphism +from Q to Q′. Thus, the converse of Lemma 4.7 does not hold even for seminormal +quasi-crystals. On the other hand, we can give a stronger version of Proposition 4.9 +in the particular case of seminormal quasi-crystals. +Proposition 4.14. Let Q and Q′ be seminormal quasi-crystals of the same type, +and let ψ : Q → Q′ be a quasi-crystal homomorphism. +For each x ∈ Q, if +ψ +� +Q(x) +� +⊆ Q′, then the restriction of ψ to Q(x) is a surjective quasi-crystal homo- +morphism from Q(x) to Q′� +ψ(x) +� +. +Proof. Let x ∈ Q be such that ψ +� +Q(x) +� +⊆ Q′. For any y ∈ Q(x) and i ∈ I, we have +that ¨ϕi(y) = ¨ϕi +� +ψ(y) +� +by Definition 3.12(2), and as Q and Q′ are seminormal, + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +17 +¨fi(y) ∈ Q if and only if ¨fi +� +ψ(y) +� +∈ Q′. +And if so, we get that ¨fi(y) ∈ Q(x) +by Definition 4.5(2), and ψ(y), ψ +� ¨fi(y) +� +∈ Q′ as ψ +� +Q(x) +� +⊆ Q′, which implies by +Definition 3.12(4) that ψ +� ¨fi(y) +� += ¨fi +� +ψ(y) +� +. Similarly, ¨ei(y) ∈ Q if and only if +¨ei +� +ψ(y) +� +∈ Q′, and if so, ¨ei(y) ∈ Q(x) and ψ +� +¨ei(y) +� += ¨ei +� +ψ(y) +� +. +Then, given +g1, . . . , gm ∈ {¨ej, ¨fj | j ∈ I}, we have that g1 · · · gm(x) is defined if and only if +g1 · · · gm +� +ψ(x) +� +is defined, in which case ψ +� +g1 · · · gm(x) +� += g1 · · · gm +� +ψ(x) +� +. This +implies that ψ +� +Q(x) +� += Q′� +ψ(x) +� +, and thus, the restriction of ψ to Q(x) induces +a well-defined surjective map from Q(x) to Q′� +ψ(x) +� +. Since ψ is a quasi-crystal +homomorphism, we obtain that the restriction of ψ to Q(x) is a surjective quasi- +crystal homomorphism from Q(x) to Q′� +ψ(x) +� +. +□ +Due to the connection between quasi-crystals and graphs, an element of a quasi- +crystal Q is also a vertex of the graph ΓQ. This leads to characterizations based +on either perspective and justifies terminology as follows. +Definition 4.15. Let Q be a quasi-crystal. An element x ∈ Q is said to be isolated +if Q(x) = {x}. +An isolated element of a quasi-crystal Q is an isolated vertex of the quasi-crystal +graph ΓQ. Thus, an element x ∈ Q is isolated if and only if ¨ei(x) = ¨fi(x) = ⊥, +for all i ∈ I, or equivalently, x is isolated if and only if x is of highest and lowest +weight. Furthermore, if Q is seminormal, then x is isolated if and only if for each +i ∈ I, either ¨εi(x) = ¨ϕi(x) = 0 or ¨εi(x) = ¨ϕi(x) = +∞. +5. Quasi-tensor product of quasi-crystals +A definition of tensor product for quasi-crystals can be given in a similar way +as it was originally done for crystals (see [Kas90, Kas91, KN94]). Such a definition +would lead to a generalization to quasi-crystals of the construction of a plactic +monoid from a crystal as in [Gui22, Ch. 5]. Since we are interested in a general +construction of the hypoplactic monoid from a quasi-crystal, in this section we +introduce a slightly different definition: the quasi-tensor product. We then study +its properties. Finally, as this notion will be used in the subsequent sections to +relate quasi-crystals and monoids, we describe a combinatorial method to compute +the quasi-crystal structure of a quasi-tensor product of quasi-crystals, which is +analogous to the signature rule for the tensor product of crystals. +5.1. Definition and results. In the following theorem we establish the founda- +tions to introduce the notion of quasi-tensor product of quasi-crystals. +Theorem 5.1. Consider a root system Φ with weight lattice Λ and index set I for +the simple roots (αi)i∈I. Let Q and Q′ be seminormal quasi-crystals of type Φ. Set +Q ¨⊗Q′ to be the Cartesian product Q×Q′ whose ordered pairs are denoted by x ¨⊗x′ +with x ∈ Q and x′ ∈ Q′. Define a map wt : Q ¨⊗ Q′ → Λ by +wt(x ¨⊗ x′) = wt(x) + wt(x′), +for x ∈ Q and x′ ∈ Q′. +And for each i ∈ I, define maps ¨ei, ¨fi : Q ¨⊗ Q′ → +(Q ¨⊗ Q′) ⊔ {⊥} and ¨εi, ¨ϕi : Q ¨⊗ Q′ → Z ∪ {−∞, +∞} as follows: +(1) if ¨ϕi(x) > 0 and ¨εi(x′) > 0, set +¨ei(x ¨⊗ x′) = ¨fi(x ¨⊗ x′) = ⊥ +and +¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞; +(2) otherwise, set +¨ei(x ¨⊗ x′) = +� +¨ei(x) ¨⊗ x′ +if ¨ϕi(x) ≥ ¨εi(x′) +x ¨⊗ ¨ei(x′) +if ¨ϕi(x) < ¨εi(x′), + +18 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +¨fi(x ¨⊗ x′) = +� ¨fi(x) ¨⊗ x′ +if ¨ϕi(x) > ¨εi(x′) +x ¨⊗ ¨fi(x′) +if ¨ϕi(x) ≤ ¨εi(x′), +¨εi(x ¨⊗ x′) = max +� +¨εi(x), ¨εi(x′) − +� +wt(x), α∨ +i +�� +, +and +¨ϕi(x ¨⊗ x′) = max +� +¨ϕi(x) + +� +wt(x′), α∨ +i +� +, ¨ϕi(x′) +� +, +where x ¨⊗ ⊥ = ⊥ ¨⊗ x′ = ⊥; +for x ∈ Q and x′ ∈ Q′. Then, Q ¨⊗ Q′ together with the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi +(i ∈ I) forms a seminormal quasi-crystal of type Φ. +Proof. Let x ∈ Q, x′ ∈ Q′ and i ∈ I. We have that ¨εi(x), ¨ϕi(x), ¨εi(x′), ¨ϕi(x′) are +all non-negative as Q and Q′ are seminormal (Definition 3.9). If ¨ϕi(x) > 0 and +¨εi(x′) > 0, it is immediate that x ¨⊗ x′ satisfies all conditions of Definition 3.1, +namely, conditions (1) and (6) which are the ones that apply to this case. So, +assume that ¨ϕi(x) = 0 or ¨εi(x′) = 0. +We have that +¨ϕi(x ¨⊗ x′) = max +� +¨ϕi(x) + +� +wt(x′), α∨ +i +� +, ¨ϕi(x′) +� += max +� +¨εi(x) + +� +wt(x), α∨ +i +� ++ +� +wt(x′), α∨ +i +� +, ¨εi(x′) + +� +wt(x′), α∨ +i +�� += max +� +¨εi(x), ¨εi(x′) − +� +wt(x), α∨ +i +�� ++ +� +wt(x), α∨ +i +� ++ +� +wt(x′), α∨ +i +� += ¨εi(x ¨⊗ x′) + +� +wt(x ¨⊗ x′), α∨ +i +� +. +Hence, condition (1) of Definition 3.1 is satisfied. +If ¨ϕi(x) = +∞ and ¨εi(x′) = 0, , then ¨εi(x) = +∞ and ¨ei(x) = ¨fi(x) = ⊥, +by (1) and (6) of Definition 3.1, which implies that ¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞, +¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ = ⊥, and ¨fi(x ¨⊗ x′) = ¨fi(x) ¨⊗ x′ = ⊥. +Analogously, +if ¨ϕi(x) = 0 and ¨εi(x′) = +∞, we have that ¨ei(x ¨⊗ x′) = ¨fi(x ¨⊗ x′) = ⊥ and +¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞. So, besides ¨ϕi(x) = 0 or ¨εi(x′) = 0, we may further +assume that ¨ϕi(x) ̸= +∞ ̸= ¨εi(x′). +We get that +� +wt(x), α∨ +i +� += ¨ϕi(x) − ¨εi(x) and +� +wt(x′), α∨ +i +� += ¨ϕi(x′) − ¨εi(x′) by +Definition 3.1(1), implying that ¨εi(x ¨⊗x′) = max +� +¨εi(x), ¨εi(x′)− ¨ϕi(x)+ ¨εi(x) +� +and +¨ϕi(x ¨⊗ x′) = max +� +¨ϕi(x) + ¨ϕi(x′) − ¨εi(x′), ¨ϕi(x′) +� +. As ¨εi(x), ¨εi(x′), ¨ϕi(x), ¨ϕi(x′) are +all non-negative where ¨ϕi(x) = 0 or ¨εi(x) = 0, we obtain that +¨εi(x ¨⊗ x′) = ¨εi(x) + ¨εi(x′) +and +¨ϕi(x ¨⊗ x′) = ¨ϕi(x) + ¨ϕi(x′). +(5.1) +Now we consider the following cases. +• Case 1: +¨ϕi(x) = ¨εi(x′) = 0. We have that ¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ and +¨fi(x ¨⊗ x′) = x ¨⊗ ¨fi(x′). Thus, ¨ei(x ¨⊗ x′) is defined if and only if ¨ei(x) ∈ Q. +If so, then we have that +wt +� +¨ei(x ¨⊗ x′) +� += wt +� +¨ei(x) +� ++ wt(x′) = wt(x) + αi + wt(x′) += wt(x ¨⊗ x′) + αi, +and since ¨ei(x) ¨⊗x′ satisfies the conditions leading to (5.1), we deduce that +¨εi +� +¨ei(x ¨⊗ x′) +� += ¨εi +� +¨ei(x) +� ++ ¨εi(x′) = ¨εi(x) − 1 + ¨εi(x′) += ¨εi(x ¨⊗ x′) − 1 +and +¨ϕi +� +¨ei(x ¨⊗ x′) +� += ¨ϕi +� +¨ei(x) +� ++ ¨ϕi(x′) = ¨ϕi(x) + 1 + ¨ϕi(x′) += ¨ϕi(x ¨⊗ x′) + 1. +Also, as ¨ϕi +� +¨ei(x) +� += ¨ϕi(x) + 1 = 1 and ¨εi(x′) = 0, we get that +¨fi +� +¨ei(x ¨⊗ x′) +� += ¨fi +� +¨ei(x) ¨⊗ x′� += ¨fi +� +¨ei(x) +� ¨⊗ x′ = x ¨⊗ x′. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +19 +On the other hand, ¨fi(x ¨⊗ x′) is defined if and only if ¨fi(x′) ∈ Q′. If so, we +have that ¨ϕi(x) = 0 and ¨εi +� ¨fi(x′) +� += ¨εi(x′) + 1 = 1, which implies that +¨ei +� ¨fi(x ¨⊗ x′) +� += ¨ei +� +x ¨⊗ ¨fi(x′) +� += x ¨⊗ ¨ei +� ¨fi(x′) +� += x ¨⊗ x′. +• Case 2: ¨ϕi(x) > 0 and ¨εi(x′) = 0. We have that ¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ +and ¨fi(x ¨⊗ x′) = ¨fi(x) ¨⊗ x′. +Thus, ¨ei(x ¨⊗ x′) is defined if and only if +¨ei(x) ∈ Q, and if so, the facts that condition (3) of Definition 3.1 holds +and ¨fi +� +¨ei(x ¨⊗ x′) +� += x ¨⊗ x′ follow as in case 1. Since Q is seminormal and +¨ϕi(x) > 0, we get that ¨fi(x) ∈ Q, which implies that ¨fi(x ¨⊗ x′) is defined. +As ¨ϕi +� ¨fi(x) +� += ¨ϕi(x) − 1 ≥ 0 and ¨εi(x′) = 0, we obtain that +¨ei +� ¨fi(x ¨⊗ x′) +� += ¨ei +� ¨fi(x) ¨⊗ x′� += ¨ei +� ¨fi(x) +� ¨⊗ x′ = x ¨⊗ x′. +• Case 3: ¨ϕi(x) = 0 and ¨εi(x′) > 0. We have that ¨ei(x ¨⊗ x′) = x ¨⊗ ¨ei(x′) +and ¨fi(x ¨⊗ x′) = x ¨⊗ ¨fi(x′). Since Q′ is seminormal and ¨εi(x′) > 0, then +¨ei(x′) ∈ Q′, which implies that ¨ei(x ¨⊗ x′) is defined. We have that +wt +� +¨ei(x ¨⊗ x′) +� += wt(x) + wt(x′) + αi = wt(x ¨⊗ x′) + αi, +and since ¨ei(x) ¨⊗ x′ satisfies the conditions leading to (5.1), we get that +¨εi +� +¨ei(x ¨⊗ x′) +� += ¨εi(x) + ¨εi(x′) − 1 = ¨εi(x ¨⊗ x′) − 1 +and +¨ϕi +� +¨ei(x ¨⊗ x′) +� += ¨ϕi(x) + ¨ϕi(x′) + 1 = ¨ϕi(x ¨⊗ x′) + 1. +Also, as ¨ϕi(x) = 0 and ¨εi +� +¨ei(x′) +� += ¨εi(x′) − 1 ≥ 0, we obtain that +¨fi +� +¨ei(x ¨⊗ x′) +� += ¨fi +� +x ¨⊗ ¨ei(x′) +� += x ¨⊗ ¨fi +� +¨ei(x′) +� += x ¨⊗ x′. +The fact that ¨ei +� ¨fi(x ¨⊗x′) +� += x ¨⊗x′, whenever ¨fi(x ¨⊗x′) is defined, follows +as in case 1. +In each case we showed that conditions (2) and (4) of Definition 3.1 are satisfied. +We also showed that if x ¨⊗ x′ lies in one of these cases, so does ¨fi(x ¨⊗ x′) when +defined. Thus, by Proposition 3.3, condition (3) of Definition 3.1 also holds. +Therefore, Q ¨⊗Q′ together with wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) forms a quasi-crystal. +It remains to prove that this quasi-crystal is seminormal (Definition 3.9). +Assume that ¨εi(x ¨⊗ x′) ̸= +∞. We have that ¨εi(x), ¨ϕi(x), ¨εi(x′), ¨ϕi(x′) ∈ Z≥0 +where ¨ϕi(x) = 0 or ¨εi(x′) = 0, and so, we have one of the three cases above. Since +Q and Q′ are seminormal, then ¨e¨εi(x) +i +(x) ∈ Q, ¨e¨εi(x)+1 +i +(x) = ⊥, ¨e¨εi(x′) +i +(x′) ∈ Q′, and +¨εi +� +¨e¨εi(x′) +i +(x′) +� += 0 ≤ ¨ϕi +� +¨e¨εi(x) +i +(x) +� +. By (5.1) and cases 1 and 3 above, we get that +¨e¨εi(x ¨⊗x′) +i +(x ¨⊗ x′) = ¨e ¨ϕi(x)+¨εi(x′) +i +(x ¨⊗ x′) = ¨e¨εi(x) +i +� +x ¨⊗ ¨e¨εi(x′) +i +(x′) +� += ¨e¨εi(x) +i +(x) ¨⊗ ¨e¨εi(x′) +i +(x′) +is defined, and +¨e¨εi(x ¨⊗x′)+1 +i +(x ¨⊗ x′) = ¨ei +� +¨e¨εi(x) +i +(x) ¨⊗ ¨e¨εi(x′) +i +(x′) +� += ¨e¨εi(x)+1 +i +(x) ¨⊗ ¨e¨εi(x′) +i +(x′) = ⊥. +Similarly, we have that ¨f ¨ϕi(x ¨⊗x′) +i +(x ¨⊗ x′) = ¨f ¨ϕi(x) +i +(x) ¨⊗ ¨f ¨ϕi(x′) +i +(x′) is defined, and +¨f ¨ϕi(x ¨⊗x′)+1 +i +(x ¨⊗ x′) = ¨f ¨ϕi(x) +i +(x) ¨⊗ ¨f ¨ϕi(x′)+1 +i +(x′) = ⊥. Hence, Q ¨⊗ Q′ together with +the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) is a seminormal quasi-crystal. +□ +Note that the quasi-crystal structure on Q ¨⊗ Q′ given in (2) of Theorem 5.1 +is similar to the original definition of the crystal structure for the tensor product +of crystals [Kas90, Kas91, KN94]. Thus, if we omitted (1) and applied (2) to all +elements, we would have obtained a generalization to quasi-crystals of the tensor +product of crystals, as remarked in the beginning of this section. Note also that + +20 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +if we apply the maps ¨ei, ¨fi, ¨εi and ¨ϕi as defined in (2) to elements of the form +x ¨⊗ x′ with ¨ϕi(x) = +∞ or ¨εi(x′) = +∞, then we get the same images as in (1). +Since Q and Q′ are seminormal, we have that ¨ϕi(x), ¨εi(x′) ∈ Z>0 if and only if +¨fi(x) ∈ Q and ¨ei(x′) ∈ Q′. Hence, condition (1) of Theorem 5.1 is specifying values +for the crystal structure on elements of the form x ¨⊗ x′, where ¨fi(x) and ¨ei(x′) are +defined, different from what they would be if the definitions in (2) would apply +to them. This is a quasi-crystal interpretation of the notion of an i-inversion in a +word, introduced in [CM17, § 5], which justifies the following terminology. +Definition 5.2. Let Q and Q′ be seminormal quasi-crystals of the same type. The +inverse-free quasi-tensor product of Q and Q′, or simply the quasi-tensor product of +Q and Q′, is the seminormal quasi-crystal defined in Theorem 5.1 and is denoted +by Q ¨⊗ Q′. +We chose to give definitions of the maps of the quasi-crystal structure of a quasi- +tensor product in Theorem 5.1 to emphasize their resemblance with the maps of the +crystal structure of a tensor product of crystals, although in the proof we deduced +alternative definitions. The following result is an immediate consequence of the +arguments that led to (5.1) and the cases that followed it. +Proposition 5.3. Let Q and Q′ be seminormal quasi-crystals of the same type. +For x ∈ Q, x′ ∈ Q′ and i ∈ I with ¨ϕi(x) = 0 or ¨εi(x′) = 0, we have that +¨ei(x ¨⊗ x′) = +� +¨ei(x) ¨⊗ x′ +if ¨εi(x′) = 0 +x ¨⊗ ¨ei(x′) +if ¨εi(x′) > 0, +¨fi(x ¨⊗ x′) = +� ¨fi(x) ¨⊗ x′ +if ¨ϕi(x) > 0 +x ¨⊗ ¨fi(x′) +if ¨ϕi(x) = 0, +¨εi(x ¨⊗ x′) = ¨εi(x) + ¨εi(x′), +¨ϕi(x ¨⊗ x′) = ¨ϕi(x) + ¨ϕi(x′). +Example 5.4. (1) The quasi-crystal A2 +3, described in Example 3.4(3) is isomorphic +to A3 ¨⊗ A3 as the map A3 × A3 → A3 ¨⊗ A3, given by (x, y) �→ x ¨⊗ y for each +x, y ∈ A3, is a quasi-crystal isomorphism. Thus, by Theorem 4.13 the quasi-crystal +graph ΓA3 ¨⊗A3 is isomorphic to the quasi-crystal graph ΓA2 +3, which is drawn in +Example 4.4(3). +(2) The quasi-crystal graph ΓC2 ¨⊗C2 of the quasi-tensor product C2 ¨⊗ C2 (see +Example 3.4(2)) is the following. +¨⊗11 +2 ¨⊗1 +2 ¨⊗1 +1 ¨⊗1 +1 ¨⊗2 +2 ¨⊗2 +2 ¨⊗2 +1 ¨⊗2 +1 ¨⊗2 +2 ¨⊗2 +2 ¨⊗2 +1 ¨⊗2 +1 ¨⊗1 +2 ¨⊗1 +2 ¨⊗1 +1 ¨⊗1 +1 +1 +2 +1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +2 +1 +1 +2 +1 +1 +where wt(x ¨⊗ y) = wt(x) + wt(y), for x, y ∈ C2. +From the previous example, we can see that the quasi-tensor product of semi- +normal crystals may not be a crystal. Indeed, if B and B′ are seminormal crystals +with elements x ∈ B and x′ ∈ B′ such that ¨fi(x) ∈ B and ¨ei(x′) ∈ B′, for some +i ∈ I, then ¨εi(x ¨⊗ x′) = +∞. Hence, apart from trivial cases, the quasi-tensor +product of seminormal crystals is not a crystal. +We only defined quasi-tensor product between quasi-crystals that are seminor- +mal, although if we did not require Q and Q′ in Theorem 5.1 to be seminormal, the +resulting structure would still be a quasi-crystal, eventually not seminormal too. +The following example shows that this condition is essential to model inversions in +words by quasi-crystals, that is, we need both ¨ei and ¨fi to be undefined on x ¨⊗ x′, +whenever ¨fi(x) and ¨ei(x′) are defined. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +21 +Example 5.5. Consider the standard quasi-crystal A2 of type A2, described in +Example 3.4(1). +Let Q be a quasi-crystal of type A2 consisting of a set Q = +{−1, −2} and maps given as follows: +x +wt(x) +¨e(x) +¨f(x) +¨ε(x) +¨ϕ(x) +−1 +−e1 +−2 +⊥ +0 +−1 +−2 +−e2 +⊥ +−1 +−1 +0 +. +Clearly, A2 is seminormal, but Q is not. Nonetheless, set a quasi-crystal structure +on A2 ¨⊗Q as defined in Theorem 5.1. Then, ¨f +� +1 ¨⊗(−1) +� += 2 ¨⊗(−1), which implies +that ¨f is defined on an element of the form x ¨⊗ y where ¨f(x) and ¨e(y) are defined. +Alternatively, let ¨e′, ¨f ′ : A2 ¨⊗ Q → (A2 ¨⊗ Q) ⊔ {⊥} be defined as follows. +x ¨⊗ y +¨e′(x ¨⊗ y) +¨f ′(x ¨⊗ y) +1 ¨⊗ (−1) +⊥ +⊥ +2 ¨⊗ (−1) +1 ¨⊗ (−1) +⊥ +1 ¨⊗ (−2) +⊥ +2 ¨⊗ (−2) +2 ¨⊗ (−2) +1 ¨⊗ (−2) +⊥ +So, ¨e′ and ¨f ′ are undefined on x ¨⊗ y whenever ¨f(x) ∈ A2 and ¨e(y) ∈ Q, otherwise +¨e′ and ¨f ′ follow the rule in Theorem 5.1(2). However, there is no quasi-crystal of +type A2 whose quasi-Kashiwara operators are ¨e′ and ¨f ′, because Definition 3.1(4) +is not satisfied, as ¨e′� +2 ¨⊗ (−1) +� += 1 ¨⊗ (−1) and ¨f ′� +1 ¨⊗ (−1) +� += ⊥. This illustrates +that requiring quasi-crystals to be seminormal is essential to give an interpretation +of an inversion on a word by the quasi-tensor product. +In the following result we show that the quasi-tensor product ¨⊗ of quasi-crystals +is an associative operation. +Theorem 5.6. Let Q1, Q2 and Q3 be seminormal quasi-crystals of the same type. +The map (Q1 ¨⊗Q2) ¨⊗Q3 → Q1 ¨⊗(Q2 ¨⊗Q3), given by (x1 ¨⊗x2) ¨⊗x3 �→ x1 ¨⊗(x2 ¨⊗x3), +is a quasi-crystal isomorphism between (Q1 ¨⊗ Q2) ¨⊗ Q3 and Q1 ¨⊗ (Q2 ¨⊗ Q3). +Proof. Define ψ : (Q1 ¨⊗ Q2) ¨⊗ Q3 → Q1 ¨⊗ (Q2 ¨⊗ Q3) by ψ((x1 ¨⊗ x2) ¨⊗ x3) = +x1 ¨⊗(x2 ¨⊗x3) for x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3. It is immediate that ψ is bijective. +Since (Q1 ¨⊗Q2) ¨⊗Q3 and Q1 ¨⊗(Q2 ¨⊗Q3) are seminormal by Theorem 5.1, to prove +that ψ is a quasi-crystal isomorphism, it suffices to show that ψ is a quasi-crystal +homomorphism by Corollary 3.18. Let x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3. Then, +wt +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += wt(x1) + wt(x2) + wt(x3) = wt +� +x1 ¨⊗ (x2 ¨⊗ x3) +� +. +Let i ∈ I. For k = 1, 2, 3, we have that ¨εi(xk), ¨ϕi(xk) ≥ 0 as Qk is seminormal. If +¨εi(xk) = +∞, for some k, then ¨εi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨εi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += ¨εi(xk) = ++∞, which implies by (1) and (6) of Definition 3.1 that ¨ϕi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += +¨ϕi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += +∞, ¨ei +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨ei +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += ⊥, and +¨fi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨fi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += ⊥. So assume that ¨εi(xk) ∈ Z≥0 for all +k = 1, 2, 3. +In the following cases we show that if ¨ϕi(xk), ¨εi(xl) > 0 for some 1 ≤ k < l ≤ 3, +then ¨εi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨εi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += +∞, and thus, it follows as above. +• Case 1: ¨ϕi(x1), ¨εi(x2) > 0. Then ¨εi(x1 ¨⊗ x2) = +∞ and ¨εi(x2 ¨⊗ x3) ≥ +¨εi(x2) > 0, which imply that ¨εi +� +(x1 ¨⊗x2) ¨⊗x3 +� += ¨εi +� +x1 ¨⊗(x2 ¨⊗x3) +� += +∞. +• Case 2: ¨ϕi(x1), ¨εi(x3) > 0. Then ¨ϕi(x1 ¨⊗x2) ≥ ¨ϕi(x1) > 0 and ¨εi(x2 ¨⊗x3) ≥ +¨εi(x3) > 0, which imply that ¨εi +� +(x1 ¨⊗x2) ¨⊗x3 +� += ¨εi +� +x1 ¨⊗(x2 ¨⊗x3) +� += +∞. +• Case 3: ¨ϕi(x2), ¨εi(x3) > 0. Then ¨ϕi(x1 ¨⊗x2) ≥ ¨ϕi(x2) > 0 and ¨εi(x2 ¨⊗x3) = ++∞, which imply that ¨εi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨εi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += +∞. +So, we further assume that ¨εi(xk), ¨ϕi(xl) > 0 implies k ≤ l. + +22 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +By Proposition 5.3, we get that +¨εi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨εi(x1) + ¨εi(x2) + ¨εi(x3) = ¨εi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� +and +¨ϕi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += ¨ϕi(x1) + ¨ϕi(x2) + ¨ϕi(x3) = ¨ϕi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� +. +We also have that +¨ei +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += + + + + + +� +¨ei(x1) ¨⊗ x2 +� ¨⊗ x3 +if ¨εi(x2) = ¨εi(x3) = 0 +� +x1 ¨⊗ ¨ei(x2) +� ¨⊗ x3 +if ¨εi(x2) > 0 = ¨εi(x3) +(x1 ¨⊗ x2) ¨⊗ ¨ei(x3) +if ¨εi(x3) > 0 +and +¨ei +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += + + + + + +¨ei(x1) ¨⊗ (x2 ¨⊗ x3) +if ¨εi(x2) = ¨εi(x3) = 0 +x1 ¨⊗ +� +¨ei(x2) ¨⊗ x3 +� +if ¨εi(x2) > 0 = ¨εi(x3) +x1 ¨⊗ +� +x2 ¨⊗ ¨ei(x3) +� +if ¨εi(x3) > 0, +implying that ψ +� +¨ei((x1 ¨⊗ x2) ¨⊗ x3) +� += ¨ei +� +ψ(x1 ¨⊗ (x2 ¨⊗ x3)) +� +. Similarly, +¨fi +� +(x1 ¨⊗ x2) ¨⊗ x3 +� += + + + + + +� ¨fi(x1) ¨⊗ x2 +� ¨⊗ x3 +if ¨ϕi(x1) > 0 +� +x1 ¨⊗ ¨fi(x2) +� ¨⊗ x3 +if ¨ϕi(x1) = 0 < ¨ϕi(x2) +(x1 ¨⊗ x2) ¨⊗ ¨fi(x3) +if ¨ϕi(x1) = ¨ϕi(x2) = 0 +and +¨fi +� +x1 ¨⊗ (x2 ¨⊗ x3) +� += + + + + + +¨fi(x1) ¨⊗ (x2 ¨⊗ x3) +if ¨ϕi(x1) > 0 +x1 ¨⊗ +� ¨fi(x2) ¨⊗ x3 +� +if ¨ϕi(x1) = 0 < ¨ϕi(x2) +x1 ¨⊗ +� +x2 ¨⊗ ¨fi(x3) +� +if ¨ϕi(x1) = ¨ϕi(x2) = 0, +implying that ψ +� ¨fi((x1 ¨⊗ x2) ¨⊗ x3) +� += ¨fi +� +ψ(x1 ¨⊗ (x2 ¨⊗ x3)) +� +. +Therefore, ψ is a quasi-crystal isomorphism. +□ +Due to the previous result, we may omit parenthesis for the quasi-tensor prod- +uct of seminormal quasi-crystals and simply write Q1 ¨⊗ Q2 ¨⊗ Q3, whose elements +are denoted by x1 ¨⊗ x2 ¨⊗ x3, for x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3. +From the +proofs of Theorems 5.1 and 5.6, we deduce the following result, which generalizes +Proposition 5.3 and describes the quasi-crystal structure of a quasi-tensor product +of an arbitrary number of seminormal quasi-crystals. +Corollary 5.7. Let Q1, . . . , Qm be seminormal quasi-crystals of the same type, and +let x1 ∈ Q1, . . . , xm ∈ Qm. Then, +wt(x1 ¨⊗ · · · ¨⊗ xm) = wt(x1) + · · · + wt(xm). +Also, for i ∈ I, by setting +p = max +� +1 ≤ k ≤ m +�� ¨εi(xk) > 0 +� +and +q = min +� +1 ≤ l ≤ m +�� ¨ϕi(xl) > 0 +� +, +we have that +(1) if p > q or ¨εi(xk) = +∞ for some 1 ≤ k ≤ m, then +¨ei(x1 ¨⊗ · · · ¨⊗ xm) = ¨fi(x1 ¨⊗ · · · ¨⊗ xm) = ⊥ +and +¨εi(x1 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(x1 ¨⊗ · · · ¨⊗ xm) = +∞; + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +23 +(2) otherwise, +¨ei(x1 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xp−1 ¨⊗ ¨ei(xp) ¨⊗ xp+1 ¨⊗ · · · ¨⊗ xm +¨fi(x1 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xq−1 ¨⊗ ¨fi(xq) ¨⊗ xq+1 ¨⊗ · · · ¨⊗ xm +¨εi(x1 ¨⊗ · · · ¨⊗ xm) = ¨εi(x1) + · · · + ¨εi(xp) +and +¨ϕi(x1 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(xq) + · · · + ¨ϕi(xm) +In the following result we show that quasi-crystal homomorphisms between semi- +normal quasi-crystals give rise to homomorphisms between quasi-tensor products +of their domains and images. +Theorem 5.8. Let Q1, Q2, Q′ +1 and Q′ +2 be seminormal quasi-crystals of the same +type, and let ψ1 : Q1 → Q′ +1 and ψ2 : Q2 → Q′ +2 be quasi-crystal homomorphisms. +The partial map ψ1 ¨⊗ψ2 : Q1 ¨⊗Q2 → Q′ +1 ¨⊗Q′ +2, given by x1 ¨⊗x2 �→ ψ1(x1) ¨⊗ψ2(x2) +for each x1 ∈ Q1 and x2 ∈ Q2 such that ψ1(x1) ∈ Q′ +1 and ψ2(x2) ∈ Q′ +2, is a +quasi-crystal homomorphism from Q1 ¨⊗ Q2 to Q′ +1 ¨⊗ Q′ +2. Moreover, if ψ1 and ψ2 +are quasi-crystal isomorphisms, then ψ1 ¨⊗ ψ2 is a quasi-crystal isomorphism. +Proof. Let x1 ∈ Q1 and x2 ∈ Q2 be such that ψ1(x1) ∈ Q′ +1 and ψ2(x2) ∈ Q′ +2. We +get that +wt +� +ψ1(x1) ¨⊗ψ2(x2) +� += wt +� +ψ1(x1) +� ++wt +� +ψ2(x2) +� += wt(x1)+wt(x2) = wt(x1 ¨⊗x2). +Let i ∈ I and k ∈ {1, 2}. By Definition 3.12(2), We have that ¨εi +� +ψk(xk) +� += ¨εi(xk) +and ¨ϕi +� +ψk(xk) +� += ¨ϕi(xk). Thus, if ¨ϕi(x1) > 0 and ¨εi(x2) > 0, then +¨ei +� +ψ1(x1) ¨⊗ ψ2(x2) +� += ¨fi +� +ψ1(x1) ¨⊗ ψ2(x2) +� += ¨ei(x1 ¨⊗ x2) = ¨fi(x1 ¨⊗ x2) = ⊥ +and +¨εi +� +ψ1(x1) ¨⊗ ψ2(x2) +� += ¨ϕi +� +ψ1(x1) ¨⊗ ψ2(x2) +� += ¨εi(x1 ¨⊗ x2) = ¨ϕi(x1 ¨⊗ x2) = +∞. +Otherwise, we get that +¨εi +� +ψ1(x1) ¨⊗ ψ2(x2) +� += max +� +¨εi +� +ψ1(x1) +� +, ¨εi +� +ψ2(x2) +� +− +� +wt +� +ψ1(x1) +� +, α∨ +i +�� += max +� +¨εi(x1), ¨εi(x2) − +� +wt(x1), α∨ +i +�� += ¨εi(x1 ¨⊗ x2). +Analogously, ¨ϕi +� +ψ1(x1) ¨⊗ ψ2(x2) +� += ¨ϕi(x1 ¨⊗ x2). By Definition 3.12(3), if ¨ei(xk) ∈ +Qk and ψk +� +¨ei(xk) +� +∈ Q′ +k, then ψk +� +¨ei(xk) +� += ¨ei +� +ψk(xk) +� +. Thus, if ¨ei(x1 ¨⊗ x2) ∈ +Q1 ¨⊗ Q2 and (ψ1 ¨⊗ ψ2) +� +¨ei(x1 ¨⊗ x2) +� +∈ Q′ +1 ¨⊗ Q′ +2, then +(ψ1 ¨⊗ ψ2) +� +¨ei(x1 ¨⊗ x2) +� += +� +ψ1 +� +¨ei(x1) +� ¨⊗ ψ2(x2) +if ¨ϕi(x1) ≥ ¨εi(x2) +ψ1(x1) ¨⊗ ψ2 +� +¨ei(x2) +� +if ¨ϕi(x1) < ¨εi(x2) += +� +¨ei +� +ψ1(x1) +� ¨⊗ ψ2(x2) +if ¨ϕi +� +ψ1(x1) +� +≥ ¨εi +� +ψ2(x2) +� +ψ1(x1) ¨⊗ ¨ei +� +ψ2(x2) +� +if ¨ϕi +� +ψ1(x1) +� +< ¨εi +� +ψ2(x2) +� += ¨ei +� +ψ1(x1) ¨⊗ ψ2(x2) +� +. +Similarly, if ¨fi(x1 ¨⊗ x2) ∈ Q1 ¨⊗ Q2 and (ψ1 ¨⊗ ψ2) +� ¨fi(x1 ¨⊗ x2) +� +∈ Q′ +1 ¨⊗ Q′ +2, then +(ψ1 ¨⊗ψ2) +� ¨fi(x1 ¨⊗x2) +� += ¨fi +� +ψ1(x1) ¨⊗ψ2(x2) +� +. Therefore, ψ1 ¨⊗ψ2 is a quasi-crystal +homomorphism from Q1 ¨⊗ Q2 to Q′ +1 ¨⊗ Q′ +2. +If ψ1 and ψ2 are quasi-crystal isomorphisms, then ψ1 ¨⊗ ψ2 is a bijective quasi- +crystal homomorphism between seminormal quasi-crystals, as proved above. Hence, +by Corollary 3.18, ψ1 ¨⊗ ψ2 is a quasi-crystal isomorphism. +□ + +24 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +5.2. The signature rule. We now describe a practical method to compute the +quasicrystal structure of the quasi-tensor product of seminormal quasi-crystals. +This method is essentially a combinatorial interpretation of Corollary 5.7, and has +a procedure similar to the signature rule for the tensor product of seminormal +crystals [HK02]. +Let Z0 be the monoid with zero defined by the following presentation ⟨−, + | +(+−, 0)⟩. So, an element of Z0, other than 0, has the form −a+b with a, b ∈ Z≥0. +Let Q be a seminormal quasi-crystal. For each i ∈ I define sgni : Q → Z0 by +sgni(x) = +� +0 +if ¨εi(x) = +∞ +−¨εi(x)+ ¨ϕi(x) +otherwise, +for each x ∈ Q. The map sgni is called the i-signature map for the quasi-tensor +product ¨⊗, and sgni(x) is called the i-signature of x ∈ Q. +In comparison with the signature map for the tensor product of crystals, we +have that the bicyclic monoid ⟨−+ | (+−, ǫ)⟩ (where ǫ denotes the empty word) +has been replaced by the monoid Z0. This allows sgni to interact with the quasi- +tensor product of seminormal quasi-crystals in the following way. +Proposition 5.9. Let Q and Q′ be seminormal quasi-crystals of the same type. +Then, +sgni(x ¨⊗ x′) = sgni(x) sgni(x′), +for all x ∈ Q, x′ ∈ Q′ and i ∈ I. +Proof. Let x ∈ Q, x′ ∈ Q′ and i ∈ I. +By Corollary 5.7, if ¨εi(x) = +∞ (or +equivalently, ¨ϕi(x) = +∞), then sgni(x) = 0 and ¨εi(x ¨⊗ x′) = +∞, which implies +that sgni(x ¨⊗ x′) = 0 = sgni(x) sgni(x′). Similarly, if ¨εi(x′) = +∞, we have that +sgni(x ¨⊗ x′) = 0 = sgni(x) sgni(x′). Thus, assume that ¨ϕi(x), ¨εi(x′) ∈ Z≥0. If +¨ϕi(x), ¨εi(x′) > 0, then ¨εi(x ¨⊗ x′) = +∞, and +sgni(x) sgni(x′) = −¨εi(x)+ ¨ϕi(x)−1+−−¨εi(x′)−1+ ¨ϕi(x′) = 0 = sgni(x ¨⊗ x′). +Finally, assume that ¨ϕi(x) = 0 or ¨εi(x′) = 0. If ¨ϕi(x) = 0, then +sgni(x) sgni(x′) = −¨εi(x)+¨εi(x′)+ ¨ϕi(x′) = −¨εi(x ¨⊗x′)+ ¨ϕi(x ¨⊗x′) = sgni(x ¨⊗ x′), +by Proposition 5.3. If ¨εi(x′) = 0, then +sgni(x) sgni(x′) = −¨εi(x)+ ¨ϕi(x)+ ¨ϕi(x′) = −¨εi(x ¨⊗x′)+ ¨ϕi(x ¨⊗x′) = sgni(x ¨⊗ x′), +by Proposition 5.3. +□ +From the previous result, given seminormal quasi-crystals Q1, Q2, . . . , Qm of the +same type and elements x1 ∈ Q1, x2 ∈ Q2, . . . , xm ∈ Qm, we can easily compute +the i-signature of x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm as +sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = sgni(x1) sgni(x2) · · · sgni(xm) +(i ∈ I). If sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = 0, then ¨εi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(x1 ¨⊗ x2 ¨⊗ +· · · ¨⊗ xm) = +∞ which implies ¨ei(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = ¨fi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = +⊥. +Otherwise, sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = −a+b, for some a, b ∈ Z≥0. +Then, +¨εi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = a and ¨ϕi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = b. From Corollary 5.7, if +a ≥ 1, then ¨ei(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xp−1 ¨⊗ ¨ei(xp) ¨⊗ xp+1 ¨⊗ · · · ¨⊗ xm, +where xp originates the right-most symbol − in sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm). Also, if +b ≥ 1, then ¨fi(x1 ¨⊗x2 ¨⊗· · · ¨⊗xm) = x1 ¨⊗· · · ¨⊗xq−1 ¨⊗ ¨fi(xq) ¨⊗xq+1 ¨⊗· · · ¨⊗xm, where +xq originates the left-most symbol + in sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm). This process is +called the signature rule for the quasi-tensor product. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +25 +Example 5.10. Consider the quasi-crystal A4 ¨⊗A4 ¨⊗A4 ¨⊗A4 ¨⊗A4, where A4 is the +standard quasi-crystal of type A4. We compute ¨e2, ¨f2, ¨ε2 and ¨ϕ2 on 3 ¨⊗1 ¨⊗2 ¨⊗2 ¨⊗3 +using the signature rule. To keep track to which element originates each − and + we +write a subscript with the position of the element, this is just an auxiliary notation +and the binary operation of Z0 should be applied ignoring the subscripts. So we +have that +sgn2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = −1+3+4−5 = −1+30 = 0, +and therefore, +¨e2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = ⊥, +¨ε2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = +∞, +¨f2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = ⊥, +¨ϕ2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = +∞. +Now we compute ¨e1, ¨f1, ¨ε1 and ¨ϕ1 on 2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1. Using the same notation +as above, we obtain that +sgn1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = −1−3+5, +and therefore, +¨e1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2 ¨⊗ 3 ¨⊗ 1 ¨⊗ 3 ¨⊗ 1, +¨ε1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2, +¨f1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 2, +¨ϕ1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 1. +6. Quasi-crystal monoids +In this section we study the algebraic framework relating quasi-crystals and +monoids, which will be used to give a general definition of hypoplactic monoid. +In Subsection 6.1, we present the definition of quasi-crystal monoid, which is the +basic concept for relating quasi-crystals and monoids. +Then in Subsection 6.2, +we introduce the definition of free quasi-crystal monoid over a seminormal quasi- +crystal, and show that free quasi-crystal monoids satisfy a universal property that +defines them up to isomorphism. Finally, in Subsection 6.3, we present the notion of +congruences on a quasi-crystal monoid, which form a lattice and allow to consider +quotients of quasi-crystal monoids, leading to the homomorphism theorems for +quasi-crystal monoids. +6.1. Quasi-crystal monoids and homomorphisms. We first introduce the fun- +damental concept relating quasi-crystals and monoids with respect to the quasi- +tensor product ¨⊗, studied in Section 5. +Definition 6.1. Let Φ be a root system with weight lattice Λ and index set I for +the simple roots (αi)i∈I. A ¨⊗-quasi-crystal monoid M of type Φ consists of a set M +together with maps wt : M → Λ, ¨ei, ¨fi : M → M⊔{⊥}, ¨εi, ¨ϕi : M → Z∪{−∞, +∞} +(i ∈ I) and a binary operation · : M × M → M satisfying the following conditions: +(1) M together with wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) forms a seminormal quasi- +crystal; +(2) M together with · forms a monoid; +(3) the map M ¨⊗ M → M, given by x ¨⊗ y �→ x · y for x, y ∈ M, induces a +quasi-crystal homomorphism from M ¨⊗ M to M. +We stated a definition of quasi-crystal monoid with respect to the quasi-tensor +product ¨⊗, because we shall see that it models the binary operation of the hy- +poplactic monoid, which we want to generalize. A similar definition can be given +by replacing the quasi-tensor product by other operation on quasi-crystals. For +instance, if we considered the tensor product instead, the subsequent would lead to +a notion of plactic monoid over a quasi-crystal. Since our goal is to introduce the +notion of hypoplactic monoid associated to a quasi-crystal, we will only consider + +26 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +quasi-crystal monoids with respect to the quasi-tensor product ¨⊗, and thus, we will +omit ¨⊗ and just say that M is a quasi-crystal monoid. +In a quasi-crystal monoid the interaction between the quasi-crystal structure +and the binary operation satisfies rules similar to those satisfied by the quasi- +crystal structure of a quasi-tensor product (see Theorem 5.1 and Proposition 5.3), +as shown in the following result. +Lemma 6.2. Let M be a quasi-crystal monoid. For x, y ∈ M, we have that +wt(xy) = wt(x) + wt(y), +and for i ∈ I, if ¨ϕi(x) > 0 and ¨εi(y) > 0, then +¨ei(xy) = ¨fi(xy) = ⊥ +and +¨εi(xy) = ¨ϕi(xy) = +∞, +otherwise, +¨ei(xy) = +� +¨ei(x) · y +if ¨εi(y) = 0 +x · ¨ei(y) +if ¨εi(y) > 0, +¨fi(xy) = +� ¨fi(x) · y +if ¨ϕi(x) > 0 +x · ¨fi(y) +if ¨ϕi(x) = 0, +¨εi(xy) = ¨εi(x) + ¨εi(y), +and +¨ϕi(xy) = ¨ϕi(x) + ¨ϕi(y), +where x⊥ = ⊥y = ⊥. +Proof. By Definition 6.1(3), let ψ : M ¨⊗ M → M be the quasi-crystal homo- +morphism given by ψ(x ¨⊗ y) = xy, for x, y ∈ M. Let x, y ∈ M and i ∈ I. By +Definition 3.12(2), we get that +wt(xy) = wt +� +ψ(x ¨⊗ y) +� += wt(x ¨⊗ y) = wt(x) + wt(y). +and similarly, ¨εi(xy) = ¨εi(x ¨⊗ y) and ¨ϕi(xy) = ¨ϕi(x ¨⊗ y). By Definition 6.1(1), M +is seminormal (Definition 3.9), and by Theorem 5.1, M ¨⊗ M is also seminormal. +Since ¨εi(xy) = ¨εi(x ¨⊗y), we have that ¨ei is defined on xy if and only if ¨ei is defined +on x ¨⊗y, and since ¨ϕi(xy) = ¨ϕi(x ¨⊗y), ¨fi is defined on xy if and only if ¨fi is defined +on x ¨⊗ y. Then, as ψ(M ¨⊗ M) ⊆ M, we obtain that ¨ei(xy) = ψ +� +¨ei(x ¨⊗ y) +� +and +¨fi(xy) = ψ +� ¨fi(x ¨⊗ y) +� +, by conditions (3) and (4) of Definition 3.12. Therefore, the +result follows directly from Theorem 5.1 and Proposition 5.3. +□ +The previous result can be generalized to get the values of the quasi-crystal +structure on an element of the form x1 · · · xm based only on their values on each +xk, k = 1, . . . , m. This leads to an analogue of Corollary 5.7. +Proposition 6.3. Let M be a quasi-crystal monoid, and let x1, . . . , xm ∈ M. +Then, +wt(x1 · · · xm) = wt(x1) + · · · + wt(xm). +Also, for i ∈ I, by setting +p = max +� +1 ≤ k ≤ m +�� ¨εi(xk) > 0 +� +and +q = min +� +1 ≤ l ≤ m +�� ¨ϕi(xl) > 0 +� +, +we have that +(1) if p > q or ¨εi(xk) = +∞ for some 1 ≤ k ≤ m, then +¨ei(x1 · · · xm) = ¨fi(x1 · · · xm) = ⊥ +and +¨εi(x1 · · · xm) = ¨ϕi(x1 · · · xm) = +∞; +(2) otherwise, +¨ei(x1 · · · xm) = x1 · · · xp−1 · ¨ei(xp) · xp+1 · · · xm, +¨fi(x1 · · · xm) = x1 · · · xq−1 · ¨fi(xq) · xq+1 · · · xm, +¨εi(x1 · · · xm) = ¨εi(x1) + · · · + ¨εi(xp), + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +27 +and +¨ϕi(x1 · · · xm) = ¨ϕi(xq) + · · · + ¨ϕi(xm). +Proof. We proceed by induction on m. If m = 1, then the result is trivial, and if +m = 2, then it coincides with Lemma 6.2. Assume as induction hypothesis (IH) +that the result holds for any x1, . . . , xk ∈ M with k ≤ m. Let y1, . . . , ym, ym+1 ∈ M. +Since · is associative, we have that y1 · · · ymym+1 = (y1 · · · ym)ym+1, where +wt +� +(y1 · · · ym)ym+1 +� += wt(y1 · · · ym)+wt(ym+1) = wt(y1)+· · ·+wt(ym)+wt(ym+1), +by Lemma 6.2 and (IH). Let i ∈ I. If ¨εi(yk) = +∞, for some k ∈ {1, . . ., m + 1}, +then we have when k ≤ m that ¨εi(y1 · · · ym) = +∞, by (IH), which implies that +¨εi(y1 · · · ymym+1) = ¨εi((y1 · · · ym) ¨⊗ ym+1) = +∞, +and by conditions (1) and (6) of Definition 3.1, ¨ei(y1 · · · ym+1) = ¨fi(y1 · · · ym+1) = +⊥ and ¨ϕi(y1 · · · ym+1) = +∞. So, assume that ¨εi(yk) ∈ Z≥0, for k = 1, . . . , m + 1. +Set +p = max +� +1 ≤ k ≤ m + 1 +�� ¨εi(xk) > 0 +� +and +q = min +� +1 ≤ l ≤ m + 1 +�� ¨ϕi(xl) > 0 +� +. +Suppose that p > q. In particular, p > 1 and q < m+1 implying that the sets where +the maximum and minimum are taken are nonempty, and thus, ¨εi(yp), ¨ϕi(yq) > 0. +By (IH), ¨ϕi(y1 · · · yp−1) ≥ ¨ϕi(yq) > 0, and ¨εi(yp · · · ym+1) ≥ ¨εi(yp) > 0. +This +implies by Lemma 6.2 that +¨εi(y1 · · · ym+1) = ¨εi +� +(y1 · · · yp−1)(yp · · · ym+1) +� += +∞, +and by conditions (1) and (6) of Definition 3.1, ¨ei(y1 · · · ym+1) = ¨fi(y1 · · · ym+1) = +⊥ and ¨ϕi(y1 · · · ym+1) = +∞. +Finally, suppose that p ≤ q. As ¨ϕi(yk) = 0, for k = 1, . . . , q − 1, we have by (IH) +that ¨εi(y1 · · · yp−1) = ¨εi(y1)+· · ·+¨εi(yp−1) and ¨ϕi(y1 · · · yp−1) = 0. By Lemma 6.2, +we get that ¨ei +� +(y1 · · · yp−1)yp +� += y1 · · · yp−1 · ¨ei(yp) (note that if ¨εi(yp) = 0, then +p = 1 and ¨ei(yp) = ⊥, as M is seminormal), and by (IH), ¨εi +� +(y1 · · · yp−1)yp +� += +¨εi(y1 · · · yp−1) + ¨εi(yp) = ¨εi(y1) + · · · + ¨εi(yp). +Also, since ¨εi(yk) = 0, for k = +p + 1, . . . , m + 1, we have by (IH) that ¨εi(yp+1 · · · ym+1) = 0. Then, we obtain that +¨ei +� +(y1 · · · yp)(yp+1 · · · ym+1) +� += ¨ei(y1 · · · yp) · yp+1 · · · ym+1 += y1 · · · yp−1 · ¨ei(yp) · yp+1 · · · ym+1 +and +¨εi +� +(y1 · · · yp)(yp+1 · · · ym+1) +� += ¨εi(y1 · · · yp) + ¨εi(yp+1 · · · ym+1) += ¨εi(y1) + · · · + ¨εi(yp), +by Lemma 6.2. Analogously, we have that +¨fi +� +(y1 · · · yq−1)(yq · · · ym+1) +� += y1 · · · yq−1 · ¨fi(yq) · yq+1 · · · ym+1 +and +¨ϕi +� +(y1 · · · yq−1)(yq · · · ym+1) +� += ¨ϕi(yq) + · · · + ¨ϕi(ym+1), +by (IH) and Lemma 6.2. +□ +In the previous result we saw how the monoid binary operation · interacts with +the quasi-crystal structure. We now show that this allows us to relate some prop- +erties of elements. First, we recall that an element x of a monoid M is called a +commutative element, also known as central element, if x commutes with every ele- +ment, that is, xy = yx, for any y ∈ M. We also recall that x is called an idempotent +element if x2 = x. + +28 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Proposition 6.4. Let M be a quasi-crystal monoid and let x ∈ M. +(1) If x is a commutative element, then x is isolated. +(2) If x is an idempotent element, then x is isolated and wt(x) = 0. +Proof. (1) Suppose that x is not an isolated element of M. +Then, take i ∈ I +such that ¨ei or ¨fi is defined on w. As M is seminormal, if ¨ei is defined on x, then +¨εi(x), ¨ϕi(x) ∈ Z≥0, where ¨εi(x) > 0, and the element y = ¨e¨εi(x) +i +(x) satisfies ¨εi(y) = 0 +and ¨ϕi(y) ∈ Z>0. By Lemma 6.2, ¨εi(xy) = ¨εi(x) ∈ Z>0 and ¨εi(yx) = +∞, which +implies that xy ̸= yx, and thus, x is not commutative. Otherwise, ¨fi is defined on +x, and since M is seminormal, we have that ¨εi(x), ¨ϕi(x) ∈ Z≥0 where ¨ϕi(x) > 0. +The element z = ¨f ¨ϕi(x) +i +(x) is such that ¨εi(z) ∈ Z>0 and ¨ϕi(z) = 0. Then, by +Lemma 6.2, ¨ϕi(xz) = +∞ and ¨ϕi(zx) = ¨ϕi(x) ∈ Z>0, which implies that xz ̸= zx, +and therefore, x is not commutative. +(2) Assume that x is idempotent. By Lemma 6.2, we have that +wt(x) = wt +� +x2� += wt(x) + wt(x), +which implies that wt(x) = 0. If ¨εi(x) ̸= +∞ (or equivalently, ¨ϕi(x) ̸= +∞), for +some i ∈ I, then +¨εi(x) = ¨εi +� +x2� += ¨εi(x) + ¨εi(x) +and +¨ϕi(x) = ¨ϕi +� +x2� += ¨ϕi(x) + ¨ϕi(x), +impliying that ¨εi(x) = ¨ϕi(x) = 0. Hence, ¨εi(x), ¨ϕi(x) ∈ {0, +∞}, for any i ∈ I. As +M is seminormal, we obtain that x is isolated. +□ +The monoid identity is in particular both a commutative and an idempotent +element, but as we show in the following result, its properties may affect the whole +quasi-crystal structure. +Proposition 6.5. Let M be a quasi-crystal monoid where the monoid identity is +denoted by 1. Then, 1 is isolated and wt(1) = 0. Moreover, for each i ∈ I, either +¨εi(1) = ¨ϕi(1) = 0 or ¨εi(x) = +∞, for all x ∈ M. +Proof. Since 1 is an idempotent element, then 1 is isolated and wt(1) = 0, by +Proposition 6.4(2). As M is seminormal and 1 is isolated, we get for each i ∈ I +that either ¨εi(1) = ¨ϕi(1) = 0 or ¨εi(1) = ¨ϕi(1) = +∞. Suppose there exists x ∈ M +such that ¨εi(x) ̸= +∞, for some i ∈ I. +Since M is seminormal, we get that +¨εi +� +¨e¨εi(x) +i +(x) +� += ¨ϕi +� ¨f ¨ϕi(x) +i +(x) +� += 0. Set y = ¨e¨εi(x) +i +(x) and z = ¨f ¨ϕi(x) +i +(x). Then, +0 ≤ ¨εi(1) ≤ ¨εi(1y) = ¨εi(y) = 0 +and +0 ≤ ¨ϕi(1) ≤ ¨ϕi(1z) = ¨ϕi(z) = 0, +by Lemma 6.2. +□ +Note that in the case where for some i ∈ I we have ¨εi(x) = +∞, for all x ∈ M, +the quasi-Kashiwara operators ¨ei and ¨fi are undefined on every element in M. Such +a case has little interest to study in the context of this paper, and we say that such +a quasi-crystal monoid is degenerate. Thus, by Proposition 6.5, we get the following +characterization. +Definition 6.6. A quasi-crystal monoid M is said to be nondegenerate if ¨εi(1) = +¨ϕi(1) = 0, for all i ∈ I. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +29 +Due to the interaction between the binary operation · of a quasi-crystal monoid +M and the quasi-crystal structure of the quasi-tensor product M ¨⊗ M required +by Definition 6.1(3), we can extend the signature rule described in Subsection 5.2 +to quasi-crystal monoids. Let x, y ∈ M and i ∈ I. Since ¨εi(xy) = ¨εi(x ¨⊗ y) and +¨ϕi(xy) = ¨ϕi(x ¨⊗ y), we have that the i-signature of xy and x ¨⊗ y coincide. Hence, +by Proposition 5.9, sgni(xy) = sgni(x) sgni(y). +Also, by Proposition 6.5, either +sgni(1) = ǫ or sgni(z) = 0, for any z ∈ M. Therefore, we obtained the following +result, which can be seen as an improvement of Proposition 5.9 for nondegenerate +quasi-crystal monoids. +Proposition 6.7. Let M be a quasi-crystal monoid, and let i ∈ I. Then, +sgni(xy) = sgni(x) sgni(y), +for any x, y ∈ M. Moreover, sgni is a monoid homomorphism from M to Z0 if and +only if ¨εi(x) ∈ Z≥0, for some x ∈ M. +The signature rule for quasi-crystal monoids follows directly from the previous +result and Proposition 6.3. Consider a quasi-crystal monoid M. Let x1, . . . , xm ∈ +M and i ∈ I. Then, we can compute the i-signature of x1 · · · xm based only in the +i-signature of each xk, k = 1, . . . , m, because +sgni(x1 · · · xm) = sgni(x1) · · · sgni(xm). +If sgni(x1 · · · xm) = 0, then ¨εi(x1 · · · xm) = ¨ϕi(x1 · · · xm) = +∞ which implies that +¨ei(x1 · · · xm) = ¨fi(x1 · · · xm) = ⊥. Otherwise, sgni(x1 · · · xm) = −a+b, for some +a, b ∈ Z≥0. Then, ¨εi(x1 · · · xm) = a and ¨ϕi(x1 · · · xm) = b. The raising quasi- +Kashiwara operator ¨ei is defined on x1 · · · xm if and only if a ≥ 1, in which case +¨ei(x1 · · · xm) = x1 · · · xp−1 · ¨ei(xp) · xp+1 · · · xm, where xp originates the right-most +symbol − in sgni(x1 · · · xm). Similarly, the lowering quasi-Kashiwara operator ¨fi is +defined on x1 · · · xm if and only if b ≥ 1, in which case ¨fi(x1 · · · xm) = x1 · · · xq−1 · +¨fi(xq) · xq+1 · · · xm, where xq originates the left-most symbol + in sgni(x1 · · · xm). +We now introduce the notion of a homomorphism between quasi-crystal monoids. +Definition 6.8. Let M and M′ be quasi-crystal monoids of the same type. A +quasi-crystal monoid homomorphism ψ from M to M′, denoted by ψ : M → M′, +is a map ψ : M → M ′ that satisfies the following conditions: +(1) ψ is a quasi-crystal homomorphism; +(2) ψ is a monoid homomorphism. +If ψ is also bijective, it is called a quasi-crystal monoid isomorphism. +Note that in the previous definition we only consider maps from M to M ′. +But as we observed after Definition 3.12, when we state that ψ is a quasi-crystal +homomorphism in condition (1) above, we mean that the map ψ′ : M ⊔ {⊥} → +M ′ ⊔ {⊥}, defined by ψ′(⊥) = ⊥ and ψ′(x) = ψ(x), for each x ∈ M, is a quasi- +crystal homomorphism from M to M′. +Also, if ψ : M → M′ is a quasi-crystal monoid isomorphism, then ψ is both +a quasi-crystal isomorphism (by Corollary 3.18) and a monoid isomorphism. The +converse is immediate, because a monoid isomorphism is bijective. Hence, a map +ψ : M → M ′ is a quasi-crystal monoid isomorphism if and only if ψ is a quasi-crystal +isomorphism and a monoid isomorphism. This implies that if ψ : M → M′ is a +quasi-crystal monoid isomorphism, then ψ−1 is a quasi-crystal monoid isomorphism +between M′ and M. +6.2. The free quasi-crystal monoid. Let Q be a seminormal quasi-crystal. For +k ≥ 1, set +Q ¨⊗k = Q ¨⊗ · · · ¨⊗ Q +� +�� +� +k times +. + +30 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +By Corollary 3.18 and Theorems 5.1 and 5.6, for k, l ≥ 1 the map Q ¨⊗k ¨⊗ Q ¨⊗l → +Q ¨⊗(k+l), given by (x1 ¨⊗ · · · ¨⊗ xk) ¨⊗ (y1 ¨⊗ · · · ¨⊗ yl) �→ x1 ¨⊗ · · · ¨⊗ xk ¨⊗ y1 ¨⊗ · · · ¨⊗ yl, +for x1, . . . , xk, y1, . . . , yl ∈ Q, is a quasi-crystal isomorphism. +Let ζ be an element that does not lie in Q. Set Q ¨⊗0 to be the seminormal quasi- +crystal of the same type as Q formed by the set Q ¨⊗0 = {ζ} and maps given by +wt(ζ) = 0, ¨ei(ζ) = ¨fi(ζ) = ⊥ and ¨εi(ζ) = ¨ϕi(ζ) = 0 (i ∈ I). Note that Q ¨⊗0 ¨⊗Q ¨⊗0 is +quasi-crystal isomorphic to Q ¨⊗0, as both quasi-crystals consist of a single element +where the quasi-crystal structure maps coincide. For any x ∈ Q and i ∈ I, by +Theorem 5.1 and Proposition 5.3, we have that wt(x ¨⊗ζ) = wt(x), ¨εi(x ¨⊗ζ) = ¨εi(x) +and ¨ϕi(x ¨⊗ζ) = ¨ϕi(x). Also, by the signature rule, since sgni(ζ) = ǫ, it is immediate +that ¨ei (or ¨fi) is defined on x ¨⊗ ζ if and only if ¨ei (resp., ¨fi) is defined on x. And +if so, ¨ei(x ¨⊗ ζ) = ¨ei(x) ¨⊗ ζ (resp., ¨fi(x ¨⊗ ζ) = ¨fi(x) ¨⊗ ζ). Therefore, the map +Q ¨⊗ Q ¨⊗0 → Q, given by y ¨⊗ ζ �→ y for each y ∈ Q, is a quasi-crystal isomorphism. +Analogously, the map Q ¨⊗0 ¨⊗Q → Q, given by ζ ¨⊗y �→ y for each y ∈ Q, is a quasi- +crystal isomorphism. Since the quasi-tensor product of quasi-crystals is associative +(Theorem 5.6), we get that Q ¨⊗k ¨⊗ Q ¨⊗0 and Q ¨⊗0 ¨⊗Q ¨⊗k are isomorphic to Q ¨⊗k, for +any k ≥ 0. +The sets Q ¨⊗k and Q ¨⊗l are disjoint, whenever k ̸= l. Thus, we can extend the +maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) defined on each Q ¨⊗k to the set +M = +� +k≥0 +Q ¨⊗k +obtaining a seminormal quasi-crystal M. By the quasi-crystal isomorphisms Q ¨⊗k ¨⊗ +Q ¨⊗l → Q ¨⊗(k+l) (k, l ≥ 0) defined above, we get that M becomes a quasi-crystal +monoid with the binary operation · : M × M → M given by +(x1 ¨⊗ · · · ¨⊗ xk) · (y1 ¨⊗ · · · ¨⊗ yl) = x1 ¨⊗ · · · ¨⊗ xk ¨⊗ y1 ¨⊗ · · · ¨⊗ yl +and +(x1 ¨⊗ · · · ¨⊗ xk) · ζ = ζ · (x1 ¨⊗ · · · ¨⊗ xk) = x1 ¨⊗ · · · ¨⊗ xk, +for x1, . . . , xk, y1, . . . , yl ∈ Q. If we identify ζ with the empty word ǫ and each +element of the form x1 ¨⊗ · · · ¨⊗ xk in M with the word x1 . . . xk over the alphabet +Q, we obtain a monoid isomorphism between M and the free monoid Q∗ over Q, +and through this identification we can also define a quasi-crystal structure on Q∗. +Therefore, we have constructed a quasi-crystal monoid that leads to the following +definition. +Definition 6.9. Let Q be a seminormal quasi-crystal. +The free quasi-crystal +monoid Q¨∗ over Q is a quasi-crystal monoid of the same type as Q consisting of +the set Q∗ of all words over Q, the usual concatenation of words, and quasi-crystal +structure maps defined as follows. For i ∈ I, set +wt(ǫ) = 0, +¨ei(ǫ) = ¨fi(ǫ) = ⊥, +and +¨εi(ǫ) = ¨ϕi(ǫ) = 0, +and for u, v ∈ Q∗, set +wt(uv) = wt(u) + wt(v), +if ¨ϕi(u) > 0 and ¨εi(v) > 0, set +¨ei(uv) = ¨fi(uv) = ⊥ +and +¨εi(uv) = ¨ϕi(uv) = +∞, +otherwise, set +¨ei(uv) = +� +¨ei(u)v +if ¨ϕi(u) ≥ ¨εi(v) +u¨ei(v) +if ¨ϕi(u) < ¨εi(v), + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +31 +¨fi(uv) = +� ¨fi(u)v +if ¨ϕi(u) > ¨εi(v) +u ¨fi(v) +if ¨ϕi(u) ≤ ¨εi(v), +¨εi(uv) = max +� +¨εi(u), ¨εi(v) − +� +wt(u), α∨ +i +�� +, +and +¨ϕi(uv) = max +� +¨ϕi(u) + +� +wt(v), α∨ +i +� +, ¨ϕi(v) +� +, +where u⊥ = ⊥v = ⊥. +As we constructed the free quasi-crystal monoid Q¨∗ based on the quasi-tensor +product ¨⊗ of quasi-crystals, we described its quasi-crystal structure based on the +definition of the quasi-crystal structure of a quasi-tensor product (see Theorem 5.1 +and Definition 5.2). Notice that we explicitly gave the values of the quasi-crystal +structure maps of Q¨∗ on ζ (which we identify with the empty word ǫ), on letters +the values follow from the quasi-crystal structure maps of Q, and on a word of the +form uv they depend only on their values on u and v. Thus, the definition of the +quasi-crystal structure above is not circular. Moreover, from Proposition 6.3, we +can obtain the values of the quasi-crystal structure maps on a word based only on +their values on its letters, which implies the following result. +Proposition 6.10. Let Q be a seminormal quasi-crystal. For any w ∈ Q∗ and +i ∈ I, if ¨ei(w) ∈ Q∗ then +��¨ei(w) +�� = |w|, and if ¨fi(w) ∈ Q∗ then +�� ¨fi(w) +�� = |w|. +Therefore, |u| = |v|, whenever u and v lie in the same connected component of Q¨∗. +Proof. Let w ∈ Q∗ and i ∈ I. If ¨fi is defined on w, then w ̸= ǫ, by Definition 6.9, +and so, w = x1 . . . , xm, for some x1, . . . , xm ∈ Q and m ≥ 1. By Proposition 6.3, +there exists q ∈ {1, . . . , m} such that +¨fi(w) = x1 . . . xq−1 ¨fi(xq)xq+1 . . . xm. +Since xq ∈ Q, then ¨fi(xq) ∈ Q, by Definition 6.9, which implies that +�� ¨fi(w) +�� = |w|. +Hence, for any w ∈ Q∗ and i ∈ I, +�� ¨fi(w) +�� = |w|, whenever ¨fi(w) ∈ Q∗. Analogously, +for any w ∈ Q∗ and i ∈ I, if ¨ei is defined on w, then +��¨ei(w) +�� = |w|. +Finally, let u and v be words lying in the same connected component of Q¨∗. Then, +by Definition 4.5, there exist g1, . . . , gk ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gk(u) = v, +and by applying recursively what we have proven above, |g1 · · · gk(u)| = |u|. +□ +From the previous result, we can deduce the following properties of the connected +components of Q¨∗ (see Definition 4.5). +Proposition 6.11. Let Q be a seminormal quasi-crystal whose underlying set Q +is finite, and let Q′ ⊆ Q∗ be a connected component of Q¨∗. Then, +(1) Q′ is finite, +(2) Q′ has at least a highest-weight element, and +(3) Q′ has at least a lowest-weight element. +Proof. (1) By Proposition 6.10, we have that words in Q′ have the same length. +Since Q is finite, there are finitely many words of a given length. Hence, Q′ is finite. +(2) Since Q′ is finite, we can take a word w ∈ Q′ whose weight wt(x) is maximal +among weights of words in Q′ with respect to the partial order given in (2.1). Then, +by Proposition 3.8(1), w is of highest weight. +(3) Analogously to (2), we can take a word w′ ∈ Q′ whose weight wt(w′) is +minimal among weights of words in Q′, and by Proposition 3.8(2), we get that w′ +is of lowest weight. +□ +We observed before Definition 5.2 that our intention with the (inverse-free) quasi- +tensor product was to allow an interpretation in terms of quasi-crystals of the notion + +32 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +of i-inversion in a word, introduced in [CM17, § 5]. In the context of quasi-crystals +of type An (see Example 3.4(1)), for i ∈ {1, . . . , n − 1}, a word w ∈ A∗ +n has an +i-inversion if it admits a decomposition of the form w = w1iw2(i + 1)w3, for some +w1, w2, w3 ∈ A∗ +n, and to accomplish our intention, the quasi-Kashiwara operators +¨ei and ¨fi should be undefined on such a word. In the following example, we check +that indeed this happens. +Example 6.12. Consider the standard quasi-crystal An of type An as described +in Example 3.4(1). The following is a direct consequence of Proposition 6.3. Let +w ∈ A∗ +n. The weight of w is given by +wt(w) = |w|1e1 + |w|2e2 + · · · + |w|nen. +Let i ∈ {1, . . . , n−1}. If w has a decomposition of the form w = w1iw2(i+1)w3, for +some w1, w2, w3 ∈ A∗ +n, then ¨εi(w) = ¨ϕi(w) = +∞. Otherwise, ¨εi(w) = |w|i+1 and +¨ϕi(w) = |w|i. The raising quasi-Kashiwara operator ¨ei is defined on w if and only if +w has a decomposition of the form w = u1(i+1)u2, for some u1, u2 ∈ A∗ +n with |u1|i = +|u2|i+1 = 0, and if so, ¨ei(w) = u1iu2. The lowering quasi-Kashiwara operator ¨fi is +defined on w if and only if w has a decomposition of the form w = v1iv2, for some +v1, v2 ∈ A∗ +n with |v1|i = |v2|i+1 = 0, and if so, ¨fi(w) = v1(i + 1)v2. Informally, +provided that w does not have a decomposition of the form w = w1iw2(i + 1)w3, +we have that ¨ei(w) is obtained by replacing the right-most symbol i + 1 in w by i, +and ¨fi(w) is obtained by replacing the left-most symbol i in w by i + 1. +From Examples 4.4(3) and 5.4(1), We have that the words of length 2 form the +following subgraph of the quasi-crystal graph of A¨∗ +3. +11 +21 +31 +12 +22 +32 +13 +23 +33 +1 +2 +1 +1 +1 +2 +2 +2 +2 +2 +The term free, used in Definition 6.9 to characterize the quasi-crystal monoid Q¨∗ +over a seminormal quasi-crystal Q, is justified by the following universal property. +Theorem 6.13. Let Q be a seminormal quasi-crystal and M be a nondegenerate +quasi-crystal monoid of the same type. Then, for each quasi-crystal homomorphism +ψ : Q → M satisfying ψ(Q) ⊆ M, there exists a unique quasi-crystal monoid +homomorphism ˆψ : Q¨∗ → M such that ˆψ(x) = ψ(x), for all x ∈ Q. +Proof. Let ψ : Q → M be a quasi-crystal homomorphism such that ψ(Q) ⊆ M. +So, we can consider ψ as a map from Q to M. It is well-known that there exists a +unique monoid homomorphism ˆψ : Q∗ → M such that ˆψ(x) = ψ(x), for all x ∈ Q. +We also have that ˆψ is given by +ˆψ(ǫ) = 1 +and +ˆψ(x1 . . . xm) = ψ(x1) · · · ψ(xm), +for x1, . . . , xm ∈ Q. It remains to show that ˆψ is a quasi-crystal homomorphism. +Let i ∈ I. By Proposition 6.5 and Definitions 6.6 and 6.9, we have that wt(ǫ) = +0 = wt(1), ¨ei(ǫ) = ¨fi(ǫ) = ⊥ = ¨ei(1) = ¨fi(1), and ¨εi(ǫ) = ¨ϕi(ǫ) = 0 = ¨εi(1) = ¨ϕi(1). +Let x1, . . . , xm ∈ Q, m ≥ 1, and set w = x1 . . . xm. +By Definition 3.12(2), we +have that wt +� +ψ(xk) +� += wt(xk), ¨εi +� +ψ(xk) +� += ¨εi(xk) and ¨ϕi +� +ψ(xk) +� += ¨ϕi(xk) for +k = 1, . . . , m. By Proposition 6.3, we obtain that +wt +� ˆψ(w) +� += wt +� +ψ(x1) +� ++ · · · + wt +� +ψ(xm) +� += wt(x1) + · · · + wt(xm) = wt(w), +and since +p = max +� +1 ≤ k ≤ m +�� ¨εi(xk) > 0 +� += max +� +1 ≤ k ≤ m +�� ¨εi +� +ψ(xk) +� +> 0 +� + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +33 +and +q = min +� +1 ≤ l ≤ m +�� ¨ϕi(xl) > 0 +� += min +� +1 ≤ l ≤ m +�� ¨ϕi +� +ψ(xl) +� +> 0 +� +, +we also get that ¨εi +� ˆψ(w) +� += ¨εi(w) and ¨ϕi +� ˆψ(w) +� += ¨ϕi(w). If ¨ei(w) ∈ Q∗, then +¨ei(xp) ∈ Q∗, more precisely ¨ei(xp) ∈ Q as xp ∈ Q, and since ψ(Q) ⊆ M we have +by Definition 3.12(3) that ψ +� +¨ei(xp) +� += ¨ei +� +ψ(xp) +� +, which implies that +ˆψ +� +¨ei(w) +� += ψ(x1) · · · ψ(xp−1) · ψ +� +¨ei(xp) +� +· ψ(xp+1) · · · ψ(xm) += ψ(x1) · · · ψ(xp−1) · ¨ei +� +ψ(xp) +� +· ψ(xp+1) · · · ψ(xm) = ¨ei +� ˆψ(w) +� +. +Analogous reasoning applies if ¨fi(w) ∈ Q∗, leading to ˆψ +� ¨fi(w) +� += ¨fi +� ˆψ(w) +� +. +□ +The property described in the previous result can be used to define the free quasi- +crystal monoid up to isomorphism among nondegenerate quasi-crystal monoids. +Corollary 6.14. Let Q be a seminormal quasi-crystal, M be a quasi-crystal monoid +of the same type, and ι : Q → M be an injective quasi-crystal homomorphism +such that for each nondegenerate quasi-crystal monoid M′ and each quasi-crystal +homomorphism ψ : Q → M′ satisfying ψ(Q) ⊆ M ′, there exists a unique quasi- +crystal monoid homomorphism ˆψ : M → M′ for which ψ = ˆψι. Then, there exists +a quasi-crystal monoid isomorphism between M and Q¨∗. +Proof. Define ψ : Q → Q∗ by ψ(x) = x, for each x ∈ Q. As the quasi-crystal +structure maps of Q¨∗ agree on words of length 1 with the quasi-crystal structure +maps of Q, we have that ψ is a quasi-crystal homomorphism from Q to Q¨∗. Thus, +there exists a unique quasi-crystal monoid homomorphism ˆψ : M → Q¨∗ such that +ˆψι(x) = x, for all x ∈ Q. Then, M is nondegenerate, because ¨εi(1) = ¨εi +� ˆψ(1) +� += +¨εi(ǫ) = 0 and ¨ϕi(1) = ¨ϕi +� ˆψ(1) +� += ¨ϕi(ǫ) = 0, for any i ∈ I. By Theorem 6.13, +there exists a unique quasi-crystal monoid homomorphism ˆι : Q¨∗ → M such that +ˆι(x) = ι(x), for all x ∈ Q. Then, ˆψˆι is a quasi-crystal monoid homomorphism +from Q¨∗ to Q¨∗ such that ˆψˆι(x) = x, for any x ∈ Q, and by Theorem 6.13, we +obtain that ˆψˆι must be the identity map on Q∗. Also, ˆι ˆψ is a quasi-crystal monoid +homomorphism from M to M such that ˆι ˆψι(x) = ι(x), for any x ∈ Q, and by +uniqueness, we get that ˆι ˆψ must be the identity map on M. Hence, ˆψ is a quasi- +crystal monoid isomorphism between M and Q¨∗. +□ +6.3. Congruences and quotients. We now study the notion of congruence on a +quasi-crystal monoid which leads to the definition of quotient quasi-crystal monoid +and to the proof of homomorphism theorems for quasi-crystal monoids. +Definition 6.15. Let M be a quasi-crystal monoid. A quasi-crystal monoid con- +gruence on M is an equivalence relation θ ⊆ M × M satisfying the conditions: +(1) if (x, y) ∈ θ, then wt(x) = wt(y), ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y) for all +i ∈ I; +(2) if (x, y) ∈ θ and ¨ei(x) ∈ M, then +� +¨ei(x), ¨ei(y) +� +∈ θ; +(3) if (x, y) ∈ θ and ¨fi(x) ∈ M, then +� ¨fi(x), ¨fi(y) +� +∈ θ; +(4) if (x1, y1), (x2, y2) ∈ θ, then (x1x2, y1y2) ∈ θ. +Let M be a quasi-crystal monoid. It is immediate from the definition that the +equality relation ∆ = +� +(x, x) +�� x ∈ M +� +is a quasi-crystal monoid congruence on M. +We have that ∆ ⊆ θ, for any quasi-crystal monoid congruence θ on M. Also, given a +nonempty family Θ of quasi-crystal monoid congruences on M, it is straightforward +to show that � Θ is a quasi-crystal monoid congruence on M. From this and the +following result, we are able to show that the quasi-crystal monoid congruences on +a quasi-crystal monoid form a lattice. + +34 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Lemma 6.16. Let M be a quasi-crystal monoid, and let R ⊆ M ×M be a relation +on M such that for any (x, y) ∈ R and i ∈ I, the following conditions are satisfied: +(1) wt(x) = wt(y), ¨εi(x) = ¨εi(y), and ¨ϕi(x) = ¨ϕi(y); +(2) if ¨ei(x) ∈ M, then +� +¨ei(x), ¨ei(y) +� +∈ R; and +(3) if ¨fi(x) ∈ M, then +� ¨fi(x), ¨fi(y) +� +∈ R. +Then, the monoid congruence θR generated by R is a quasi-crystal monoid congru- +ence on M. +Proof. We check that in every step of constructing θR from R, properties (1) to (3) +are preserved. Set +R1 = +� +(ux, uy) +�� (x, y) ∈ R, u ∈ M +� +. +For (x, y) ∈ R, u ∈ M and i ∈ I, since wt(x) = wt(y), ¨εi(x) = ¨εi(y) and ¨ϕi(x) = +¨ϕi(y), we get that wt(ux) = wt(uy), ¨εi(ux) = ¨εi(uy) and ¨ϕi(ux) = ¨ϕi(uy), by +Lemma 6.2. As M is seminormal, we have that ¨ei(ux) ∈ M if and only if ¨ei(uy) ∈ +M. And if so, we obtain when ¨εi(x) = 0 that +� +¨ei(ux), ¨ei(uy) +� += +� +¨ei(u) · x, ¨ei(u) · +y +� +∈ R1, and when ¨εi(x) > 0 that +� +¨ei(ux), ¨ei(uy) +� += +� +u · ¨ei(x), u · ¨ei(y) +� +∈ R1, +because +� +¨ei(x), ¨ei(y) +� +∈ R, by (2). Similarly, if ¨fi(ux) ∈ M, we get when ¨ϕi(u) > +0 that +� ¨fi(ux), ¨fi(uy) +� += +� ¨fi(u) · x, ¨fi(u) · y +� +∈ R1, and when ¨ϕi(u) = 0 that +� ¨fi(ux), ¨fi(uy) +� += +� +u · ¨fi(x), u · ¨fi(y) +� +∈ R1, because +� ¨fi(x), ¨fi(y) +� +∈ R, by (3). +Hence, R1 satisfies conditions (1) to (3). +We can analogously deduce that R2 = +� +(xv, yv) +�� (x, y) ∈ R1, v ∈ M +� +satisfies +conditions (1) to (3). +It is immediate that the reflexive closure R3 = R2 ∪ +� +(x, x) +�� x ∈ M +� +of R2 +satisfies conditions (1) to (3). +Set R4 = R3 ∪ +� +(x, y) +�� (y, x) ∈ R3 +� +, which correspondes to the symmetric +closure of R3. Since R3 satisfies condition (1), so it does R4. For (x, y) ∈ R3, if +¨ei(y) ∈ M, then ¨ei(x) ∈ M, because ¨εi(x) = ¨εi(y) and M is seminormal, and so, +� +¨ei(x), ¨ei(y) +� +∈ R3, by (2). Similarly, if ¨fi(y) ∈ M, then +� ¨fi(x), ¨fi(y) +� +∈ R3, by (3). +Hence, R4 also satisfies conditions (2) and (3). +Finally, θR corresponds to the transitive closure of R4. For (x, y) ∈ θR, there +exist x0, x1, . . . , xm ∈ M such that x = x0, y = xm and (xk−1, xk) ∈ R4, for +k = 1, . . . , m. For i ∈ I, since R4 satisfies condition (1), we get that +wt(x) = wt(x0) = wt(x1) = · · · = wt(xm) = wt(y), +and similarly, ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y). As R4 satisfies condition (2), if +¨ei(x) ∈ M, we get that +� +¨ei(x), ¨ei(x1) +� +∈ R4, and recursively, +� +¨ei(xk−1), ¨ei(xk) +� +∈ +R4, for k = 1, . . . , m, which implies that +� +¨ei(x), ¨ei(y) +� +∈ θR. Analogously, as R4 +satisfies condition (3), if ¨fi(x) ∈ M, then +� ¨fi(x), ¨fi(y) +� +∈ θR. Therefore, θR satisfies +conditions (1) to (3). Moreover, θR satisfies Definition 6.15(4), as by construction +θR is a monoid congruence on M, and thus, θR is a quasi-crystal monoid congruence +on M. +□ +Theorem 6.17. Let M be a quasi-crystal monoid. Then, the quasi-crystal monoid +congruences on M form a lattice with respect to the partial order ⊆ of inclusion. +Proof. Let Θ be the set of all quasi-crystal monoid congruences on M. Since the +equality relation ∆ lies in Θ, we have that Θ is nonempty. Clearly, ⊆ is a partial +order on Θ. Let θ, σ ∈ Θ. Since θ ∩ σ is a quasi-crystal monoid congruence and the +largest set contained in θ and σ, we have that θ ∩ σ is the infimum of θ and σ in +Θ. Finally, the relation R = θ ∪ σ satisfies the conditions of Lemma 6.16, and so, +the monoid congruence θR generated by R is a quasi-crystal monoid congruence on +M. Also, θR is the smallest equivalence relation on M satisfying Definition 6.15(4) +and containing R. Hence, θR is the supremum of θ and σ in Θ. +□ + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +35 +Let θ be a quasi-crystal monoid congruence on a quasi-crystal monoid M. From +Definition 6.15, it follows that the quasi-crystal structure maps wt, ¨ei, ¨fi, ¨εi and +¨ϕi (i ∈ I), and the monoid binary operation · of M give rise in a natural way to +a quasi-crystal monoid whose underlying set is the set of all θ-equivalence classes +M/θ. +For each x ∈ M, denote the θ-equivalence class of x by [x]θ, or simply, +[x]. If x, y ∈ M are such that [x] = [y], then wt(x) = wt(y), ¨εi(x) = ¨εi(y) and +¨ϕi(x) = ¨ϕi(y), by Definition 6.15(1). As M is seminormal and ¨εi(x) = ¨εi(y), we +have that ¨ei is defined on x if and only if ¨ei is defined on y. And if so, we have +by Definition 6.15(2) that [¨ei(x)] = [¨ei(y)]. Similarly, by Definition 6.15(3), if ¨fi is +defined on x or y, then [ ¨fi(x)] = [ ¨fi(y)]. Finally, if x1, x2, y1, y2 ∈ M are such that +[x1] = [y1] and [x2] = [y2], then [x1x2] = [y1y2], by Definition 6.15(4). Therefore, +we obtain the following construction. +Definition 6.18. Let θ be a congruence on a quasi-crystal monoid M. The quotient +quasi-crystal monoid of M by θ is a quasi-crystal monoid M/θ of the same type +as M consisting of the set M/θ and maps given by +wt([x]) = wt(x) +¨ei([x]) = [¨ei(x)] +¨fi([x]) = [ ¨fi(x)] +¨εi([x]) = ¨εi(x) +¨ϕi([x]) = ¨ϕi(x) +[x] · [y] = [x · y], +where [⊥] = ⊥, for x, y ∈ M and i ∈ I. +The following result follows directly from the previous definition. +Lemma 6.19. Let θ be a congruence on a quasi-crystal monoid M. Then, the map +π : M → M/θ, given by π(x) = [x] for each x ∈ M, is a surjective quasi-crystal +monoid homomorphism from M to M/θ. +This leads to the following result that relates congruences and homomorphisms +on quasi-crystal monoids. +Theorem 6.20. Let M be a quasi-crystal monoid, and let θ ⊆ M × M. Then, θ +is a congruence on M if and only if there exist a quasi-crystal monoid M′ and a +quasi-crystal monoid homomorphism ψ : M → M′ such that θ = ker ψ. +Proof. Let θ be a quasi-crystal monoid congruence on M. Set π : M → M/θ +to be the quasi-crystal monoid homomorphism defined in Lemma 6.19. Then, for +x, y ∈ M, we have that π(x) = π(y) if and only if (x, y) ∈ θ, which implies that +θ = ker π. +Conversely, let ψ : M → M′ be a quasi-crystal monoid homomorphism. It is +immediate that ker ψ is an equivalence relation on M. Let (x, y) ∈ ker ψ and i ∈ I. +We have that +wt(x) = wt +� +ψ(x) +� += wt +� +ψ(y) +� += wt(y), +and similarly, ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y). If ¨ei(x) ∈ M, then ¨ei(y) ∈ M, +because M is seminormal and ¨εi(x) = ¨εi(y). Also, since ψ(M) ⊆ M ′, we get that +ψ +� +¨ei(x) +� +, ψ +� +¨ei(y) +� +∈ M ′, and so, +ψ +� +¨ei(x) +� += ¨ei +� +ψ(x) +� += ¨ei +� +ψ(y) +� += ψ +� +¨ei(y) +� +, +which implies that +� +¨ei(x), ¨ei(y) +� +∈ ker ψ. Analogously, if ¨fi(x) ∈ M, then we obtain +that +� ¨fi(x), ¨fi(y) +� +∈ ker ψ. Finally, for (x1, y1), (x2, y2) ∈ ker ψ, we have that +ψ(x1x2) = ψ(x1)ψ(x2) = ψ(y1)ψ(y2) = ψ(y1y2), +which implies that (x1x2, y1y2) ∈ ker ψ. Hence, ker ψ is a quasi-crystal monoid +congruence on M. +□ +Now, we introduce the homomorphism theorems for quasi-crystal monoids. + +36 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Theorem 6.21. Let M and M′ be quasi-crystal monoids of the same type, and +let ψ : M → M′ be a quasi-crystal monoid homomorphism. Then, for each quasi- +crystal monoid congruence θ on M satisfying θ ⊆ ker ψ, there exists a unique +quasi-crystal monoid homomorphism ˆψ : M/θ → M′ such that ˆψ([x]θ) = ψ(x) for +any x ∈ M. +Furthermore, if ψ is surjective, then there exists a unique quasi-crystal monoid +isomorphism ˆψ : M/ ker ψ → M′ such that ˆψ([x]) = ψ(x), for all x ∈ M. +Proof. Let θ be a quasi-crystal monoid congruence on M such that θ ⊆ ker ψ. For +x, y ∈ M, if [x] = [y], then ψ(x) = ψ(y), because θ ⊆ ker ψ. Thus, we can define a +map ˆψ : M/θ → M ′ by ˆψ([x]) = ψ(x), for each x ∈ M. Since ψ is a quasi-crystal +monoid homomorphism from M to M′, it is immediate from Definition 6.18 that +ˆψ is a quasi-crystal monoid homomorphism from M/θ to M′. +Assume that ψ is surjective and take θ = ker ψ. Then, given x′ ∈ M ′, there +exists x ∈ M such that ψ(x) = x′, and so, ˆψ([x]) = x. +Also, if y, z ∈ M are +such that ˆψ([y]) = ˆψ([z]), then ψ(y) = ψ(z), or equivalently, (y, z) ∈ ker ψ, which +implies that [y] = [z]. Hence, ˆψ is bijective, and therefore, a quasi-crystal monoid +isomorphism. +□ +Theorem 6.22. Let θ and σ be congruences on a quasi-crystal monoid M such +that θ ⊆ σ. Define a relation σ/θ on M/θ by +σ/θ = +� +([x]θ, [y]θ) +�� (x, y) ∈ σ +� +. +Then, σ/θ is a quasi-crystal monoid congruence on M/θ. +Moreover, the map +(M/θ)/(σ/θ) → M/σ, given by [[x]θ]σ/θ �→ [x]σ for each x ∈ M, is a quasi-crystal +monoid isomorphism between (M/θ)/(σ/θ) and M/σ. +Proof. Define a map π : M → M/σ by π(x) = [x]σ, for each x ∈ M. +By +Lemma 6.19, π is a quasi-crystal monoid homomorphism from M to M/σ. Since +θ ⊆ σ = ker π, by Theorem 6.21, we have a quasi-crystal monoid homomorphism +ˆπ : M/θ → M/σ given by ˆπ([x]θ) = [x]σ, for each x ∈ M. As π is surjective, +then ˆπ is also surjective. +For x, y ∈ M, we have that ˆπ([x]θ) = ˆπ([y]θ) if and +only if (x, y) ∈ σ. Hence, ker ˆπ = θ/σ. By Theorem 6.21, we obtain that the map +(M/θ)/(σ/θ) → M/σ, given by [[x]θ]σ/θ �→ [x]σ for each x ∈ M, is a quasi-crystal +monoid isomorphism between (M/θ)/(σ/θ) and M/σ. +□ +7. The hypoplactic congruence +This section is devoted to study the hypoplactic congruence on a free quasi- +crystal monoid. +We start by proving that it results in a quasi-crystal monoid +congruence. Based on this, we give the definition of hypoplactic monoid associated +to a seminormal quasi-crystal. We then characterize the commutative elements of +such a monoid. +Definition 7.1. Let Q be a seminormal quasi-crystal. The hypoplactic congruence +on Q¨∗ is a relation ¨∼ on Q∗ given as follows. For u, v ∈ Q∗, u ¨∼ v if and only if +there exists a quasi-crystal isomorphism ψ : Q¨∗(u) → Q¨∗(v) such that ψ(u) = v. +To prove that ¨∼ is a quasi-crystal monoid congruence on Q¨∗ we first show the +following result. +Lemma 7.2. Let Q be a seminormal quasi-crystal. For each u, v ∈ Q∗, the map +� +Q∗(u) ¨⊗ Q∗(v) +� +(u ¨⊗ v) → Q∗(uv), given by x ¨⊗ y �→ xy, is a quasi-crystal isomor- +phism between +� +Q¨∗(u) ¨⊗ Q¨∗(v) +� +(u ¨⊗ v) and Q¨∗(uv). + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +37 +Proof. Let u, v ∈ Q∗. Since Q¨∗ is a quasi-crystal monoid, the map Q∗ ¨⊗ Q∗ → Q∗, +defined by x ¨⊗ y �→ xy for each x, y ∈ Q∗, is a quasi-crystal homomorphism, +by Definition 6.1(3). Then, by Proposition 4.14 we have a surjective quasi-crystal +homomorphism ψ : +� +Q¨∗ ¨⊗ Q¨∗� +(u ¨⊗ v) → Q¨∗(uv) given by ψ(x ¨⊗ y) = xy, for each +x ¨⊗ y ∈ (Q∗ ¨⊗ Q∗)(u ¨⊗ v). +We now show that +� +Q¨∗ ¨⊗Q¨∗� +(u ¨⊗v) and +� +Q¨∗(u) ¨⊗Q¨∗(v) +� +(u ¨⊗v) correspond to the +same quasi-crystal. Since they are formed by connected components of Q¨∗ ¨⊗Q¨∗ and +Q¨∗, it suffices to prove that their underlying sets coincide. As Q∗(u), Q∗(v) ⊆ Q∗, +we get that +� +Q∗(u) ¨⊗Q∗(v) +� +(u ¨⊗v) ⊆ (Q∗ ¨⊗Q∗)(u ¨⊗v). Let x, y ∈ Q∗ be such that +x ¨⊗ y ∈ (Q∗ ¨⊗ Q∗)(u ¨⊗ v). By Definition 4.5, there exist g1, . . . , gm ∈ {¨ei, ¨fi | i ∈ I} +such that x ¨⊗ y = g1 · · · gm(u ¨⊗ v), and by Definition 5.2, x = g′ +1 · · · g′ +k(u) and +y = g′′ +1 · · · g′′ +l (v) for some g′ +1, . . . , g′ +k, g′′ +1, . . . , g′′ +l ∈ {g1, . . . , gm}. Hence, x ∈ Q∗(u) +and y ∈ Q∗(v), and so, (Q∗ ¨⊗ Q∗)(u ¨⊗ v) = +� +Q∗(u) ¨⊗ Q∗(v) +� +(u ¨⊗ v) Therefore, ψ is +a surjective quasi-crystal homomorphism from +� +Q¨∗(u) ¨⊗ Q¨∗(v) +� +(u ¨⊗ v) to Q¨∗(uv). +Finally, we show that ψ is injective. Let x1 ¨⊗y1, x2 ¨⊗y2 ∈ +� +Q∗(u) ¨⊗Q∗(v) +� +(u ¨⊗v). +We have by Proposition 6.10 that |x1| = |x2| and |y1| = |y2|. Thus, if x1y1 = x2y2, +then x1 = x2 and y1 = y2 implying x1 ¨⊗ y1 = x2 ¨⊗ y2. Hence, ψ is injective. By +Corollary 3.18, ψ is a quasi-crystal isomorphism between +� +Q¨∗(u) ¨⊗ Q¨∗(v) +� +(u ¨⊗ v) +and Q¨∗(uv). +□ +Theorem 7.3. Let Q be a seminormal quasi-crystal. Then, the hypoplactic con- +gruence ¨∼ on Q¨∗ is a quasi-crystal monoid congruence on Q¨∗. +Proof. It is straightforward to see that ¨∼ is an equivalence relation. +Let u, v ∈ Q∗ with u ¨∼ v. +Then there exists a quasi-crystal isomorphism +ψ : Q¨∗(u) → Q¨∗(v) such that ψ(u) = v. Let i ∈ I. Since ψ is a quasi-crystal +isomorphism, we get that +wt(u) = wt +� +ψ(u) +� += wt(v), +and similarly, ¨εi(u) = ¨εi(v) and ¨ϕi(u) = ¨ϕi(v). If ¨ei(u) ∈ Q∗, then +ψ +� +¨ei(u) +� += ¨ei +� +ψ(u) +� += ¨ei(v), +and since Q∗� +¨ei(u) +� += Q∗(u) and Q∗� +¨ei(v) +� += Q∗(v), we obtain ¨ei(u) ¨∼ ¨ei(v). +Analogously, if ¨fi(u) ∈ Q∗, then ¨fi(u) ¨∼ ¨fi(v). +Additionally, let u′, v′ ∈ Q∗ with u′ ¨∼ v′. Then we also have a quasi-crystal +isomorphism ψ′ : Q¨∗(u′) → Q¨∗(v′) such that ψ(u′) = v′. +By Lemma 7.2, we +have quasi-crystal isomorphisms ψ1 : Q¨∗(uv) → +� +Q¨∗(u) ¨⊗ Q¨∗(v) +� +(u ¨⊗ v) and ψ2 : +� +Q¨∗(u′) ¨⊗ Q¨∗(v′) +� +(u′ ¨⊗ v′) → Q¨∗(u′v′) such that ψ1(uv) = u ¨⊗ v and ψ2(u′ ¨⊗ +v′) = u′v′. By Theorem 5.8, we have a quasi-crystal isomorphism ψ ¨⊗ ψ′ between +Q¨∗(u) ¨⊗Q¨∗(v) and Q¨∗(u′) ¨⊗Q¨∗(v′) satisfying (ψ ¨⊗ψ′)(u ¨⊗v) = u′ ¨⊗v′. Set ψ3 to be +the restriction of ψ ¨⊗ψ′ to +� +Q∗(u) ¨⊗Q∗(v) +� +(u ¨⊗v). By Proposition 4.9, ψ3 is a quasi- +crystal isomorphism between +� +Q¨∗(u) ¨⊗Q¨∗(v) +� +(u ¨⊗v) and +� +Q¨∗(u′) ¨⊗Q¨∗(v′) +� +(u′ ¨⊗v′). +Then, ψ2ψ3ψ1 is a quasi-crystal isomorphism between Q¨∗(uv) and Q¨∗(u′v′) that +satisfies ψ2ψ3ψ1(uv) = u′v′. Hence, uv ¨∼ u′v′. Therefore, ¨∼ is a quasi-crystal +monoid congruence on Q¨∗. +□ +We have now set up the framework to present the following definition. +Definition 7.4. Let Q be a seminormal quasi-crystal, and let ¨∼ be the hypoplactic +congruence on Q¨∗. The quotient quasi-crystal monoid Q¨∗/ ¨∼ is called the hypoplac- +tic quasi-crystal monoid, or simply the hypoplactic monoid, associated to Q, and is +denoted by hypo(Q). + +38 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Although hypo(Q) is a quasi-crystal monoid, we are interested in studying its +properties as a monoid, and thus, we just refer it as the hypoplactic monoid asso- +ciated to Q. However, we will be constantly considering its quasi-crystal structure, +as it plays a fundamental role in the construction of hypo(Q), and consequently, in +its properties. +This terminology will make more sense in the following section, where we see +how the classical hypoplactic monoid can be placed in context as the hypoplactic +monoid associated to the standard quasi-crystal of type An. +In a hypoplactic monoid the converse of Proposition 6.4(1) also holds, because +the isolated elements (Definition 4.15) are commutative, which is a consequence of +the following result. +Theorem 7.5. Let Q be a seminormal quasi-crystal, and let u, v ∈ Q∗ be such that +uv is an isolated element of Q¨∗. Then, +uvw ¨∼ uwv ¨∼ wuv, +for any w ∈ Q∗. +Proof. By Proposition 6.3, we have that +wt(uvw) = wt(u) + wt(v) + wt(w) = wt(uwv), +for any w ∈ Q∗. Since uv is isolated, we have that ¨ei(uv) = ¨fi(uv) = ⊥, for all +i ∈ I. Then, for each i ∈ I, either ¨εi(uv) = ¨ϕi(uv) = 0 or ¨εi(uv) = ¨ϕi(uv) = +∞, +because Q¨∗ is seminormal. Set J = +� +i ∈ I +�� ¨εi(uv) = 0 +� +. By Lemma 6.2, for +any i ∈ I \ J, since ¨εi(uv) = +∞, we have that ¨εi(u) = +∞, ¨εi(v) = +∞, or +¨ϕi(u), ¨εi(v) ∈ Z>0. This implies by Proposition 6.3 that, for any w ∈ Q∗, +¨εi(uvw) = ¨ϕi(uvw) = ¨εi(uwv) = ¨ϕi(uwv) = +∞, +and so, ¨ei and ¨fi are undefined on uvw and on uwv. By Lemma 6.2, for any j ∈ J, +we have that ¨εj(u) = ¨εj(v) = ¨ϕj(u) = ¨ϕj(v) = 0, because ¨εj(uv) = ¨ϕj(uv) = 0. +Then, by Proposition 6.3, for any w ∈ Q∗, we get that +¨ej(uvw) = uv¨ej(w), +¨fj(uvw) = uv ¨fj(w), +¨εj(uvw) = ¨εj(w), +¨ϕj(uvw) = ¨ϕj(w), +¨ej(uwv) = u ¨fj(w)v, +¨fj(uwv) = u ¨fj(w)v, +¨εj(uwv) = ¨εj(w), +¨ϕj(uwv) = ¨ϕj(w). +In particular, given w ∈ Q∗ and g1, . . . , gm ∈ {¨ei, ¨fi | i ∈ I}, we have that g1 · · · gm +is defined on uvw if and only if g1 · · · gm is defined on uwv if and only if g1, . . . , gm ∈ +{¨ej, ¨fj | j ∈ J} and g1 · · · gm is defined on w. +Let w ∈ Q∗. Define +X = +� +w′ ∈ Q∗ �� w′ = g1 · · · gm(w), for some g1, . . . , gm ∈ {¨ej, ¨fj | j ∈ J} +� +. +Then, +Q∗(uvw) = {uvw′ | w′ ∈ X} +and +Q∗(uwv) = {uw′v | w′ ∈ X}. +Also, the map ψ : Q∗(uvw) → Q∗(uwv) given by ψ(uvw′) = uw′v, for each +w′ ∈ X, is a bijective quasi-crystal homomorphism from Q¨∗(uvw) to Q¨∗(uwv). By +Corollary 3.18, ψ is a quasi-crystal isomorphism between Q¨∗(uvw) and Q¨∗(uwv). +As ψ(uvw) = uwv, we obtain that uvw ¨∼ uwv. +The fact that uwv ¨∼ wuv follows analogously. +□ +We now show that the converse of Proposition 6.4(2) holds for hypoplactic +monoids. +This leads to a characterization of the idempotent elements of a hy- +poplactic monoid. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +39 +Theorem 7.6. Let Q be a seminormal quasi-crystal, and let w ∈ Q∗. +Then, +w ¨∼ w2 if and only if w is an isolated element of Q¨∗ and wt(w) = 0. +Proof. The direct implication follows from Proposition 6.4(2). So, assume that w +is an isolated element of Q¨∗ and wt(w) = 0. We get that +wt +� +w2� += wt(w) + wt(w) = 0. +Let i ∈ I. As Q¨∗ is seminormal and w is isolated, we have that either ¨εi(w) = +¨ϕi(w) = 0 or ¨εi(w) = ¨ϕi(w) = +∞. If ¨εi(w) = ¨ϕi(w) = 0, then +¨εi +� +w2� += ¨εi(w) + ¨εi(w) = 0 +and +¨ϕi +� +w2� += ¨ϕi(w) + ¨ϕi(w) = 0, +implying that ¨ei and ¨fi are undefined on w2. Otherwise, ¨εi(w) = ¨ϕi(w) = +∞, +we get by Lemma 6.2 that ¨εi +� +w2� += ¨ϕi +� +w2� += +∞, which implies that ¨ei and +¨fi are undefined on w2, by Definition 3.1(6). Hence, w2 is isolated and the map +ψ : q∗(w) → Q∗� +w2� +, defined by ψ(w) = w2, is a quasi-crystal isomorphism between +Q¨∗(w) and Q¨∗� +w2� +. Therefore, w ¨∼ w2. +□ +The following result is a direct consequence of Theorems 7.5 and 7.6. +Corollary 7.7. Let Q be a seminormal quasi-crystal. In the hypoplactic monoid +hypo(Q), the idempotent elements commute. +8. Crystallizing the classical hypoplactic monoid +In this section we prove that the classical hypoplactic monoid hypon of rank n +arises as the hypoplactic monoid hypo(An) associated to the standard quasi-crystal +An of type An. This is accomplished by showing that the direct approach in [CM17] +can be placed in the context developped in the previous sections. +Recall that Kashiwara crystals [Kas90, Kas91] give rise to a plactic monoid +anti-isomorphic to the original one [LS81]. Since we introduced the quasi-tensor +product of quasi-crystals (Section 5) based on the tensor product of crystals defined +by Kashiwara, it is natural to expect the hypoplactic monoid obtained from quasi- +crystals to be anti-isomorphic to the original one [KT97]. Therefore, the results +in this section concerning the classical hypoplactic monoid are adaptations of the +original results. +Definition 8.1. Let n ≥ 1. The classical hypoplactic monoid hypon of rank n is +given by the presentation ⟨An | R1 ∪ R2 ∪ R3 ∪ R4⟩ where +R1 = +� +(yzx, yxz), (xzy, zxy) +�� x < y < z +� +, +R2 = +� +(xyx, xxy), (xyy, yxy) +�� x < y +� +, +R3 = +� +(xzty, zxyt) +�� t ≤ x < y ≤ z +� +, +and +R4 = +� +(ytzx, tyxz) +�� t < x ≤ y < z +� +. +The Knuth relations consist of R1 ∪ R2, and the quartic relations consist of R3 ∪ +R4. The classical hypoplactic congruence ∼hypon is the monoid congruence on A∗ +n +generated by R1 ∪ R2 ∪ R3 ∪ R4. +The Knuth and quartic relations given above are respectively the reverse of the +ones given in [Knu70] and [KT97]. This is part of the adaptations we pointed out +in the beginning of this section. +In the rest of this section, fix the root system associated to Cartan type An. +The maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi, i = 1, . . . , n − 1, always refer to the quasi-crystal +structure of A¨∗ +n, and ¨∼ always denote the hypoplactic congruence on A¨∗ +n. + +40 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +As we saw in Example 6.12, for w ∈ A∗ +n and i ∈ {1, . . . , n − 1}, ¨ei is defined on +w if and only if w does not have an i-inversion and |w|i+1 > 0, and if so, ¨ei(w) is +obtained from w by replacing the right-most symbol i + 1 by i. Also, ¨fi is defined +on w if and only if w does not have an i-inversion and |w|i > 0, and if so, ¨fi(w) is +obtained from w by replacing the left-most symbol i by i+1. Therefore, we can use +the quasi-crystal structure of A¨∗ +n to construct a graph similar to the one in [CM17, +§ 5]. +Definition 8.2. Let Γ′ +n denote the Λ-weighted {1, . . . , n − 1}-labelled directed +graph consisting of the vertex set A∗ +n, the weight map of A¨∗ +n, and for each u, v ∈ A∗ +n +and i ∈ {1, . . . , n − 1}, an edge u +i +−−−→ v whenever ¨fi(u) = v. +Note that Γ′ +n can be obtained from the graph constructed in [CM17, § 5] by +reversing the words on each vertex. This is one of the adaptations pointed out in +the beginning of this section. +Theorem 8.3 ([CM17, Theorem 6.11]). Let u, v ∈ A∗ +n. +Then, u ∼hypon v if +and only if there exists a (weight-preserving labelled directed) graph isomorphism ψ +between Γ′ +n(u) and Γ′ +n(v) such that ψ(u) = v. +We now proceed to prove that ¨∼ and ∼hypon are the same relation on A∗ +n. +Lemma 8.4. The graph Γ′ +n coincides with the graph that results from ΓA¨∗n by +removing all loops. +Proof. From Definitions 4.2 and 8.2, we have that the vertex sets and weight maps +of Γ′ +n and ΓA¨∗n coincide. For u, v ∈ A∗ +n and i ∈ {1, . . ., n − 1}, if ¨fi(u) = v, then +wt(u) > wt(v), by Proposition 3.5, which implies that u ̸= v, and so, Γ′ +n is simple. +Finally, we have that u +i +−−−→ v is an edge of Γ′ +n if and only if ¨fi(u) = v if and only +if u +i +−−−→ v is an edge of ΓA¨∗n and u ̸= v. +□ +Lemma 8.5. Let u, v ∈ A∗ +n be such that u ∼hypon v. Then, for i ∈ {1, . . . , n − 1}, +u has an i-inversion if and only if v has an i-inversion. +Proof. Let i ∈ {1, . . ., n−1}. Suppose that u has an i-inversion and v does not have +an i-inversion. By Theorem 8.3, there exists a graph isomorphism ψ between Γ′ +n(u) +and Γ′ +n(v) such that ψ(u) = v. Since u has an i-inversion, we have that |u|i ≥ 1, +and since ψ preserves weights, wt(u) = wt(v), which implies that i occurs in v. As +v does not have an i-inversion, ¨fi is defined on v, and so, v +i +−−−→ ¨fi(v) is an edge of +Γ′ +n. Since ψ is a graph isomorphism and ψ(u) = v, we get that u +i +−−−→ ψ−1� ¨fi(v) +� +is an edge of Γ′ +n, which is a contradiction, because ¨fi is undefined on u, as u has +an i-inversion. The other direction is similar. +□ +Theorem 8.6. Let u, v ∈ A∗ +n. Then, u ¨∼ v if and only if u ∼hypon v. Therefore, +hypo(An) and hypon are isomorphic monoids. +Proof. Assume that u ∼hypon v. By Definitions 7.1, there exists a quasi-crystal +isomorphism ψ : A¨∗ +n(u) → A¨∗ +n(v) such that ψ(u) = v. By Proposition 4.9, ψ is a +graph isomorphism between ΓA¨∗ +n(u) and ΓA¨∗ +n(v). By Lemma 8.4, Γ′ +n(u) and Γ′ +n(v) +can be obtained from ΓA¨∗n(u) and ΓA¨∗n(v), respectively, by removing all loops. +Then, ψ is a graph isomorphism between Γ′ +n(u) and Γ′ +n(v) satisfying ψ(u) = v, +which implies by Theorem 8.3 that u ∼hypon v. +Conversely, assume that u ∼hypon v. +By Theorem 8.3, there exists a graph +isomorphism ψ′ between Γ′ +n(u) and Γ′ +n(v) such that ψ′(u) = v. To prove that ψ′ is +a graph isomorphism between ΓA¨∗n(u) and ΓA¨∗n(v), by Lemma 8.4 it just remains + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +41 +to show that ψ′ and (ψ′)−1 preserve loops. Let w ∈ A∗ +n and i ∈ {1, . . ., n − 1} such +that w has an i-labelled loop in ΓA¨∗ +n(u). Then, ¨εi(w) = +∞, which implies that u +has an i-inversion (see Example 6.12). By Lemma 8.4, w lies in Γ′ +n(u), and since +w ∼hypon ψ′(w) by Theorem 8.3, we get by Lemma 8.5 that ψ′(w) also has an i- +inversion. Hence, ¨εi +� +ψ′(w) +� += +∞, which implies that ψ′(w) has an i-labelled loop +in ΓA¨∗n. Analogously, if w′ ∈ A∗ +n has an i-labelled loop in ΓA¨∗n(v), then (ψ′)−1(w′) +also has an i-labelled loop in ΓA¨∗n, because (ψ′)−1 is a graph isomorphism between +Γ′ +n(v) and Γ′ +n(u). Therefore, ψ′ is a graph isomorphism between ΓA¨∗n(u) and ΓA¨∗n(v) +We have by Theorem 4.13 that ψ′ is a quasi-crystal isomorphism between A¨∗ +n(u) +and A¨∗ +n(v) satisfying ψ′(u) = v, which implies that u ¨∼ v. +□ +The previous result justifies the term hypoplactic used in Definitions 7.1 and 7.4, +since the classical hypoplactic monoid can be obtained as the hypoplactic monoid +associated to the standard quasi-crystal An of type An. Moreover, it shows that the +theory of quasi-crystals presented in Sections 3 to 7 gives rise to a genuine general- +ization of the classical hypoplactic monoid. We now have a process of crystallizing +the hypoplactic monoid which allows the construction of the classical hypoplactic +monoid from the standard quasi-crystal An of type An, and allows the analogous +construction of a monoid based on any other seminormal quasi-crystal. +9. The hypoplactic monoid of type Cn +We described in Section 7 a method of obtaining a monoid from a seminormal +quasi-crystal. We then showed in Section 8 that for the standard quasi-crystal An +of type An it results in the classical hypoplactic monoid of rank n. A natural way +of proceeding is to study the monoids that are obtained for other quasi-crystals. +In this section, we make a detailed study of the hypoplactic monoid hypo(Cn). +We start in Subsection 9.1 by presenting a description of the free quasi-crystal +monoid C¨∗ +n over Cn, from which hypo(Cn) emerges. In Subsections 9.2 and 9.3, we +characterize the highest-weight and isolated words of C¨∗ +n, which allows us to identify +the commutative and idempotent elements of hypo(Cn). +In Subsection 9.4, we +investigate whether hypo(Cn) satisfies some well-known relations, such as the Knuth +relations. In Subsection 9.5, we show that hypo(C2) satisfies nontrivial identities, +and describe some of the properties any such identity must have. We also show +that hypo(Cn) does not satisfy nontrivial identities, for n ≥ 3. In Subsection 9.6, +we prove that hypo(Cn) does not admit a finite presentation, but we identify the +connected components of C¨∗ +2 up to isomorphism, leading to a class of representatives +for the elements of hypo(C2). Finally, in Subsections 9.7 and 9.8 we describe monoid +embeddings of hypo(An−1) and hypo(Cn−1) into hypo(Cn). +9.1. The definition of hypo(Cn). From Examples 3.4(2) and 4.4(2), we have that +the standard quasi-crystal Cn of type Cn is a seminormal quasi-crystal consisting of +an ordered set +Cn = +� +1 < 2 < · · · < n < n < n − 1 < · · · < 1 +� +. +Its quasi-crystal graph is +1 +1 +−−−→ 2 +2 +−−−→ · · · +n−1 +−−−→ n +n +−−−→ n +n−1 +−−−→ n − 1 +n−2 +−−−→ · · · +1 +−−−→ 1, +where the weight map wt : Cn → Zn is defined by +wt(x) = ex +and +wt(x) = −ex, +for x ∈ {1, . . . , n}. + +42 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Notice that for x, y ∈ Cn and i ∈ {1, . . ., n − 1}, if ¨ϕi(x) > 0 and ¨εi(y) > 0, +then x ∈ +� +i, i + 1 +� +and y ∈ +� +i + 1, i +� +. If ¨ϕn(x) > 0 and ¨εn(y) > 0, then x = n and +y = n. +To avoid constant division into cases where i ̸= n and i = n, for brevity, in +the rest of this section we formally consider n + 1 and n + 1 to be symbols that +never appear in any word. Thus, the observation in the previous paragraph can be +simply re-stated as follows: for any i ∈ {1, . . ., n}, if ¨ϕi(x) > 0 and ¨εi(y) > 0, then +x ∈ +� +i, i + 1 +� +and y ∈ +� +i + 1, i +� +. This leads to the concept of an i-inversion for +words over the alphabet Cn. +Definition 9.1. Let i ∈ {1, . . . , n}. A word w ∈ C∗ +n is said to have an i-inversion +if w admits a decomposition of the form w = w1xw2yw3, for some w1, w2, w3 ∈ C∗ +n, +x ∈ +� +i, i + 1 +� +and y ∈ +� +i + 1, i +� +. +A word w ∈ C∗ +n is said to be i-inversion-free if w does not have an i-inversion. +By Definition 6.1 and Proposition 6.3, we obtain the following description of the +free quasi-crystal monoid C¨∗ +n over Cn. +Definition 9.2. The free quasi-crystal monoid C¨∗ +n over Cn consists of the set C∗ +n +of all words over Cn, under the operation of concatenation of words, and a quasi- +crystal structure given as follows. For w ∈ C∗ +n, the weight of w is +wt(w) = +� +|w|1 − |w|1, |w|2 − |w|2, . . . , |w|n − |w|n +� +. +For i ∈ {1, . . ., n}, if w has an i-inversion, then +¨εi(w) = ¨ϕi(w) = +∞, +otherwise, +¨εi(w) = |w|i+1 + |w|i +and +¨ϕi(w) = |w|i + |w|i+1. +The raising quasi-Kashiwara operator ¨ei is defined on w if and only if ¨εi(w) ∈ +Z>0. +The lowering quasi-Kashiwara operator ¨fi is defined on w if and only if +¨ϕi(w) ∈ Z>0. When they are defined, the quasi-Kashiwara operators can be com- +puted (as in Proposition 6.3) as follows: Let i ∈ {1, . . . , n}, and let w ∈ C∗ +n be +i-inversion-free. If i + 1 or i occurs in w, let x be the right-most i + 1 or i occurring +in w, then ¨ei(w) is obtained from w by replacing x by ¨ei(x). If i or i + 1 occurs in +w, let y be the left-most i or i + 1 occurring in w, then ¨fi(w) is obtained from w +by replacing y by ¨fi(y). +Example 9.3. Consider n = 5. Take w = 253553234. We have that wt(w) = +(0, 2, −1, 1, 1). Since w admits a decomposition of the form w = w12w23w3, we +get that w has a 2-inversion. It has a 3-inversion, as it admits a decomposition of +the form w = w13w23w3. It has a 4-inversion, as it admits a decomposition of the +form w = w15w25w3. Therefore, for i ∈ {2, 3, 4}, the quasi-Kashiwara operators +¨ei and ¨fi are undefined on w. +On the other hand, w is 1-inversion-free and 5- +inversion-free. We have that ¨f1 is undefined on w, as neither 1 nor 2 occurs in w, +¨e1(w) = 253553134, ¨e5(w) = 253553234, and ¨f5(w) = 253553234. +In Section 4, we showed that a seminormal quasi-crystal can be described by +its quasi-crystal graph. Thus, to study the hypoplactic monoid hypo(Cn), we will +frequently resort to the quasi-crystal graph ΓC¨∗ +n of C¨∗ +n +Example 9.4. The empty word ǫ is an isolated vertex in ΓC¨∗ +n without loops. The +set of letters Cn forms a connected component which is described in Example 4.4(2). +We now turn our attention to the case n = 2. Words of length 2 form a subgraph of +ΓC¨∗ +2 that is isomorphic to the quasi-crystal graph of C2 ¨⊗ C2, which is described in + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +43 +Example 5.4(2). The connected component ΓC¨∗ +2 (121) of ΓC¨∗ +2 containing 121 is the +following: +121 +121 +221 +211 +2 11 +212 +2 12 +2 1 2 +2 +1 +1 +1 +2 +2 +2 +1 +2 +1 +1 +1 +The connected component ΓC¨∗ +2 (212) of ΓC¨∗ +2 containing 212 is the following: +212 +212 +212 +112 +112 +122 +121 +1 2 1 +2 +2 +1 +2 +1 +1 +2 +1 +1 +1 +2 +1 +By Definitions 7.1 and 7.4, the hypoplactic monoid hypo(Cn) is the quotient +monoid of C∗ +n by the hypoplactic congruence ¨∼. Although we omitted the weight +map wt in the example above, recall that ΓC¨∗ +n is a weighted labelled graph, from +which ¨∼ can be obtained, since, for two words u, v ∈ C∗ +n, we have by Theorem 4.13 +that u ¨∼ v if and only if there exists a graph isomorphism ψ between the connected +components ΓC¨∗ +n(u) and ΓC¨∗ +n(v) of ΓC¨∗ +n such that ψ(u) = v. +The following result shows that the hypoplactic congruence ¨∼ respects inversions. +Proposition 9.5. Let u, v ∈ C∗ +n with u ¨∼ v, and let i ∈ {1, . . . , n}. Then, u has +an i-inversion if and only if v has an i-inversion. +Proof. If u has an i-inversion, then ¨εi(u) = +∞. Since u ¨∼ v, then ¨εi(v) = ¨εi(u) = ++∞, which implies that v has an i-inversion. The converse follows from the fact +that u ¨∼ v implies v ¨∼ u. +□ +This result is analogous to one obtained for the classical hypoplactic monoid in +Lemma 8.5, where, for each i ∈ {1, . . . , n−1}, either all words in a congruence class +of the hypoplactic congruence have an i-inversion, or all of them are i-inversion-free. +From Definition 8.2, Lemma 8.4 and [CM17, Proposition 5.2], we can deduce a +construction of the quasi-crystal graph ΓA¨∗n from the crystal graph over A∗ +n of type +An. An analogous construction of the quasi-crystal graph ΓC¨∗ +n from the crystal +graph over C∗ +n of type Cn [KN94, Lec02] can also be given. First, note that the +weight maps coincide. By Definition 6.9, we have that when the quasi-Kashiwara +operators are defined, they coincide with the Kashiwara operators. Thus, the quasi- +crystal graph ΓC¨∗ +n is obtained from the crystal graph over C∗ +n of type Cn by deleting +all i-labelled edges starting or ending on a word with an i-inversion, and then, +adding i-labelled loops on all words with an i-inversion, for i = 1, . . . , n. +The empty word ǫ and the word 11 are related by the plactic congruence on +C∗ +n [Lec02]. Since 11 has a 1-inversion, we get by Proposition 9.5 that 11 ̸¨∼ ǫ. +Hence, the plactic congruence on C∗ +n is not contained in the hypoplactic congruence +on C∗ +n. This contrasts with the well-known result for type An, where the plactic +congruence on A∗ +n is contained in the hypoplactic congruence on A∗ +n. +Finally, notice that a word w in C∗ +n may have unbarred symbols and barred +symbols. For the sake of simplicity, we introduce the following notation. +Definition 9.6. Set ǫ = ǫ. For each x ∈ {1, . . . , n}, set x = x. Given a word +w = x1x2 . . . xm, with x1, x2, . . . , xm ∈ Cn, set w = xm xm−1 . . . x1. + +44 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +The following result shows that this notation preserves inversions. +Lemma 9.7. Let w ∈ C∗ +n and i ∈ {1, . . ., n}. Then, w has an i-inversion if and +only if w has an i-inversion. +Proof. If w has an i-inversion, then w = w1xw2yw3, for some w1, w2, w3 ∈ C∗ +n, +x ∈ +� +i, i + 1 +� +and y ∈ +� +i + 1, i +� +. Thus, w = w3 y w2 x w1, where y ∈ +� +i + 1, i +� +and +x ∈ +� +i, i + 1 +� +, and so, w has an i-inversion. +The converse is immediate since w = w. +□ +In the quasi-crystal monoid C¨∗ +n we have the following relation between a word +w ∈ C∗ +n and its barred version w. +Proposition 9.8. Let w ∈ C∗ +n, and let i ∈ {1, . . ., n}. Then, wt(w) = − wt(w), +¨εi(w) = ¨ϕi(w), and ¨ϕi(w) = ¨εi(w). Furthermore, if ¨ei(w) or ¨fi(w) are defined, +then ¨ei(w) = ¨fi(w), and if ¨fi(w) or ¨ei(w) are defined, then ¨fi(w) = ¨ei(w). +Proof. From Definition 9.2, we have that +wt(w) = +� +|w|1 − |w|1, . . . , |w|n − |w|n +� += +� +|w|1 − |w|1, . . . , |w|n − |w|n +� += − wt(w). +By Lemma 9.7, we get that ¨εi(w) = ¨ϕi(w) = +∞ if and only if ¨εi(w) = ¨ϕi(w) = ++∞. And if so, the result follows trivially. Thus, assume that neither w nor w has +an i-inversion. For i ̸= n, we have that +¨εi(w) = |w|i+1 + |w|i = |w|i+1 + |w|i = ¨ϕi(w); +furthermore, ¨εn(w) = |w|n = |w|n = ¨ϕn(w). +As C¨∗ +n is seminormal, ¨ei(w) is defined if and only if ¨fi(w) is defined. +If so, +then there exist w1, w2 ∈ C∗ +n and x ∈ +� +i, i + 1 +� +such that |w1|i = |w1|i+1 = 0, +w = w1xw2, and ¨fi(w) = w1 ¨fi(x)w2. Since w = w2 x w1, where x ∈ +� +i + 1, i +� +and |w1|i+1 = |w1|i = 0, we get that ¨ei(w) = w2¨ei(x)w1. +Note that if x = i, +then ¨ei(x) = i + 1 = ¨fi(x), and otherwise, x = i + 1 and ¨ei(x) = i = ¨fi(x). +Hence, ¨ei(w) = ¨fi(w). Analogously, ¨fi(w) = ¨ei(w), whenever ¨fi(w) or ¨ei(w) are +defined. +□ +The following results are straightforward consequences of the previous result: +Corollary 9.9. Given u, v ∈ C∗ +n, there is an edge u +i +−−−→ v in the quasi-crystal +graph ΓC¨∗ +n if and only if there is an edge v +i +−−−→ u. Thus, for any w ∈ C∗ +n, +C∗ +n(w) = +� +u +�� u ∈ Cn(w) +� +. +Corollary 9.10. Let u, v ∈ C∗ +n. Then, u ¨∼ v if and only if u ¨∼ v. +9.2. Highest-weight words. In the study of plactic monoids for the infinite Car- +tan types [Lec02, Lec03], words of highest weight are extremely relevant as they are +used to index connected components of crystal graphs. An analogous relation was +proven in [CM17] for the classical hypoplactic monoid. Thus, we now characterize +the highest-weight words of C¨∗ +n, and check whether they satisfy properties similar +to highest-weight words in the mentioned contexts. +From Definition 3.6(1), we have that a word w ∈ C∗ +n is of highest weight if ¨ei is +undefined on w, for all i ∈ {1, . . ., n}. Equivalently, w is of highest weight if the +only edges in ΓC¨∗ +n ending on w are loops. +Proposition 9.11. Let w ∈ C∗ +n. Then, w is of highest weight if and only if for +each letter x ∈ Cn occurring in w, the following conditions are satisfied: + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +45 +(1) if x ∈ {2, . . . , n}, then w has an (x − 1)-inversion; +(2) if x ∈ +� +n, . . . , 1 +� +, then w has an x-inversion. +Proof. Suppose that w is of highest weight. Let x ∈ Cn be a letter occurring in +w. If x ∈ {2, . . . , n}, then ¨εx−1(w) > 0, and since C¨∗ +n is seminormal and ¨ex−1 is +undefined on w, we get that ¨εx−1(w) = +∞, or equivalently, w has an (x − 1)- +inversion. If x ∈ +� +n, . . . , 1 +� +, then ¨εx(w) > 0, which implies as in the previous case +that w has an x-inversion. +Conversely, suppose that w is not of highest weight. Then, take i ∈ {1, . . ., n} +such that ¨ei is defined on w. By Definition 9.2, we have that ¨ei(w) = w1¨ei(x)w2, +for some w1, w2 ∈ C∗ +n and x +� +i + 1, i +� +. Therefore, the letter i + 1 or the letter i +occurs in w, and w does not have an i-inversion. +□ +Example 9.12. Consider n = 4. In C¨∗ +4, the following words are of highest weight: +1, 12, 11, 332, 333, and 123443 2 1. +In the crystal graphs studied in [KN94], which led to the construction of the plac- +tic monoids for the infinite Cartan types [Lec02, Lec03], each connected component +has exactly one highest-weight element and exactly one lowest-weight element. Due +to the results in [CM17] and in Section 8, we also have that the connected compo- +nents of the free quasi-crystal monoid A¨∗ +n have exactly one highest-weight word and +exactly one lowest-weight word. The free quasi-crystal monoid C¨∗ +n does not have +this property, as we can see in Example 9.4 that 212 and 112 are highest-weight +words of C¨∗ +2 which belong to the same connected component. The same happens in +C¨∗ +n for any n ≥ 2, because +¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1 ¨fn ¨fn−1 · · · ¨f2(212) += ¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1 ¨fn(n12) += ¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1(n12) += ¨e2 ¨f1 ¨f2 +� +212 +� += ¨e2 ¨f1 +� +21 ¨f2(2) +� += ¨e2 +� +11 ¨f2(2) +� += 112. +We can also see that 2 11 and 2 1 2 are lowest-weight words of C¨∗ +n which belong to the +same connected component. Although we have that a connected component of C¨∗ +n +may have more than one highest-weight word or more then one lowest-weight word, +we can guarantee by Proposition 6.11 that it has at least one of each. Moreover, in +the following result we describe a one-to-one correspondence between highest-weight +words and lowest-weight words of C¨∗ +n. +Proposition 9.13. Let w ∈ C∗ +n. Then, w is of highest weight if and only if w is +of lowest weight. Also, w is of lowest weight if and only if w is of highest weight. +Proof. For any i ∈ {1, . . ., n}, we have by Corollary 9.9 that ¨ei (or ¨fi) is defined +on w if and only if ¨fi (resp., ¨ei) is defined on w. This implies that w is of highest +(resp., lowest) weight if and only if w is of lowest (resp., highest) weight. +□ +From Example 9.12, we have that 123443 2 1 is a highest-weight word in C¨∗ +4. +Also, wt +� +12344 3 2 1 +� += 0 and ¨εi +� +12344 3 2 1 +� += +∞, for all i ∈ {1, 2, 3, 4}. Thus, +for any w ∈ C∗ +4, we get that 12344 3 2 1w is a highest-weight word with weight +wt(w). In the following result, we generalize this reasoning, from which we can +see that highest weights (Definition 3.7) in C¨∗ +n do not identify a relevant subset of +weights. + +46 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Proposition 9.14. Any element λ ∈ Zn is a highest weight in C¨∗ +n. +Proof. Let λ = (λ1, . . . , λn) ∈ Zn. For each i ∈ {1, . . ., n}, if λi ≥ 0, set ai = λi + 1 +and bi = 1, otherwise, set ai = 1 and bi = −λi + 1. The word +w = 1a12a2 . . . nannbnn − 1 +bn−1 . . . 1 +b1 +is such that +wt(w) = (a1 − b1, a2 − b2, . . . , an − bn) = λ +and ¨εi(w) = +∞, for any i ∈ {1, . . . , n}, because w has a decomposition of the +form w = w1iw2iw3, for some w1, w2, w3 ∈ C∗ +n. Hence, w is a highest-weight word +with weight λ, which implies that λ is a highest weight. +□ +The previous result together with Propositions 9.8 and 9.13 implies that any +element λ ∈ Zn is a lowest weight in C¨∗ +n. +9.3. Isolated words. By Proposition 6.4 and Theorem 7.5, we have that the com- +mutative elements of the hypoplactic monoid hypo(Cn) correspond to the hypoplac- +tic congruence classes of isolated words of C¨∗ +n. By Theorem 7.6, we also have that +the idempotent elements of the hypoplactic monoid hypo(Cn) correspond to the +hypoplactic congruence classes of isolated words of C¨∗ +n with weight 0. Therefore, we +now turn our attention to characterizing the isolated words in C¨∗ +n, and consequently, +obtain some relations in hypo(Cn). +By Definition 4.15, a word w ∈ C∗ +n is isolated if it is an isolated vertex in the +quasi-crystal graph ΓC¨∗ +n. In other words, w is isolated if and only if it is both of +highest and of lowest weight in C¨∗ +n. +Proposition 9.15. Let w ∈ C∗ +n. Then, w is an isolated word if and only if w is +an isolated word. Also, w is an isolated word if and only if both w and w are of +highest weight. +Proof. By Proposition 9.13, w is of highest and of lowest weight if and only if w is +of highest and of lowest weight. Also, w and w are of highest weight if and only if +w and w are of lowest weight. And so, the result follows. +□ +Proposition 9.16. Let w ∈ C∗ +n. Then, w is an isolated word if and only if both of +the following conditions hold: +(1) w has a 1-inversion if 1 or 1 occurs in w; +(2) w has an (i − 1)-inversion and an i-inversion, for all i ∈ {2, . . . , n} such +that i or i occurs in w. +Proof. Suppose that w is an isolated word. By Proposition 9.15, w and w are of +highest weight. Let i ∈ {1, . . . , n}. If i occurs in w, or equivalently, i occurs in w, +then we have by Proposition 9.11 that w has an (i − 1)-inversion when i ≥ 2, and +that w has an i-inversion which implies by Lemma 9.7 that w has an i-inversion. +Analogously, if i occurs in w, or equivalently, i occurs in w, then w has an (i − 1)- +inversion and an i-inversion. +Conversely, suppose that w is not an isolated word. Take i ∈ {1, . . . , n} such +that ¨ei or ¨fi is defined on w. By Definition 9.2, w does not have an i-inversion, and +some letter among i, i + 1, i + 1 and i occurs in w. +□ +Example 9.17. Consider n = 4. In C¨∗ +4, the following are isolated words: 11, 122, +22 1, 333, and 123443 2 1. +In the following result, we show how to obtain commutative and idempotent +elements of hypo(Cn) from each word in C∗ +n. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +47 +Proposition 9.18. Let w ∈ C∗ +n. Then, www and wwww are isolated words in C¨∗ +n. +Therefore, www is a commutative element of hypo(Cn), and wwww is a commuta- +tive and idempotent element of hypo(Cn). +Proof. For each i ∈ {1, . . ., n}, if i occurs in w, then www and wwww have decom- +positions of the form w1iw2iw3iw4, for some w1, w2, w3, w4 ∈ C∗ +n, which implies +that www and wwww have an (i − 1)-inversion and an i-inversion. If i occurs in +w, then www and wwww have decompositions of the form w1iw2iw3iw4, for some +w1, w2, w3, w4 ∈ C∗ +n, which implies that www and wwww have an (i − 1)-inversion +and an i-inversion. By Proposition 9.16, www and wwww are isolated words, and +by Theorem 7.5, they are commutative elements of hypo(Cn). By Propositions 6.3 +and 9.8, we have that +wt(wwww) = wt(w) − wt(w) + wt(w) − wt(w) = 0, +and by Theorem 7.6, we get that wwww is an idempotent element of hypo(Cn). +□ +To give a complete characterization of the commutative and idempotent elements +of hypo(Cn), we first introduce the following notation. +Definition 9.19. For each word w ∈ C∗ +n, define an n-tuple inv(w) = (δ1, . . . , δn) ∈ +{0, 1}n, where, for i ∈ {1, . . . , n}, δi = 1 if and only if w has an i-inversion. +Lemma 9.20. Let u, v ∈ C∗ +n be isolated words in C¨∗ +n. Then, u ¨∼ v if and only if +wt(u) = wt(v) and inv(u) = inv(v). +Proof. Since u and v are isolated words, we get that C∗ +n(u) = {u} and C∗ +n(v) = +{v}, which implies that ¨εi(u), ¨ϕi(u), ¨εi(v), ¨ϕi(v) ∈ {0, +∞}, for all i ∈ {1, . . ., n}, +because C¨∗ +n is seminormal. By Proposition 9.5, for each i ∈ {1, . . . , n}, we have +that ¨εi(u) = ¨ϕi(u) = +∞ (or ¨εi(u) = ¨ϕi(v) = +∞) if and only if u (resp., v) has +an i-inversion. Therefore, the map ψ : C∗ +n(u) → C∗ +n(v), given by ψ(u) = v, is a +quasi-crystal isomorphism between C¨∗ +n(u) and C¨∗ +n(v) if and only if wt(u) = wt(v) +and inv(u) = inv(v). +□ +Theorem 9.21. The map that sends each isolated word w ∈ C∗ +n to +� +wt(w), inv(w) +� +induces a bijection between the set of commutative elements of hypo(Cn) and the set +of pairs (λ, δ) with λ = (λ1, . . . , λn) ∈ Zn and δ = (δ1, . . . , δn) ∈ {0, 1}n satisfying +the following conditions: +(1) if λi ̸= 0, for some i ∈ {1, . . ., n}, then δi = 1, and δi−1 = 1 when i ≥ 2; +(2) if δi = 1, for some i ∈ {2, . . . , n}, then δi−1 = 1, or δi+1 = 1 when i ≤ n−1. +Proof. By Proposition 6.4 and Theorem 7.5, we have that the commutative ele- +ments of hypo(Cn) correspond to the hypoplactic congruence classes of isolated +words of C¨∗ +n. +By Lemma 9.20, the map that sends each isolated word w ∈ C∗ +n +to +� +wt(w), inv(w) +� +induces a well-defined injective map from the commutative ele- +ments of hypo(Cn) to Zn × {0, 1}n. +We now show that the pairs +� +wt(w), inv(w) +� +, where w ∈ C∗ +n is an isolated word +of C¨∗ +n, satisfy conditions (1) and (2). Let w ∈ C∗ +n be an isolated word of C¨∗ +n. For +each i ∈ {1, . . ., n}, set λi = |w|i − |w|i, and if w has an i-inversion, take δi = 1, +otherwise, take δi = 0. So, wt(w) = (λ1, . . . , λn) and inv(w) = (δ1, . . . , δn). If +λi ̸= 0, for some i ∈ {1, . . ., n}, then i or i occurs in w implying that w has an +(i − 1)-inversion (if i ≥ 2) and an i-inversion, by Proposition 9.16, and so, δi−1 = 1 +(if i ≥ 2) and δi = 1. If δi = 1, for some i ∈ {2, . . ., n}, then some letter among +i, i + 1, i + 1 and i occurs in w implying that w has an (i − 1)-inversion, or when +i ≤ n − 1, an (i + 1)-inversion, by Proposition 9.16, and thus, δi−1 = 1 or δi+1 = 1. +Therefore, the pair +� +wt(w), inv(w) +� +satisfies conditions (1) and (2). + +48 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Finally, we show that for each pair (λ, δ) ∈ Zn ×{0, 1}n satisfying conditions (1) +and (2), there exists an isolated word w ∈ C∗ +n such that wt(w) = λ and inv(w) = δ. +Let λ = (λ1, . . . , λn) ∈ Zn and δ = (δ1, . . . , δn) ∈ {0, 1}n satisfying conditions (1) +and (2). If δ1 = 1, set w1 = 11, otherwise, set w1 = ǫ. For each i ∈ {2, . . ., n}, +if δi−1 = δi = 1, take wi = iiii, otherwise, take wi = ǫ. By (2), if δi = 1, for +some i ∈ {2, . . . , n}, then wi ̸= ǫ or wi+1 ̸= ǫ, which implies that wiwi+1 has an +i-inversion. Also, for i ∈ {2, . . ., n}, we have that ¨εi−1(wi) = ¨εi(wi) = +∞, and +¨εj(wi) = 0, whenever j ∈ {1, . . ., n} \ {i − 1, i}. Then, the word w′ = w1w2 . . . wn +has a 1-inversion if and only if δ1 = 1, and for i ∈ {2, . . . , n}, w′ has an i-inversion +if and only if δi = 1. Hence, inv(w′) = δ. Since wt(wi) = 0, for any i ∈ {1, . . ., n}, +we get that wt(w′) = 0. +For each i ∈ {1, . . . , n}, if λi ≥ 0, set ai = λi and bi = 0, otherwise, set ai = 0 +and bi = −λi. Let +w = w′1a12a2 . . . nannbnn − 1 +bn−1 . . . 1 +b1. +By (1), if ai ̸= 0 or bi ̸= 0, for some i ∈ {1, . . ., n}, then δi−1 = 1 when i ≥ 2, +and δi = 1, implying that w′ has an (i − 1)-inversion (if i ≥ 2) and an i-inversion. +Hence, inv(w) = inv(w′) = δ. Since wt(w′) = 0, we get that +wt(w) = (a1 − b1, a2 − b2, . . . , an − bn) = λ. +Therefore, +� +wt(w), inv(w) +� += (λ, δ). +□ +Corollary 9.22. The map that sends each isolated word w ∈ C∗ +n to inv(w) induces +a bijection between the set of idempotent elements of hypo(Cn) and the set of n- +tuples δ = (δ1, . . . , δn) ∈ {0, 1}n such that for each i ∈ {2, . . ., n}, if δi = 1, then +δi−1 = 1, or δi+1 = 1 when i ≤ n − 1. +Proof. By Theorem 7.6, the idempotent elements of hypo(Cn) correspond to the +hypoplactic congruence classes of isolated words of C¨∗ +n with weight 0. Thus, the +result follows directly from Theorem 9.21. +□ +9.4. Relations. In this subsection, we first prove some results for hypo(C2) that +allow a deeper understanding of this monoid, which will be necessary to deduce +some properties in the following subsections. Motivated by the fact that the plactic +monoid of type Cn satisfies the Knuth relations (see Definition 8.1) with the restric- +tion that x ̸= z [Lec02, Definition 3.2.1], we then study whether the hypoplactic +monoid hypo(Cn) satisfies the Knuth relations. In fact, we show that the Knuth +relations only hold for one choice of generators. +Lemma 9.23. Let m1, m2, p1, p2 ∈ Z≥0 and n1, n2 ∈ Z>0. Then, 1m12n11p1 ¨∼ +1m22n21p2 in C¨∗ +2 implies that m1 = m2, n1 = n2 and p1 = p2. +Proof. Assume that 1m12n11p1 ¨∼ 1m22n21p2. Then, m1+p1 = m2+p2 and n1 = n2, +because wt(1m12n11p1) = wt(1m22n21p2). +Suppose m1 ̸= m2. +Without loss of +generality, assume m1 < m2. Set +u = ¨f m2+n1−1 +1 +¨f n1 +2 (1m12n11p1) = ¨f m2+n1−1 +1 +� +1m12 +n11p1� += 2m11 +n12m2−m1−11p1−m2+m1+1 +and +v = ¨f m2+n2−1 +1 +¨f n2 +2 (1m22n21p2) = ¨f m2+n2−1 +1 +� +1m22 +n21p2� += 2m21 +n2−121p2. +Since 1m12n11p1 ¨∼ 1m22n21p2 and n1 = n2, we get that u ¨∼ v. By Proposition 9.5, +this is a contradiction, because u is 2-inversion-free and v has a 2-inversion. +□ + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +49 +Lemma 9.24. Let n1, n2, p1, p2, ∈ Z≥0 and m1, m2, q1, q2 ∈ Z>0. Then, in C¨∗ +2, +1m12n11p12q1 ¨∼ 1m22n21p22q2 if and only if m1+p1 = m2+p2 and n1+q1 = n2+q2. +Proof. If we first suppose that 1m12n11p12q1 ¨∼ 1m22n21p22q2 then we get that +wt(1m12n11p12q1) = wt(1m22n21p22q2), which implies that m1 + p1 = m2 + p2 and +n1 + q1 = n2 + q2. +We now show that 12k1l2 ¨∼ 1l+12k+1, for any k, l ∈ Z≥0. The aim is to show that +each connected component ΓC¨∗ +2 +� +12k1l2 +� +and ΓC¨∗ +2 +� +1l+12k+1� +is a path with 2k+2l+5 +vertices. This will allow us to define a bijection ψ : C∗ +2 +� +12k1l2 +� +→ C∗ +2 +� +1l+12k+1� +that maps each word w ∈ C∗ +2 +� +12k1l2 +� +to the word ψ(w) ∈ C∗ +2 +� +1l+12k+1� +such that +the position of w in ΓC¨∗ +2 +� +12k1l2 +� +is the same as ψ(w) in ΓC¨∗ +2 +� +1l+12k+1� +. +The paths ΓC¨∗ +2 +� +12k1l2 +� +and ΓC¨∗ +2 +� +1l+12k+1� +start in 12k1l2 and 1l+12k+1, which +are of highest weight. From these starting-points, there is a sequence of k +1 edges +labelled by 2, each of which transforms a symbol 2 to a symbol 2, in order from +left to right through the word; at each step except the last, there is a 1-inversion +in the word and so a loop labelled by 1 at that vertex. There are then k + l + 2 +edges labelled by 1, each of which transforms a symbol 1 to a symbol 2 or a symbol +2 to a symbol 1, in order from left to right through the word; again, in each step +except the first and the last, there is a 2-inversion in the word so a loop labelled +by 2 at that vertex. Finally, there is a sequence of l + 1 edges labelled by 2, each +transforming a symbol 2 to a symbol 2, in order from left to right throughout the +word; again, there is a loop labelled by 1 at each vertex. +Hence, u +i +−−−→ v is an edge in ΓC¨∗ +2 +� +12k1l2 +� +if and only if ψ(u) +i +−−−→ ψ(v) is an +edge of ΓC¨∗ +2 +� +1l+12k+1� +. And since ψ +� +12k1l2 +� += 1l+12k+1, where +wt +� +12k1l2 +� += (l + 1, k + 1) = wt +� +1l+12k+1� +, +we have that ψ preserves weights. Therefore, by Theorem 4.13, ψ is a quasi-crystal +isomorphism, which implies that 12k1l2 ¨∼ 1l+12k+1. +Finally, as ¨∼ is a monoid congruence, we can iterately apply 12k1l2 ¨∼ 1l+12k+1 +to see that 1m12n11p12q1 ¨∼ 1m1+p12n1+q1 and 1m22n21p22q2 ¨∼ 1m2+p22n2+q2. If +m1 + p1 = m2 + p2 and n1 + q1 = n2 + q2, we then obtain that 1m12n11p12q1 ¨∼ +1m22n21p22q2. +□ +Proposition 9.25. Let w ∈ {1, 2}∗. Then, w ¨∼ 2m11m22m31m4 in C¨∗ +2, for some +m1, m2, m3, m4 ∈ Z≥0. +Proof. Suppose that w ̸= 2m11m22m31m4, for any m1, m2, m3, m4 ∈ Z≥0. Then, +w = 2q01p12q11p22q2 . . . 1pk2qk1pk+1, +for some pk+1, q0 ∈ Z≥0 and p1, . . . , pk, q1, . . . , qk ∈ Z>0. By Lemma 9.24, we have +that 1p12q11p22q2 ¨∼ 1p1+p22q1+q2, and by iterating this process, we obtain that +1p12q11p22q2 . . . 1pk2qk ¨∼ 1p1+p2+···+pk2q1+q2+···+qk, +which implies that w ¨∼ 2q01p1+p2+···+pk2q1+q2+···+qk1pk+1. +□ +We will see in Theorem 9.34 that the previous result does not hold in C¨∗ +n when +n ≥ 3. +We now study some properties satisfied by words u, v ∈ C∗ +n that are +hypoplactic congruent u ¨∼ v in C¨∗ +n, for any n ≥ 2. +Lemma 9.26. Let u, v ∈ C∗ +n with u ¨∼ v in C¨∗ +n, and let x ∈ {1, . . . , n}. +(1) If x ≤ n − 1 and u ∈ {x, x + 1}∗, then v ∈ {x, x + 1}∗, |u|x = |v|x and +|u|x+1 = |v|x+1. +(2) If x ≤ n − 1 and u ∈ +� +x + 1, x +�∗, then v ∈ +� +x + 1, x +�∗, |u|x+1 = |v|x+1 +and |u|x = |v|x. + +50 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +(3) If u ∈ {x}∗, then v ∈ {x}∗ and |u|x = |v|x. +(4) If u ∈ {x}∗, then v ∈ {x}∗ and |u|x = |v|x. +(5) If x ̸= 2 and u ∈ {x, x}∗, then v ∈ {x, x}∗ and |u|x − |u|x = |v|x − |v|x. +Proof. (1) Assume x ≤ n − 1 and u ∈ {x, x + 1}∗. For i ∈ {1, . . . , n} \ {x, x + 1}, +since neither i nor i + 1 occurs in u, we have that u is i-inversion-free, and as u ¨∼ v, +|v|i ≤ ¨ϕi(v) = ¨ϕi(u) = |u|i + |u|i+1 = 0, +which implies that i does not occur in v. And since wt(u) = wt(v), we get that +−|v|i = |u|i − |u|i = 0, which implies that i does not occur in v. +Hence, v ∈ +� +x, x + 1, x + 1, x +� +. Since neither x + 2 nor x + 1 occurs in u, we have that u is +(x + 1)-inversion-free, and so, +|v|x+1 ≤ ¨εx+1(v) = ¨εx+1(u) = |u|x+2 + |u|x+1 = 0, +implying that x + 1 does not occur in v. As wt(u) = wt(v), we get that |u|x+1 = +|v|x+1. Then, u′ ¨∼ v′ where +u′ = ¨f |u|x+1 +x+2 +¨f |u|x+1 +x+3 +· · · ¨f |u|x+1 +n−1 +¨f |u|x+1 +n +¨f |u|x+1 +n−1 +· · · ¨f |u|x+1 +x+1 +(u) +and +v′ = ¨f |v|x+1 +x+2 +¨f |v|x+1 +x+3 +· · · ¨f |u|x+1 +n−1 +¨f |v|x+1 +n +¨f |v|x+1 +n−1 +· · · ¨f |v|x+1 +x+1 +(v). +Note that u′ and v′ are respectively obtained from u and v by replacing each x + 1 +by x + 1. In particular, u′ ∈ +� +x, x + 1 +� +and v′ ∈ +� +x, x + 1, x +� +. Since neither x + 1 +nor x occurs in u′, we get that u′ is x-inversion-free, and since u′ ¨∼ v′, +|v|x = |v′|x ≤ ¨εx(v′) = ¨εx(u′) = |u′|x+1 + |u′|x = 0, +which implies that x does not occur in v. Therefore, v ∈ {x, x + 1} which implies +that |u|x = |v|x and |u|x+1 = |v|x+1, because wt(u) = wt(v). +(2) If x ≤ n − 1 and u ∈ +� +x + 1, x +�∗, then u ∈ {x, x + 1}∗, and as u ¨∼ v by +Corollary 9.10, we get by (1) that v ∈ {x, x + 1}∗, |u|x = |v|x and |u|x+1 = |v|x+1. +This implies that v ∈ +� +x + 1, x +�∗, |u|x+1 = |v|x+1 and |u|x = |v|x. +(3) Suppose u ∈ {x}∗. If x > 1 then u lies in {x − 1, x}∗, which implies by (1) +that v lies in {x−1, x}∗ , where |u|x = |u|x and |v|x−1 = |v|x+1 = 0 and so v ∈ {x}∗. +If x = 1, then u ∈ {1, 2}∗ and the result follows similarly from (1). +(4) If u ∈ {x}∗, then u ∈ {x}∗ and the result follows from Corollary 9.10 and (3). +(5) Assume x ̸= 2 and u ∈ {x, x}∗. As u ¨∼ v, we have that wt(u) = wt(v), which +implies that |u|x − |u|x = |v|x − |v|x. For i ∈ {1, . . . , n} \ {x − 1, x}, since neither i +nor i + 1 occurs in u, we have that u is i-inversion-free, and as u ¨∼ v, +|v|i ≤ ¨ϕi(v) = ¨ϕi(u) = |u|i + |u|i+1 = 0, +which implies that i does not occur in v. And since wt(u) = wt(v), we get that +−|v|i = |u|i − |u|i = 0, which implies that i does not occur in v. If x = 1, then the +result is proven. Otherwise, x ≥ 3, we have that +|v|x−1 ≤ ¨εx−2(v) = ¨εx−2(u) = |u|x−1 + |u|x−2 = 0, +and then, −|v|x−1 = |u|x−1 − |u|x−1 = 0, implying that neither x − 1 nor x − 1 +occurs in v. Therefore, v ∈ {x, x}∗. +□ +Proposition 9.27. Let x, y ∈ {1, . . . , n} with {x, y} ̸= +� +2, 2 +� +, and let u, v ∈ C∗ +n +with u ¨∼ v in C¨∗ +n. If u ∈ {x, y}∗, then v ∈ {x, y}∗. +Proof. Assume that u ∈ {x, y}∗. If x = y, the result follows from items (3) and (4) +of Lemma 9.26. If x = y, then x ̸= 2, because {x, y} ̸= +� +2, 2 +� +, and so, the result +follows from Lemma 9.26(5). Thus, in the following, we assume that x ̸= y and +x ̸= y. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +51 +Take i, j ∈ {1, . . . , n} such that x ∈ +� +i, i +� +and y ∈ +� +j, j +� +. Since x ̸= y and x ̸= y, +we have that i ̸= j. Without loss of generality, we assume that i < j. We thus +consider the following cases. +• Case 1: x = i and y = j. If j > i + 1, then u is (j − 1)-inversion-free, +because neither j − 1 nor j occurs in u. In this case, as u ¨∼ v, we get that +u′ ¨∼ v′ for +u′ = ¨e|u|j +i+1¨e|u|j +i+2 · · · ¨e|u|j +j−1(u) +and +v′ = ¨e|u|j +i+1¨e|u|j +i+2 · · · ¨e|u|j +j−1(v). +Note that u′ is obtained from u by replacing each j by i + 1. If j = i + 1, +we take u′ = u and v′ = v; trivially, u′ ¨∼ v′. In either case, u′ ∈ {i, i + 1}∗. +We get by Lemma 9.26(1) that v′ ∈ {i, i + 1}∗ and |v′|i+1 = |u′|i+1 = |u|j. +In the case j = i + 1, this establishes the result immediately since v = v′. +In the case j > i + 1, +v = ¨f |u|j +j−1 ¨f |u|j +j−2 · · · ¨f |u|j +i+1 (v′), +we have that v is obtained from v′ by replacing each i + 1 by j, as |v′|i+1 = +|u|j, and so, v ∈ {i, j}∗. +• Case 2: x = i and y = j. Note that u is j-inversion-free, because neither j +nor j + 1 occurs in u, as i < j. Since u ¨∼ v, we get that u′ ¨∼ v′ for +u′ = ¨e +|u|j +i+1¨e +|u|j +i+2 · · · ¨e +|u|j +n−1¨e +|u|j +n +¨e +|u|j +n−1 · · · ¨e +|u|j +j +(u) +and +v′ = ¨e +|u|j +i+1¨e +|u|j +i+2 · · · ¨e +|u|j +n−1¨e +|u|j +n +¨e +|u|j +n−1 · · · ¨e +|u|j +j +(v). +Note that u′ is obtained from u by replacing each j by i + 1. +With a +reasoning analogous to case 1, we obtain that v ∈ +� +i, j +�∗. +• Case 3: x = i and y = j. If j > i + 1, then u is (j − 1)-inversion-free, as +neither j nor j − 1 occurs in u, and since u ¨∼ v, we get that u′ ¨∼ v′ for +u′ = ¨f +|u|j +i+1 ¨f +|u|j +i+2 · · · ¨f +|u|j +j−1(u) +and +v′ = ¨f +|u|j +i+1 ¨f +|u|j +i+2 · · · ¨f +|u|j +j−1(v). +Note that u′ is obtained from u by replacing each j by i + 1. If j = i + 1, +we take u′ = u and v′ = v; trivially, u′ ¨∼ v′. In either case, u′ ∈ +� +i + 1, i +�∗. +We get by Lemma 9.26(2) that v′ ∈ +� +i + 1, i +�∗ and |v′|i+1 = |u′|i+1 = |u|j. +In the case j = i + 1, this establishes the result immediately since v = v′. +In the case j > i + 1, +v = ¨e +|u|j +j−1¨e +|u|j +j−2 · · · ¨f +|u|j +i+1 (v′), +we have that v is obtained from v′ by replacing each i + 1 by j, as |v′|i+1 = +|u|j, and thus, v ∈ +� +j, i +�∗. +• Case 4: x = i and y = j. As i < j, note that u is j-inversion-free, because +neither j + 1 nor j occurs in u. Since u ¨∼ v, we get that u′ ¨∼ v′ for +u′ = ¨f |u|j +i+1 ¨f |u|j +i+2 · · · ¨f |u|j +n−1 ¨f |u|j +n +¨f |u|j +n−1 · · · ¨f |u|j +j +(u) +and +v′ = ¨f |u|j +i+1 ¨f |u|j +i+2 · · · ¨f |u|j +n−1 ¨f |u|j +n +¨f |u|j +n−1 · · · ¨f |u|j +j +(v). +Note that u′ is obtained from u by replacing each j by i + 1. +With a +reasoning analogous to case 3, we obtain that v ∈ +� +j, i +�∗. +In either case, we get that v ∈ {x, y}∗. +□ +Proposition 9.28. Let m ∈ Z≥0, and let i, j ∈ {1, . . . , n} with i < j. In C¨∗ +n, for +any u, v ∈ +� +1, 2, . . ., i, j, i − 1, i − 2, . . . , 1 +�∗, we have that +(1) jmiu ̸¨∼ jm+1v; and + +52 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +(2) ujim ̸¨∼ vim+1. +Proof. (1) Suppose there exist u, v ∈ +� +1, 2, . . . , i, j, i − 1, i − 2, . . . , 1 +�∗ such that +jmiu ¨∼ jm+1v in C¨∗ +n. So, wt(jmiu) = wt +� +jm+1v +� +which implies that |v|i = |u|i + 1 +and |u|j = |v|j + 1. In particular, j occurs in u. Since neither j + 1 nor j occurs in +u or v, then jmiu and jm+1v are j-inversion-free. Set +w1 = ¨f |u|j+m +i+1 +¨f |u|j+m +i+2 +· · · ¨f |u|j+m +n−1 +¨f |u|j+m +n +¨f |u|j+m +n−1 +· · · ¨f |u|j+m +j +(jmiu) += i + 1 +mi +� ¨f |u|j +i+1 ¨f |u|j +i+2 · · · ¨f |u|j +n−1 ¨f |u|j +n +¨f |u|j +n−1 · · · ¨f |u|j +j +(u) +� += i + 1 +miu′ +and +w2 = ¨f |u|j+m +i+1 +¨f |u|j+m +i+2 +· · · ¨f |u|j+m +n−1 +¨f |u|j+m +n +¨f |u|j+m +n−1 +· · · ¨f |u|j+m +j +� +jm+1v +� += i + 1 +m+1� ¨f |u|j−1 +i+1 +¨f |u|j−1 +i+2 +· · · ¨f |u|j−1 +n−1 +¨f |u|j−1 +n +¨f |u|j−1 +n−1 +· · · ¨f |u|j−1 +j +(v) +� += i + 1 +m+1v′. +As jmiu ¨∼ jm+1v, we get that w1 ¨∼ w2. +Note that u′ is obtained from u by +replacing each j by i + 1, and as |v|j = |u|j − 1, v′ is obtained from v by replacing +each j by i + 1. In particular, i + 1 occurs in u′, as j occurs in u. Since neither +i + 1 nor i occurs in w1 or w2, we have that w1 and w2 are i-inversion-free. Set +w′ +1 = ¨f m+1 +i +(w1) = i +m(i + 1)u′ +and +w′ +2 = ¨f m+1 +i +(w2) = i +m+1v′. +As w1 ¨∼ w2, we get that w′ +1 ¨∼ w′ +2. +Since i + 1 occurs in u′, then w′ +1 has an +(i + 1)-inversion. And since neither i + 1 nor i + 2 occurs in w′ +2, then w′ +2 is (i + 1)- +inversion-free. By Proposition 9.5, this is a contradiction, because we obtained that +w′ +1 has an (i + 1)-inversion, w′ +2 is (i + 1)-inversion-free, and w′ +1 ¨∼ w′ +2. +(2) Suppose there exist u, v ∈ +� +1, 2, . . . , i, j, i − 1, i − 2, . . . , 1 +�∗ such that ujim ¨∼ +vim+1 in C¨∗ +n. So, wt(ujim) = wt +� +vim+1� +which implies that |u|i = |v|i + 1 and +|v|j = |u|j + 1. As justified in (1), set +w1 = ¨f |u|j+1 +i+1 +¨f |u|j+1 +i+2 +· · · ¨f |u|j+1 +n−1 +¨f |u|j+1 +n +¨f |u|j+1 +n−1 +· · · ¨f |u|j+1 +j +(ujim) = u′i + 1im +w′ +1 = ¨f |u|i+|u|j+m+1 +i +(w1) = u′′i(i + 1)m +w′′ +1 = ¨f |u|i+m +j+1 +¨f |u|i+m +j+2 +· · · ¨f |u|i+m +n−1 +¨f |u|i+m +n +¨f |u|i+m +n−1 +· · · ¨f |u|i+m +i+1 +(w′ +1) = u′′′ij +m +and +w2 = ¨f |u|j+1 +i+1 +¨f |u|j+1 +i+2 +· · · ¨f |u|j+1 +n−1 +¨f |u|j+1 +n +¨f |u|j+1 +n−1 +· · · ¨f |u|j+1 +j +� +vim+1� += v′im+1 +w′ +2 = ¨f |u|i+|u|j+m+1 +i +(w2) = v′′(i + 1)m+1 +w′′ +2 = ¨f |u|i+m +j+1 +¨f |u|i+m +j+2 +· · · ¨f |u|i+m +n−1 +¨f |u|i+m +n +¨f |u|i+m +n−1 +· · · ¨f |u|i+m +i+1 +(w′ +2) = v′′′j +m+1. +As ujim ¨∼ vim+1, we get that w′′ +1 ¨∼ w′′ +2. Note that w′′ +1 is obtained from ujim by +replacing each i by j and each j by i, and since |v|i = |u|i − 1 and |v|j = |u|j + 1, +w′′ +2 is obtained from vim+1 by replacing each i by j and each j by i. In particular, +u′′′, v′′′ ∈ +� +1, 2, . . ., i − 1, j, i, i − 1, . . . , 1 +�∗. We have by Corollary 9.10 that +jmiu′′′ = w′′ +1 ¨∼ w′′ +2 = jm+1v′′′, +which is a contradiction by (1). +□ +From Propositions 9.27 and 9.28, we have for words u, v ∈ C∗ +n with length at +most 2 that u ¨∼ v implies u = v. In the following result, we identify which words +of length 3 are hypoplactic congruent, and obtain that for distinct words, it comes +under the statement of Theorem 7.5. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +53 +Theorem 9.29. Let x, y, z, x′, y′, z′ ∈ Cn. Then, xyz ¨∼ x′y′z′ in C¨∗ +n if and only if +xyz = x′y′z′ or xyz, x′y′z′ ∈ +� +a11, 1a1, 11a +� +, for some a ∈ Cn. +Proof. As 11 is an isolated word, we have by Theorem 7.5 that a11 ¨∼ 1a1 ¨∼ 11a, +for any a ∈ Cn. And so, the converse implication holds. +Assume that xyz ¨∼ x′y′z′, that is, there exists a quasi-crystal isomorphism +between the connected components C¨∗ +n(xyz) and C¨∗ +n(x′y′z′) mapping xyz to x′y′z′. +By Propositions 6.10 and 6.11(2), we have that all words in C¨∗ +n(xyz) have exactly +three letters, and C¨∗ +n(xyz) has at least one highest-weight word. So, we first suppose +that xyz is a highest-weight word. As xyz ¨∼ x′y′z′, we get that if xyz is isolated, +so is x′y′z′, and if xyz is of highest weight but not isolated, so is x′y′z′. So, in the +following, we consider this two cases separately. +By Proposition 9.16, the isolated words in C¨∗ +n with three letters are 111, 11 1, +122, 22 1, or of the form ttt, for t ∈ Cn. If xyz and x′y′z′ are among these words and +xyz ̸= x′y′z′, then as wt(xyz) = wt(x′y′z′), we must have that xyz and x′y′z′ lie +in +� +111, 122, 111 +� +or +� +11 1, 22 1, 111 +� +. Since both 122 and 221 have a 2-inversion, +we get by Corollary 9.10 that 111 ̸¨∼ 122 ̸¨∼ 111 and 11 1 ̸¨∼ 22 1 ̸¨∼ 111. Thus, if xyz +and x′y′z′ are isolated and xyz ̸= x′y′z′, then they lie in +� +111, 111 +� +or +� +11 1, 111 +� +. +By Propositions 9.11 and 9.16, the highest-weight words in C¨∗ +n, which are not +isolated and consist of three letters, are 111, 112, 121, 122, 212, 123 (if n ≥ 3), 112, +121, 211, of the form ii(i + 1), for i ∈ {1, . . . , n − 1}, or of the form (j + 1)j + 1 j, +for j ∈ {2, . . . , n − 1}. If xyz and x′y′z′ are among these words and xyz ̸= x′y′z′, +then as wt(xyz) = wt(x′y′z′), we must have that xyz and x′y′z′ lie in {112, 121}, +{122, 212}, +� +112, 121, 211, 112 +� +. +We get by Proposition 9.28(1) that 212 ̸¨∼ 122 +and by Proposition 9.28(2) that 112 ̸¨∼ 121. Since 112 is 1-inversion-free, we have +by Corollary 9.10 that 112 ̸¨∼ w, for any w ∈ +� +112, 121, 211 +� +. Thus, if xyz and +x′y′z′ are of highest weight, but not isolated, and xyz ̸= x′y′z′, then they lie in +� +112, 121, 211 +� +. +We have that +C∗ +n +� +112 +� += +� +11a +�� a ∈ Cn, 2 ≤ a ≤ 2 +� +, +C∗ +n +� +121 +� += +� +1a1 +�� a ∈ Cn, 2 ≤ a ≤ 2 +� +, +and +C∗ +n +� +211 +� += +� +a11 +�� a ∈ Cn, 2 ≤ a ≤ 2 +� +. +And so, if xyz and x′y′z′ lie in some of these connected components, then as +wt(xyz) = wt(x′y′z′), we obtain that xyz and x′y′z′ lie in +� +11a, 1a1, a11 +� +, for +some a ∈ Cn with 2 ≤ a ≤ 2. +Therefore, for any x, y, z, x′, y′, z′ ∈ Cn such that xyz ̸= x′y′z′ and xyz ¨∼ x′y′z′, +we have that xyz and x′y′z′ lie in +� +11a, 1a1, a11 +� +, for some a ∈ Cn. +□ +From the previous result, we get that the Knuth relations (Definition 8.1) only +hold in hypo(Cn) for instances that come under the statement of Theorem 7.5. +Corollary 9.30. Let x, y, z ∈ Cn. Then, yzx ¨∼ yxz in C¨∗ +n if and only if x = y = z +or (y, x) = +� +1, 1 +� +) or (y, z) = +� +1, 1 +� +). Also, xzy ¨∼ zxy in C¨∗ +n if and only if x = y = z +or (x, y) = +� +1, 1 +� +) or (z, y) = +� +1, 1 +� +). +9.5. Identities. We start by checking some properties of the identities satisfied by +hypo(C2). +Theorem 9.31. Let X be a finite alphabet, and let u, v ∈ X∗. If hypo(C2) satisfies +the identity u = v, then the following conditions are satisfied: +(1) |u|x = |v|x, for all x ∈ X; + +54 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +(2) until the first occurrence of a letter x ∈ X in u and v, each letter of X +occurs exactly the same number of times in u and v, that is, if u = u1xu2 +and v = v1xv2, where u1, u2, v1, v2 ∈ X∗ are such that |u1|x = |v1|x = 0, +then |u1|y = |v1|y, for all y ∈ X; +(3) after the last occurrence of a letter x ∈ X in u and v, each letter of X +occurs exactly the same number of times in u and v, that is, if u = u1xu2 +and v = v1xv2, where u1, u2, v1, v2 ∈ X∗ are such that |u2|x = |v2|x = 0, +then |u2|y = |v2|y, for all y ∈ X. +Proof. Let x ∈ X. +(1) If we consider the map from X to C∗ +2 that sends x to 1 and each other letter +of X to ǫ, we obtain that 1|u|x ¨∼ 1|v|x. So, wt +� +1|u|x� += wt +� +1|v|x� +which implies +that |u|x = |v|x. +(2) Since |u|x = |v|x, we have that x occurs in u if and only if x occurs in v. And +if so, there exist u1, u2, v1, v2 ∈ X∗ such that u = u1xu2, v = v1xv2, and x does not +occur in u1 or v1. Given y ∈ X, consider the monoid homomorphism ψ : X∗ → C∗ +2 +induced by ψ(x) = 1, ψ(y) = 2 and ψ(z) = ǫ, for each z ∈ X \ {x, y}. Then, +2|u1|y1ψ(u2) = ψ(u) ¨∼ ψ(v) = 2|v1|y1ψ(v2), +which implies that |u1|y = |v1|y, by Proposition 9.28(1). +(3) Since |u|x = |v|x, we have that x occurs in u if and only if x occurs in v. And +if so, there exist u1, u2, v1, v2 ∈ X∗ such that u = u1xu2, v = v1xv2, and x does not +occur in u2 or v2. Given y ∈ X, consider the monoid homomorphism ψ : X∗ → C∗ +2 +induced by ψ(x) = 2, ψ(y) = 1 and ψ(z) = ǫ, for each z ∈ X \ {x, y}. Then, +ψ(u1)21|u2|y = ψ(u) ¨∼ ψ(v) = ψ(v1)21|v2|y, +which implies that |u2|y = |v2|y, by Proposition 9.28(2). +□ +Theorem 9.32. The hypoplactic monoid hypo(C2) satisfies the identity xyxyxy = +xyyxxy, that is, uvuvuv ¨∼ uvvuuv in C¨∗ +2, for any u, v ∈ C∗ +2. +Proof. Let u, v ∈ C∗ +2. We first assume that uvuvuv is of highest weight, that is, +¨ε1, (uvuvuv), ¨ε2(uvuvuv) ∈ {0, +∞}. If 2 occurs in uvuvuv, then uvuvuv has a +2-inversion, because it is of highest weight. Hence 2 also occurs in uvuvuv. In this +case, 2 and 2 occur in uv and vu, implying that both uvuvuv and uvvuuv have +a decomposition of the form w12w22w32w4, for some w1, w2, w3, w4 ∈ C∗ +2. Hence, +uvuvuv and uvvuuv are isolated words as they have 1- and 2-inversions. Since +wt(uvuvuv) = 3 wt(u) + 3 wt(v) = wt(uvvuuv), +we get by Lemma 9.20 that uvuvuv ¨∼ uvvuuv. So, we now assume that 2 does not +occur in uvuvuv. +If 1 occurs in uvuvuv, then uvuvuv has a 1-inversion, because it is of highest +weight. Since 2 does not occur in uvuvuv, we get that 1 and 1 occur in uv and +vu, implying that both uvuvuv and uvvuuv have a decomposition of the form +w11w21w3, for some w1, w2, w3 ∈ +� +1, 2, 1 +�∗. As 11 is an isolated word, we have by +Theorem 7.5 that +uvuvuv ¨∼ 13|u|1+3|v|123|u|2+3|v|21 +3|u|1+3|v|1 ¨∼ uvvuuv. +Thus, we further assume that 1 does not occur in uvuvuv. +If 2 occurs in uvuvuv, then uvuvuv has a 1-inversion, because it is of highest +weight. Since 2 and 1 do not occur in uvuvuv, we get that uv, vu ∈ {1, 2}∗, and 1 +and 2 occur in uv, implying that there exist w1, w2, w3, w4 ∈ {1, 2}∗ such that uv = + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +55 +w11w2 and uv = w32w4. As |w2uvw3|1 = |w2vuw3|1 and |w2uvw3|2 = |w2vuw3|2, +we have by Lemma 9.24 that +uvuvuv = w11w2uvw32w4 ¨∼ w11|w2uvw3|1+12|w2uvw3|2+1w4 +¨∼ w11w2vuw32w4 = uvvuuv. +Finally, if we also assume that 2 does not occur in uvuvuv, then u = 1|u| and +v = 1|v|, which implies that uvuvuv = uvvuuv. +Therefore, we obtain that uvuvuv ¨∼ uvvuuv, for any u, v, ∈ C∗ +2 such that uvuvuv +is of highest weight. We now show that this also holds when uvuvuv is not of highest +weight. +Suppose there exist u, v ∈ C∗ +2 such that uvuvuv ̸¨∼ uvvuuv. The set +W = +� +u′v′u′v′u′v′ �� u′, v′ ∈ C∗ +2, u′v′u′v′u′v′ ̸¨∼ u′v′v′u′u′v′, |u′| = |u|, |v′| = |v| +� +is nonempty and finite, so we can take words u′, v′ ∈ C∗ +2 such that u′v′u′v′u′v′ ̸¨∼ +u′v′v′u′u′v′, |u′| = |u|, |v′| = |v|, and u′v′u′v′u′v′ has maximal weight among +weights of words in W, that is, if w ∈ W and wt(w) ≥ wt(u′v′u′v′u′v′), then +wt(w) = wt(u′v′u′v′u′v′). +As shown above, we have that u′v′u′v′u′v′ is not of +highest weight. +Take i ∈ {1, 2} such that ¨ei is defined on u′v′u′v′u′v′. +Then, +u′v′u′v′u′v′ does not have an i-inversion, ¨εi(u′v′u′v′u′v′) = 3¨εi(u′) + 3¨εi(v′) ∈ Z>0, +and +¨e¨εi(u′v′u′v′u′v′) +i +(u′v′u′v′u′v′) += ¨e¨εi(u′) +i +(u′)¨e¨εi(v′) +i +(v′)¨e¨εi(u′) +i +(u′)¨e¨εi(v′) +i +(v′)¨e¨εi(u′) +i +(u′)¨e¨εi(v′) +i +(v′). +Set u′′ = ¨e¨εi(u′) +i +(u′) and v′′ = ¨e¨εi(v′) +i +(v′). By Proposition 3.5, wt(u′′v′′u′′v′′u′′v′′) > +wt(u′v′u′v′u′v′), and by Proposition 6.10, |u′′| = |u′| = |u| and |v′′| = |v′| = |v|. +As the weight of u′v′u′v′u′v′ is maximal among weights of words in W, we get that +u′′v′′u′′v′′u′′v′′ /∈ W, which implies that u′′v′′u′′v′′u′′v′′ ¨∼ u′′v′′v′′u′′u′′v′′. We have +by Definition 3.1(4) that ¨fi is defined on u′′v′′u′′v′′u′′v′′, and so, u′′v′′u′′v′′u′′v′′ +does not have an i-inversion. Since u′′v′′u′′v′′u′′v′′ ¨∼ u′′v′′v′′u′′u′′v′′, we get that +¨ϕi(u′′v′′v′′u′′u′′v′′) = ¨ϕi(u′′v′′u′′v′′u′′v′′) = 3 ¨ϕi(u′′) + 3 ¨ϕi(v′′), and as +¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) +i +(u′′v′′u′′v′′u′′v′′) += ¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) +i +� +¨e¨εi(u′v′u′v′u′v′) +i +(u′v′u′v′u′v′) +� += u′v′u′v′u′v′ +and +¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) +i +(u′′v′′v′′u′′u′′v′′) += ¨f ¨ϕi(u′′) +i +(u′′) ¨f ¨ϕi(v′′) +i +(v′′) ¨f ¨ϕi(v′′) +i +(v′′) ¨f ¨ϕi(u′′) +i +(u′′) ¨f ¨ϕi(u′′) +i +(u′′) ¨f ¨ϕi(v′′) +i +(v′′) += u′v′v′u′u′v′, +we obtain that u′v′u′v′u′v′ ¨∼ u′v′v′u′u′v′, which is a contradiction. +Therefore, +for any u, v ∈ C∗ +2, we have that uvuvuv ¨∼ uvvuuv, that is, hypo(C2) satisfies the +identity xyxyxy = xyyxxy. +□ +We now turn our attention for whether hypo(Cn) satisfies identities when n ≥ 3. +In the following results, we prove that hypo(Cn) does not satisfy any nontrivial +identity, for n ≥ 3. This is achieved by showing that it contains free submonoids +with more than one generator. +Lemma 9.33. Consider n ≥ 3. Let u, v ∈ +� +1, 2 +�∗. Then, u ¨∼ v in C¨∗ +n if and only +if u = v. + +56 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +Proof. Suppose that u ¨∼ v and u ̸= v. Since wt(u) = wt(v), we have that |u|1 = |v|1 +and |u|2 = |v|2, and as u ̸= v, both 1 and 2 occur in u and v. Take u′, v′, w ∈ +� +1, 2 +�∗ +such that u = w1u′ and v = w2v′. Since |u|1 = |v|1 and |u|2 = |v|2, we have that +|v′|1 = |u′|1 + 1 and |u′|2 = |v′|2 + 1. +The words u and v do not have 2-inversions, because neither 2 nor 3 occurs in +them. Set +u1 = ¨e +|v′|2 +2 +(u) = w1¨e +|v′|2 +2 +(u′) +and +v1 = ¨e +|v′|2 +2 +(v) = w2¨e +|v′|2 +2 +(v′). +Note that ¨e +|v′|2 +2 +(v′) is obtained from v′ by replacing each 2 by 3, and as |v′|2 = +|u′|2 − 1, ¨e +|v′|2 +2 +(u′) is obtained from u′ by replacing each 2 by 3 except for the left- +most 2 that remains unchanged. In particular, 2 occurs in ¨e +|v′|2 +2 +(u′) and does not +occur in ¨e +|v′|2 +2 +(v′). As u ¨∼ v, we also have that u1 ¨∼ v1. +The words u1 and v1 do not have 1-inversions, because neither 2 nor 1 occurs in +them. Set +u2 = ¨f |w|+1 +1 +(u1) = ¨f |w| +1 +(w)2¨e +|v′|2 +2 +(u′) +and +v2 = ¨f |w|+1 +1 +(v1) = ¨f |w| +1 +(w)1¨e +|v′|2 +2 +(v′). +Note that ¨f |w| +1 +(w) is obtained from w by replacing each 1 by 2 and each 2 by 1. +Since 2 occurs in ¨e +|v′|2 +2 +(u′), we have that u2 has a 2-inversion, and since neither 3 +nor 2 occurs in v2, we have that v2 does not have a 2-inversion. By Proposition 9.5, +this is a contradiction, because u1 ¨∼ v1 implies that u2 ¨∼ v2. +□ +Theorem 9.34. Consider n ≥ 3. Let a, b ∈ Cn with a ̸= b, and let u, v ∈ {a, b}∗. +Then, u ¨∼ v in C¨∗ +n if and only if u = v. +Proof. Take i, j ∈ {1, . . ., n} such that a ∈ +� +i, i +� +and b ∈ +� +j, j +� +. Since a ̸= b and the +case a = b is trivial, without loss of generality we assume i < j. By Corollary 9.10, +we have that u ¨∼ v if and only if u ¨∼ v, and so, without loss of generality, we +further assume that a = i. +Suppose that u ¨∼ v. +As wt(u) = wt(v) and i ̸= j, we get that |u|a = |v|a +and |u|b = |v|b. If i = 1, set u′ = u and v′ = v, otherwise, u and v do not have +(i − 1)-inversions, because neither i − 1 nor i occurs in them as i < j, and so, set +u′ = ¨e|u|a +1 +¨e|u|a +2 +· · · ¨e|u|a +i−1(u) +and +v′ = ¨e|v|a +1 +¨e|v|a +2 +· · · ¨e|v|a +i−1(v). +As a = i, note that u′ and v′ are respectively obtained from u and v by replacing +each a by 1. +In particular, u′, v′ ∈ {1, b}∗ and |u′|b = |v′|b. +Since u ¨∼ v and +|u|a = |v|a, we get that u′ ¨∼ v′. +In the following cases, we obtain words u′′, v′′ ∈ +� +1, 2 +� +by applying the same and +in the same order quasi-Kashiwara operators to u′ and v′. +• Case 1: b = j. +The words u′ and v′ do not have j-inversions, because +neither j + 1 nor j occurs in them as 1 ≤ i < j. Set +u′′ = ¨f |u′|b +j +¨f |u′|b +j+1 · · · ¨f |u′|b +n +¨f |u′|b +n−1 · · · ¨f |u′|b +2 +(u′) +and +v′′ = ¨f |v′|b +j +¨f |v′|b +j+1 · · · ¨f |v′|b +n +¨f |v′|b +n−1 · · · ¨f |v′|b +2 +(v′). +• Case 2: b = j. If j = 2, set u′′ = u′ and v′′ = v′, otherwise, we have that +j ≥ 3, as j > i ≥ 1, and the words u′ and v′ do not have (j − 1)-inversions, +because neither j nor j − 1 occurs in them, and so, set +u′′ = ¨f |u′|b +j−1 ¨f |u′|b +j−2 · · · ¨f |u′|b +2 +(u′) +and +v′′ = ¨f |v′|b +j−1 ¨f |v′|b +j−2 · · · ¨f |v′|b +2 +(v′). + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +57 +In either case, note that u′′ and v′′ are respectively obtained from u′ and v′ by +replacing each b by 2. In particular, u′′, v′′ ∈ +� +1, 2 +�∗. Since u′ ¨∼ v′ and |u′|b = |v′|b, +we get that u′′ ¨∼ v′′. By Lemma 9.33, we obtain that u′′ = v′′, and since the quasi- +Kashiwara operators are injective when defined (Definition 3.1(4)), we deduce that +u = v. +□ +From the previous result we have when n ≥ 3 that for a, b ∈ Cn with a ̸= b, the +set {a, b} is free on hypo(Cn). This marks a difference when compared to hypo(C2), +where {1, 2} is not free, as shown in Lemma 9.24. This also implies the following +result. +Corollary 9.35. Let n ≥ 3. Then, hypo(Cn) does not satisfy nontrivial identities. +9.6. Presentations. We first show that the hypoplactic monoid hypo(C2) does not +admit a finite presentation. +Theorem 9.36. The hypoplactic congruence ¨∼ on the free monoid C∗ +2 is not finitely +generated. Therefore, hypo(C2) has no finite presentation. +Proof. Let R be a finite subset of ¨∼, and denote by ∼R the monoid congruence on +C∗ +2 generated by R. Thus, ∼R ⊆ ¨∼. As R is finite, set +m = 1 + max +(u,v)∈R +� +|u|, |v| +� +. +We first show that for any subword u of 121m2 with length at most m (that is, +any word u consisting of at most m consecutive letters of 121m2) and v ∈ C∗ +2, if +u ¨∼ v, then u = v. Let u, v ∈ C∗ +2 be such that u ¨∼ v and u is a subword of 121m2. +Then, u ∈ {1, 2}∗, and by Lemma 9.26, v ∈ {1, 2}∗. As u is a subword of 121m2 +with length at most m, we get that u lies in one of the following cases. +• Case 1: u = 1p where p ∈ Z≥0. Since u ¨∼ v, we have that wt(u) = wt(v), +and since v ∈ {1, 2}∗, we get that |v|1 = |u|1 = p and |v|2 = |u|2 = 0. +Hence, v = 1p = u. +• Case 2: u = 1p121p2 where p1, p2 ∈ Z≥0. As in the previous case, we have +that |v|1 = p1 + p2 and |v|2 = 1. So, v = 1q121q2, for some q1, q2 ∈ Z≥0 +such that q1 + q2 = p1 + p2. By Lemma 9.23, 1p121p2 ¨∼ 1q121q2 implies +that p1 = q1 and p2 = q2. Hence, v = u. +Since R ⊆ ¨∼ and every pair (u, v) ∈ R satisfies |u| < m and |v| < m, we get +that if (u, v) ∈ R and u or v are subwords of 121m2, then u = v. As R generates +∼R, we obtain for w ∈ C∗ +2 that 121m2 ∼R w if and only if w = 121m2. On the +other hand, we have that 121m2 ̸= 1m+122, and by Lemma 9.24, 121m2 ¨∼ 1m+122. +Hence, ∼R ̸= ¨∼. And therefore, ¨∼ is not finitely generated. +□ +Although, hypo(C2) does not have a finite presentation, we are able to describe +connected components of ΓC¨∗ +2 , and thus, find representatives for the hypoplactic +congruence classes. +Lemma 9.37. Let u, v, w ∈ +� +1, 2, 1 +�∗. Then, 2u1v1w2 ¨∼ 1m+12p+21 +q+1 in C¨∗ +2, for +some m, p, q ∈ Z≥0 where m = 0 or q = 0. +Proof. Set m = max +� +0, |u|1 + |v|1 + |w|1 − |u|1 − |v|1 − |w|1 +� +, p = |u|2 + |v|2 + |w|2, +and q = max +� +0, |u|1 + |v|1 + |w|1 − |u|1 − |v|1 − |w|1 +� +. We first show that each +connected component ΓC¨∗ +2 +� +2u1v1w2 +� +and ΓC¨∗ +2 +� +1m+12p+21 +q+1� +is a path with p + 3 +vertices. The paths ΓC¨∗ +2 +� +2u1v1w2 +� +and ΓC¨∗ +2 +� +1m+12p+21 +q+1� +start respectively in +2u1v1w2 and 1m+12p+21 +q+1, which are highest-weight words without 2-inversions, +as 2 does not occur in them. From these starting-points, there is a sequence of p+2 +edges labelled by 2, each of which transforms a symbol 2 to a symbol 2, in order + +58 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +from left to right through the word. The end-points of the paths ΓC¨∗ +2 +� +2u1v1w2 +� +and ΓC¨∗ +2 +� +1m+12p+21 +q+1� +are 2 ¨f |u|2 +2 +(u)1 ¨f |v|2 +2 +(v)1 ¨f |w|2 +2 +(w)2 and 1m+12 +p+21 +q+1, re- +spectively, which are lowest-weight words. Also, each vertex of the paths has a +loop labelled by 1, because it admits a decomposition of the form w11w22w3 or +w12w21w3, for some w1, w2, w3 ∈ C∗ +2. +Define a bijection ψ : C∗ +2 +� +2u1v1w2 +� +→ C∗ +2 +� +1m+12p+21 +q+1� +that maps each word +w ∈ C∗ +2 +� +2u1v1w2 +� +to the word ψ(w) ∈ C∗ +2 +� +1m+12p+21 +q+1� +such that the position +of w in ΓC¨∗ +2 +� +2u1v1w2 +� +is the same as ψ(w) in ΓC¨∗ +2 +� +1m+12p+21 +q+1� +. +As shown +above, for u, v ∈ C∗ +2 +� +2u1v1w2 +� +and i ∈ {1, 2}, we have that u +i +−−−→ v is an edge +in ΓC¨∗ +2 +� +2u1v1w2 +� +if and only if ψ(u) +i +−−−→ ψ(v) is an edge of ΓC¨∗ +2 +� +1m+12p+21 +q+1� +. +And since ψ +� +2u1v1w2 +� += 1m+12p+21 +q+1, where +wt +� +2u1v1w2 +� += (m − q, p + 2) = wt +� +1m+12p+21 +q+1� +, +we get that ψ preserves weights. Therefore, by Theorem 4.13, ψ is a quasi-crystal +isomorphism, which implies that 2u1v1w2 ¨∼ 1m+12p+21 +q+1. +□ +Theorem 9.38. Any connected component of C¨∗ +2 is quasi-crystal isomorphic to one +and only one of the following: +(1) C¨∗ +2 +� +1m� +, m ∈ Z≥0; +(2) C¨∗ +2 +� +2m11m2+12m3+11m4� +, m1, m2, m3, m4 ∈ Z≥0; +(3) C¨∗ +2 +� +1m1+12m21 +m3+1� +, m1, m2, m3 ∈ Z≥0 with m1 = 0 or m3 = 0; +(4) C¨∗ +2 +� +1m1+12m2+12 +m3+11 +m4+1� +, m1, m2, m3, m4 ∈ Z≥0 with m1 = 0 or m4 = +0, and m2 = 0 or m3 = 0. +Therefore, the elements in these connected components form a minimal set of rep- +resentatives for the hypoplactic congruence classes on C∗ +2. +Proof. By Proposition 6.11(2) any connected component of C¨∗ +2 has at least a highest- +weight word. Let w ∈ C∗ +2 be a highest-weight word of C¨∗ +2. If 2 occurs in w, then w +has a 2-inversion, because it is of highest weight. This implies that 2 occurs in w, +and as w is of highest weight, w has a 1-inversion. Hence, w is an isolated word. +For each i ∈ {1, 2}, if |w|i ≥ |w|i, set mi = |w|i − |w|i and m5−i = 0, otherwise, set +mi = 0 and m5−i = |w|i − |w|i. Then, +wt(w) = (m1 − m4, m2 − m3) = wt +� +1m1+12m2+12 +m3+11 +m4+1� +, +which implies by Lemma 9.20 that w ¨∼ 1m1+12m2+12 +m3+11 +m4+1, and so, we have +a quasi-crystal isomorphism between C¨∗ +2(w) and C¨∗ +2 +� +1m1+12m2+12 +m3+11 +m4+1� +. So, +in the following, we assume that 2 does not occur in w. +If 1 occurs in w, then w has a 1-inversion, because it is of highest weight. Since +2 does not occur in w, then we have in w that a 1 appears to the right of a 1, or +otherwise, 2 appears to the right of a 1. In this second case, as 1 occurs in w, we +may have that a 1 appears to the right of a 2, or otherwise, every 1 occurs to the +left of any 1 and 2. Based on these decompositions, we get that w lies in one of the +following cases. +• Case 1: w = w11w21w3, for some w1, w2, w3 ∈ +� +1, 2, 1 +�∗. Since 11 is an +isolated word, we get by Theorem 7.5 that w ¨∼ 1w1w2w31, and by iterating +this process, w ¨∼ 1|w|12|w|21 +|w|1. +Set m2 = |w|2. +If |w|1 ≥ |w|1, set +m1 = |w|1 − |w|1 and m3 = 0, otherwise, set m1 = 0 and m3 = |w|1 − |w|1. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +59 +Since wt +� +11 +� += 0, we get by Theorems 7.5 and 7.6 that +w ¨∼ 1|w|12m21 +|w|1 ¨∼ 1|w|11 +|w|12m2 ¨∼ 1m1+11 +m3+12m2 +¨∼ 1m1+12m21 +m3+1. +In particular, there exists a quasi-crystal isomorphism between C¨∗ +2(w) and +C¨∗ +2 +� +1m1+12m21 +m3+1� +. +• Case 2: w = w12w21w31w42w5, for some w1, w2, w3, w4, w5 ∈ +� +1, 2, 1 +�∗. +By Lemma 9.37, we have that 2w21w31w42 ¨∼ 1p1+12p2+21 +p3+1, for some +p1, p2, p3 ∈ Z≥0. As in the previous case, we get that +w = w11p1+12p2+21 +p3+1w5 ¨∼ 1m1+12m21 +m3+1, +for some m1, m2, m3 ∈ Z≥0 with m1 = 0 or m3 = 0. Hence, there exists a +quasi-crystal isomorphism between C¨∗ +2(w) and C¨∗ +2 +� +1m1+12m21 +m3+1� +. +• Case 3: w = 1 +pu, for some u ∈ {1, 2}∗ and p ∈ Z>0. By Proposition 9.25, +we have that u ¨∼ 2q11q22q31q4, for some q1, q2, q3, q4 ∈ Z≥0. In particular, +|w|2 = |u|2 = q1 + q3. Since w has a 1-inversion, then u has a 1-inversion, +which implies by Proposition 9.5 that q2 > 0 and q3 > 0. Note that +¨ep+q1+q3 +2 +¨ep +1 ¨f q1+q3 +2 +� +1 +p2q11q22q31q4� += ¨ep+q1+q3 +2 +¨ep +1 +� +1 +p2 +q11q22 +q31q4� += ¨ep+q1+q3 +2 +� +2 +p2 +q11q22 +q31q4� += 2p+q11q22q31q4. +Since u ¨∼ 2q11q22q31q4, we get that w ¨∼ 1 +p2q11q22q31q4, and as |w|2 = +q1 + q3, we obtain that +¨e|w|2+p +2 +¨ep +1 ¨f |w|2 +2 +(w) ¨∼ 2p+q11q22q31q4. +Set m1 = p+q1, m2 = q2 −1, m3 = q3 −1 and m4 = q4. Then, there exists +a quasi-crystal isomorphism between C¨∗ +2(w) and C¨∗ +2 +� +2m11m2+12m3+11m4� +. +So, we further assume that 1 does not occur in w. +If 2 occurs in w, then w has a 1-inversion, because it is of highest weight. As +neither 1 nor 2 occurs in w, we have that w ∈ {1, 2}∗, and by Proposition 9.25, +w ¨∼ 2p11p22p31p4, for some p1, p2, p3, p4 ∈ Z≥0. Since w has a 1-inversion, we get +by Proposition 9.5 that p2 > 0 and p3 > 0. Set m1 = p1, m2 = p2 − 1, m3 = p3 − 1 +and m4 = p4. Then, there exists a quasi-crystal isomorphism between C¨∗ +2(w) and +C¨∗ +2 +� +2m11m2+12m3+11m4� +. +Finally, if 2 does not occur in w, then w = 1m, where m = |w|. And so, C¨∗ +2(w) +coincides with C¨∗ +2(1m). +We have thus proved that any connected component of C¨∗ +2 is quasi-crystal iso- +morphic to some connected component lying in (1) to (4). It remains to show that +it is quasi-crystal isomorphic to only one of such connected components. So, we +now show that there are no quasi-crystal isomorphic connected components among +the ones in (1) to (4). +Each connected component C¨∗ +2 +� +1m1+12m2+12 +m3+11 +m4+1� +in (4) consists of an +isolated word with weight (m1 − m4, m2 − m3), a 1-inversion and a 2-inversion. By +the condition that m1 = 0 or m4 = 0, and m2 = 0 or m3 = 0, we have that all +words in (4) have different weights, which implies by Lemma 9.20 that they are not +hypoplactic congruent. Also, all connected components in (1) to (3) contain a word +without a 2-inversion, and so, there is no quasi-crystal isomorphism between any +of them and some connected component in (4). +Each connected component C¨∗ +2 +� +1m1+12m21 +m3+1� +in (3) is formed by m2 + 1 +words of the form 1m1+12 +p12p21 +m3+1, for p1, p2 ∈ Z≥0 such that p1 + p2 = m2. + +60 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +In particular, each connected component has exactly one highest-weight word, +namely: 1m1+12m21 +m3+1. So, if C¨∗ +2 +� +1m1+12m21 +m3+1� +and C¨∗ +2 +� +1m′ +1+12m′ +21 +m′ +3+1� +are +quasi-crystal isomorphic connected components lying in (3), then we have that +1m1+12m21 +m3+1 ¨∼ 1m′ +1+12m′ +21 +m′ +3+1, which implies that m1 − m3 = m′ +1 − m′ +3 and +m2 = m′ +2, and by the condition that m1 = 0 or m3 = 0 and the condition that +m′ +1 = 0 or m′ +3 = 0, we obtain that m1 = m′ +1 and m3 = m′ +3. Hence, there are no +quasi-crystal isomorphic connected components among the ones in (3). Also, all +words lying in the connected components in (3) have 1-inversions, each connected +component in (1) or (2) have at least one word without a 1-inversion (respectively, +1m or 2 +m11m2+12 +m3+11m4), and so, there is no quasi-crystal isomorphism between +some connected component in (3) and some connected component in (1) or (2). +The only word lying in a connected component C¨∗ +2 +� +2m11m2+12m3+11m4� +in (2) +and the set {1, 2}∗ is 2m11m2+12m3+11m4. +So, if C¨∗ +2 +� +2m11m2+12m3+11m4� +and +C¨∗ +2 +� +2m′ +11m′ +2+12m′ +3+11m′ +4� +are quasi-crystal isomorphic connected components in (2), +then we have by Lemma 9.26 that 2m11m2+12m3+11m4 +¨∼ 2m′ +11m′ +2+12m′ +3+11m′ +4, +which implies by Proposition 9.28 that m1 = m′ +1, m2 = m′ +2, m3 = m′ +3 and +m4 = m′ +4. +Hence, there are no quasi-crystal isomorphic connected components +among the ones in (2). Also, each connected component in (2) contains at least one +word with a 2-inversion (for instance, 1 +m12m2+12 +m3+11m4), while no word lying +in some connected component in (1) has a 2-inversion, and so, there is no quasi- +crystal isomorphism between some connected component in (2) and some connected +component in (1). +Finally, note that the only highest-weight word lying in a connected component +C¨∗ +2 +� +1m� +in (1) and the set {1, 2}∗ is 1m. If C¨∗ +2 +� +1m� +and C¨∗ +2 +� +1m′� +are quasi-crystal +isomorphic connected components in (1), then we have by Lemma 9.26 that 1m ¨∼ +1m′, which implies that m = m′. Hence, there is no quasi-crystal isomorphism +between distinct connected components among the ones in (1). +□ +Finally, we show that hypo(Cn) does not admit a finite presentation for n ≥ 3. +Theorem 9.39. Consider n ≥ 3. The hypoplactic congruence ¨∼ on C∗ +n is not +finitely generated. Therefore, hypo(Cn) has no finite presentation. +Proof. Let R be a finite subset of ¨∼, and denote by ∼R the monoid congruence on +C∗ +n generated by R. Thus, ∼R ⊆ ¨∼. As R is finite, take +m = 1 + max +(u,v)∈R +� +|u|, |v| +� +. +Set w = 1m2m1m2 +m1 +m. If u is a subword of w with length at most m (that is, a +word u consisting of at most m consecutive letters of w), then u lies in {a, b}∗, for +some a, b ∈ +� +1, 2, 2, 1 +� +with a ̸= b, and if v ∈ C∗ +n is such that u ¨∼ v, we get by +Proposition 9.27 that v ∈ {a, b}∗, and then, we obtain by Theorem 9.34 that u = v. +Since R ⊆ ¨∼ and every pair (u, v) ∈ R satisfies |u| < m and |v| < m, we get that +if (u, v) ∈ R and u or v are subwords of w, then u = v. As R generates ∼R, we +have for w′ ∈ C∗ +n that w ∼R w′ if and only if w′ = w. On the other hand, we have +that w ̸= 12m2m2 +m1 +m, and as 1m2m2 +m1 +m is an isolated word, w ¨∼ 12m2m2 +m1 +m, +by Theorem 7.5. +This implies that ∼R ̸= ¨∼. +And therefore, ¨∼ is not finitely +generated. +□ +9.7. From hypo(An−1) to hypo(Cn). We first show that an embedding of hypo(An) +into hypo(Cn) cannot map each letter of An to a letter of Cn. +Proposition 9.40. For m ≥ 2 and n ≥ 2, there exists no injective monoid ho- +momorphism ψ : hypo(Am) → hypo(Cn) such that ψ(x), ψ(y) ∈ Cn, for some +x, y ∈ Am with x ̸= y. + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +61 +Proof. Suppose ψ : hypo(Am) → hypo(Cn) is an injective monoid homomorphism +such that ψ(x), ψ(y) ∈ Cn, for some x, y ∈ Am with x ̸= y. Without loss of gener- +ality assume x < y. Since xyx = xxy in hypo(Am), we get that ψ(x)ψ(y)ψ(x) = +ψ(x)ψ(x)ψ(y) in hypo(Cn), which implies by Corollary 9.30 that ψ(x) = ψ(y), or +ψ(x) = 1 and ψ(y) = 1. As ψ is injective, we must have that ψ(x) = 1 and ψ(y) = 1. +Since 11 is an isolated word of C¨∗ +n and wt +� +11 +� += 0, we get by Theorem 7.6 that +ψ(xyxy) = 1111 = 11 = ψ(xy) +in hypo(Cn), which is a contradiction as ψ is injective. +□ +We now show that an injective map between the relevant alphabets cannot be ex- +tended to a (not necessarily injective) homomorphism from the hypoplactic monoid +of type An to that of type Cn. +Proposition 9.41. For m ≥ 3 and n ≥ 2, no injective map from Am to Cn can be +extended to a monoid homomorphism from hypo(Am) to hypo(Cn). +Proof. Suppose ψ : hypo(Am) → hypo(Cn) is a monoid homomorphism where its +restriction ψ|Am is an injective map from Am to Cn. So, we can take y ∈ {2, 3} +such that ψ(y) ̸= 1 and ψ(y) ̸= ψ(1). Since 1y1 = 11y in hypo(Am), we obtain +that ψ(1)ψ(y)ψ(1) = ψ(1)ψ(1)ψ(y) in hypo(Cn), contradicting Corollary 9.30. +□ +Now, we show that hypo(An−1) can be embedded in hypo(Cn). +Theorem 9.42. Let n ≥ 3. Define a map ψ : A∗ +n−1 → C∗ +n by +ψ(w) = +� +ǫ +if w = ǫ +wnnnn +otherwise, +for each w ∈ A∗ +n−1. Then ψ factors to give an injective monoid homomorphism +from hypo(An−1) to hypo(Cn). +Proof. Denote the quasi-crystal structure of A¨∗ +n−1 by wtA, ¨eA +i , ¨f A +i , ¨εA +i and ¨ϕA +i (i ∈ +{1, . . . , n − 2}), and denote the hypoplactic congruence on A¨∗ +n−1 by ¨∼A. Similarly, +denote the quasi-crystal structure of C¨∗ +n by wtC, ¨eC +i , ¨f C +i , ¨εC +i and ¨ϕC +i (i ∈ {1, . . ., n}), +and denote the hypoplactic congruence on C¨∗ +n by ¨∼C. +From Example 6.12 and +Definition 9.2, it is immediate that gA(w) = gC(w), for any w ∈ A∗ +n−1 and g ∈ +� +¨ei, ¨fi, ¨εi, ¨ϕi +�� i ∈ {1, . . ., n − 2} +� +. +We now show that for u, v ∈ A∗ +n−1, u ¨∼A v if and only if ψ(u) ¨∼C ψ(v), which +implies that ψ induces a well-defined injective map from hypo(An−1) to hypo(Cn). +First, note that if u ¨∼A ǫ, for some u ∈ A∗ +n−1, then wtA(u) = 0, which implies that +u = ǫ. If u′ ¨∼C ǫ, for some u′ ∈ C∗ +n, then +|u′|i + |u′|i+1 ≤ ¨ϕC +i (u′) = ¨ϕC +i (ǫ) = 0, +for any i ∈ {1, . . . , n}, implying that u′ = ǫ. Let w ∈ A∗ +n−1 with w ̸= ǫ. We have +that +wtC(wnnnn) = +� +|w|1, |w|2, . . . , |w|n−1, 0 +� +, +which implies for w′ ∈ A∗ +n−1 that wtC(wnnnn) = wtC(w′nnnn) if and only if +wtA(w) = wtA(w′). Also, +¨εC +n−1(wnnnn) = ¨εC +n(wnnnn) = +∞, +which implies that ¨eC +n−1, ¨f C +n−1, ¨eC +n and ¨f C +n are undefined on wnnnn, and for i ∈ +{1, . . . , n − 2}, +¨εC +i (wnnnn) = ¨εC +i (w) = ¨εA +i (w) +and +¨ϕC +i (wnnnn) = ¨ϕC +i (w) = ¨ϕA +i (w), + +62 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +because ¨εC +i (nnnn) = ¨ϕC +i (nnnn) = 0. Since A¨∗ +n−1 and C¨∗ +n are seminormal, we get +that ¨f C +i is defined on wnnnn if and only if ¨f A +i +is defined on w, and if so, +¨f C +i (wnnnn) = ¨f C +i (w)nnnn = ¨f A +i (w)nnnn. +Hence, for u, v ∈ A∗ +n−1 and i ∈ {1, . . ., n − 2}, we have an edge u +i +−−−→ v in +ΓA¨∗ +n−1 if and only if we have an edge unnnn +i +−−−→ vnnnn in ΓC¨∗ +n. This implies that +ΓC¨∗ +n(wnnnn) is obtained from ΓA¨∗ +n−1(w) by concatenating nnnn to each vertex, and +adding (n−1)-labelled and n-labelled loops to each vertex. Equivalently, ΓA¨∗ +n−1(w) +is obtained from ΓC¨∗ +n(wnnnn) by removing the last four letters of each vertex, and +removing all (n − 1)-labelled and n-labelled loops. Therefore, for any u, v ∈ A∗ +n−1, +there exists a graph isomorphism between ΓA¨∗ +n−1(u) and ΓA¨∗ +n−1(v) mapping u to v if +and only if there exists a graph isomorphism between ΓC¨∗ +n(unnnn) and ΓC¨∗ +n(vnnnn) +mapping unnnn to vnnnn. By Theorem 4.13, we have that u ¨∼A v if and only if +unnnn ¨∼C vnnnn. +To obtain that ψ induces an injective monoid homomorphism from hypo(An−1) +to hypo(Cn), it remains to prove that ψ(uv) ¨∼C ψ(u)ψ(v), for any u, v ∈ A∗ +n−1, since +ψ(ǫ) = ǫ follows from the definition of ψ. As shown above, if uv ¨∼A ǫ, then uv = ǫ, +which implies that u = v = ǫ, and so, ψ(uv) ¨∼C ψ(u)ψ(v). By Proposition 9.18, +we have that nnnn is a commutative and idempotent element of hypo(Cn). So, for +any u′, v′ ∈ C∗ +n, we get that +u′nnnnv′nnnn ¨∼C u′v′(nnnn)2 ¨∼C u′v′nnnn, +in particular, for u, v ∈ A∗ +n−1 with u ̸= ǫ or v ̸= ǫ, we obtain that ψ(uv) ¨∼C +ψ(u)ψ(v). Therefore, we get that ψ induces an injective monoid homomorphism +from hypo(An−1) to hypo(Cn). +□ +9.8. From hypo(Cn−1) to hypo(Cn). The following result shows that we have a +monoid embedding from hypo(Cn−1) to hypo(Cn). +Theorem 9.43. Let n ≥ 3. Consider ψ to be the monoid homomorphism from +C∗ +n−1 to C∗ +n such that +ψ(x) = (x + 1)11 +and +ψ(x) = +� +x + 1 +� +11, +for each x ∈ {1, . . . , n − 1}. Then, ψ induces an injective monoid homomorphism +from hypo(Cn−1) to hypo(Cn). +Proof. Let τ be the monoid homomorphism from C∗ +n−1 to C∗ +n such that +τ(x) = x + 1 +and +τ(x) = x + 1, +for each x ∈ {1, . . ., n − 1}. Equivalently, for w ∈ C∗ +n−1, τ(w) is obtained from w +by replacing each x by x + 1 and each x by x + 1, for x ∈ {1, . . ., n − 1}. +For each m ∈ {n−1, n}, denote the quasi-crystal structure of C¨∗ +m by wt(m), ¨e(m) +i +, +¨f (m) +i +, ¨ε(m) +i +and ¨ϕ(m) +i +(i ∈ {1, . . . , m}), and denote the hypoplactic congruence on +C¨∗ +m by ¨∼(m). From Definition 9.2, for w ∈ C∗ +n−1 and i ∈ {1, . . ., n − 1}, note that w +has an i-inversion if and only if τ(w) has an (i + 1)-inversion, and so, we get that +¨ε(n) +i+1 +� +τ(w) +� += ¨ε(n−1) +i +(w) and ¨ϕ(n) +i+1 +� +τ(w) +� += ¨ϕ(n−1) +i +(w). Moreover, ¨e(n) +i+1 is defined +on τ(w) if and only if ¨e(n−1) +i +is defined on w, and if so, ¨e(n) +i+1 +� +τ(w) +� += τ +� +¨e(n−1) +i +(w) +� +. +Analogously, ¨f (n) +i+1 is defined on τ(w) if and only if ¨f (n−1) +i +is defined on w, and if so, +¨f (n) +i+1 +� +τ(w) +� += τ +� ¨f (n−1) +i +(w) +� +. +Since ψ is a monoid homomorphism from C∗ +n−1 to C∗ +n, to prove that ψ induces an +injective monoid homomorphism from hypo(Cn−1) to hypo(Cn), it suffices to show + +QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS +63 +that for any u, v ∈ C∗ +n−1, u ¨∼(n−1) v if and only if ψ(u) ¨∼(n) ψ(v). Note that for +m ∈ {n − 1, n} and u ∈ C∗ +m, if u ¨∼(m) ǫ, then +|u|i + |u|i+1 ≤ ¨ϕ(m) +i +(u) = ¨ϕ(m) +i +(ǫ) = 0, +for any i ∈ {1, . . . , m}, implying that u = ǫ. Let w ∈ C∗ +n−1 with w ̸= ǫ. Take +x1, . . . , xk ∈ Cn−1 such that w = x1 . . . xk. Since 11 is an isolated word of C¨∗ +n and +wt +� +11 +� += 0, we have by Theorems 7.5 and 7.6 that +ψ(w) = τ(x1)11τ(x2)11 . . . τ(xk)11 += τ(x1)τ(x2) . . . τ(xk) +� +11 +�k = τ(w)11. +We have that +wt(n)� +ψ(w) +� += +� +0, |w|1 − |w|1, |w|2 − |w|2, . . . , |w|n−1 − |w|n−1 +� +, +which implies that for w′ ∈ C∗ +n−1, wt(n)� +ψ(w) +� += wt(n)� +ψ(w′) +� +if and only if +wt(n−1)(w) = wt(n−1)(w′). Also, ¨ε(n) +1 +� +ψ(w) +� += +∞, which implies that ¨e(n) +1 +and +¨f (n) +1 +are undefined on ψ(w), and for i ∈ {2, . . . , n}, +¨ε(n) +i +� +ψ(w) +� += ¨ε(n) +i +� +τ(w) +� += ¨ε(n−1) +i−1 +(w) +and +¨ϕ(n) +i +� +ψ(w) +� += ¨ϕ(n) +i +� +τ(w) +� += ¨ϕ(n−1) +i−1 +(w), +because ¨ε(n) +i +� +11 +� += ¨ϕ(n) +i +� +11 +� += 0. Since C¨∗ +n−1 and C¨∗ +n are seminormal, we have that +¨f (n) +i +is defined on ψ(w) if and only if ¨f (n−1) +i−1 +is defined on w, and if so, +¨f (n) +i +� +ψ(w) +� += ¨f (n) +i +� +τ(w) +� +11 = τ +� ¨f (n−1) +i−1 +(w) +� +11 = ψ +� ¨f (n−1) +i−1 +(w) +� +. +Hence, for u, v ∈ C∗ +n−1 and i ∈ {1, . . ., n − 1}, we have an edge u +i +−−−→ v in ΓC¨∗ +n−1 if +and only if we have an edge ψ(u) +i +−−−→ ψ(v) in ΓC¨∗ +n. This implies that ΓC¨∗ +n +� +ψ(w) +� +is +obtained from ΓC¨∗ +n−1(w) by applying ψ to each vertex, and adding 1-labelled loops +to each vertex. As ψ is injective, ΓC¨∗ +n−1(w) can also be obtained from ΓC¨∗ +n +� +ψ(w) +� +by +reversing the described process. Therefore, for any u, v ∈ C∗ +n−1, there exists a graph +isomorphism between ΓC¨∗ +n−1(u) and ΓC¨∗ +n−1(v) mapping u to v if and only if there +exists a graph isomorphism between ΓC¨∗ +n +� +ψ(u) +� +and ΓC¨∗ +n +� +ψ(v) +� +mapping ψ(u) to +ψ(v). By Theorem 4.13, we have that u ¨∼(n−1) v if and only if ψ(u) ¨∼(n) ψ(v). +□ +By composing the homomorphisms from the previous result, we get the following. +Corollary 9.44. Let n > m ≥ 2. Consider ψ to be the monoid homomorphism +from C∗ +m to C∗ +n such that +ψ(x) = (x + n − m)12 . . .(n − m)(n − m) +� +n − m − 1 +� +. . . 1 +and +ψ(x) = (x + n − m)12 . . . (n − m)(n − m) +� +n − m − 1 +� +. . . 1, +for each x ∈ {1, . . . , m}. Then, ψ induces an injective monoid homomorphism from +hypo(Cm) to hypo(Cn). + +64 +ALAN J. CAIN, RICARDO P. GUILHERME, AND ANTÓNIO MALHEIRO +References +[Bol98] +B. Bollobás. Modern graph theory, volume 184 of Graduate Texts in Mathematics. +Springer-Verlag, New York, 1998. doi:10.1007/978-1-4612-0619-4. +[Bou02] +N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics +(Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by +Andrew Pressley. doi:10.1007/978-3-540-89394-3. +[BS17] +D. Bump and A. 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Humaines, +(140):5–10, 1997. +Center for Mathematics and Applications (NovaMath), FCT NOVA, 2829–516 Ca- +parica, Portugal +Email address: a.cain@fct.unl.pt +Center for Mathematics and Applications (NovaMath), FCT NOVA, and Department +of Mathematics, FCT NOVA, 2829–516 Caparica, Portugal +Email address: rj.guilherme@campus.fct.unl.pt +Center for Mathematics and Applications (NovaMath), FCT NOVA, and Department +of Mathematics, FCT NOVA, 2829–516 Caparica, Portugal +Email address: ajm@fct.unl.pt + diff --git a/2dAyT4oBgHgl3EQfbvf7/content/tmp_files/load_file.txt b/2dAyT4oBgHgl3EQfbvf7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7355aae8d3d39438a3ac722623010a5495ab8a85 --- /dev/null +++ b/2dAyT4oBgHgl3EQfbvf7/content/tmp_files/load_file.txt @@ -0,0 +1,3642 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf,len=3641 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='00271v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='CO] 31 Dec 2022 QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS AND ASSOCIATED GENERALIZATIONS OF THE HYPOPLACTIC MONOID ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic monoid was introduced by Krob and Thibon through a presentation and through quasi-ribbon tableaux and an insertion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Just as Kashiwara crystals enriched the structure of the plactic monoid and allowed its generalization, the first and third authors of this paper introduced a construction of the hypoplactic monoid by identifying vertices in a quasi-crystal graph derived from the crystal graph associated to the gen- eral linear Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Although this construction is based on Kashiwara’s work, it cannot be extended to other crystal graphs, since the analogous quasi- Kashiwara operators on words do not admit a recursive definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This paper addresses these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A general notion of quasi-crystal is introduced, fol- lowed by a study of its properties and relation with crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A combinatorial study of quasi-crystals is then made by associating a quasi-crystal graph to each quasi-crystal, which for the class of seminormal quasi-crystals results in a one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To model the binary operation of the hypoplactic monoid by quasi-crystals, a notion of quasi-tensor product of quasi-crystals is introduced, along with a combinatorial way of computing it similar to the signature rule for the tensor product of crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This framework allows the generalization of the classical hypoplactic monoid to a family of hypoplactic monoids associated to the various simple Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quasi-crystal struc- ture is then used to establish algebraic properties of the hypoplactic monoid associated to the symplectic Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Preliminaries 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystals and homomorphisms 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal graphs 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-tensor product of quasi-crystals 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition and results 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The signature rule 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal monoids 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal monoids and homomorphisms 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The free quasi-crystal monoid 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Congruences and quotients 33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic congruence 36 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Primary 05E16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Secondary 05E10, 20M05, 20M10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal, hypoplactic monoid, crystal, plactic monoid, Kashiwara operator, weight labelled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The second author was funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', under grant reference SFRH/BD/121819/2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For all three authors, this work was funded by national funds through the FCT – Fun- dação para a Ciência e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications), and under the scope of the SemiComb project PTDC/MAT-PUR/31174/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 2 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Crystallizing the classical hypoplactic monoid 39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic monoid of type Cn 41 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The definition of hypo(Cn) 41 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Highest-weight words 44 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Isolated words 46 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Relations 48 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Identities 53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Presentations 57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From hypo(An−1) to hypo(Cn) 60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From hypo(Cn−1) to hypo(Cn) 62 References 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Introduction The plactic monoid, formally introduced by Lascoux and Schützenberger [LS81], is an algebraic object of great interest, with connections to several fields such as representation theory, combinatorics [Ful97], symmetric functions, and Schubert polynomials [LS85, LS89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It was also used to give a first rigorous proof of the Littlewood–Richardson rule [LR34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This led Schützenberger [Sch97] to consider it one of the most fundamental monoids in algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' There are numerous ways of obtaining the plactic monoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' we highlight three of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' First, it originally emerged from Young tableaux and the Schensted insertion algorithm [Sch61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Sec- ond, it also has a presentation by the so-called Knuth relations [Knu70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Third, it can be obtained by identifying words in the same position of isomorphic connected components of a certain crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Kashiwara [Kas90, Kas91, Kas94] introduced crystal bases for modules of quan- tized universal enveloping algebras, discovered independently by Drinfel’d [Dri85] and Jimbo [Jim85], and showed that the plactic monoid arises from the crystal ba- sis associated with the vector representation of the quantized universal enveloping general linear Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This result allowed a deeper study of the plactic monoid and its generalization, because the underlying construction still results in a monoid for crystal bases associated with other quantized universal enveloping algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, Kashiwara and Nakashima [KN94] studied crystal graphs for the Cartan types An, Bn, Cn and Dn, leading to a notion of Kashiwara–Nakashima tableaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Based on this, Lecouvey [Lec02, Lec03] presented comprehensive descriptions of the plactic monoids for the Cartan types Bn, Cn, and Dn, which later appeared in a survey [Lec07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In recent works, Cain, Gray and Malheiro [CGM15a, CGM19] presented rewriting systems and biautomatic structures for these monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In an independent work and by an alternative approach, Hage [Hag15] described a finite convergent presentation of the plactic monoid for type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic monoid was introduced by Krob and Thibon [KT97] from representation-theoretical interpretations of quasi-symmetric functions and non- commutative symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It emerged from a noncommutative realization of quasi-symmetric functions analogous to the realization of symmetric functions by the plactic monoid presented by Lascoux and Schützenberger [LS81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This led to a construction of the hypoplactic monoid through quasi-ribbon tableaux and an insertion algorithm, and to a presentation consisting of the Knuth relations and the quartic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A detailed study of the hypoplactic monoid was done by Novelli [Nov00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A comparative study with other monoids was done by Cain, Gray and Malheiro in [CGM15b], where a rewriting system and a biautomatic structure for the hypoplactic monoid is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Recently, following the work in [Rib22], QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 3 a complete description of the identities satisfied by the hypoplactic monoid was presented by Cain, Malheiro and Ribeiro [CMR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A first notion of quasi-crystal graph was introduced by Krob and Thibon [KT99] to encode the full structure of the modules that give rise to the hypoplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The vertex set of such a graph is formed by the quasi-ribbon words over the alphabet {12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, which also form a complete set of representatives for the hypoplactic congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, these quasi-crystal graphs do not allow a construction of the hypoplactic monoid analogous to the construction of the plactic monoid from crystal graphs, because they do not have isomorphic connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To overcome the limitations of the first notion of quasi-crystal graph, Cain and Malheiro [CM17] described a new quasi-crystal graph, derived from the crystal graph for type An, that allows a construction of the hypoplactic monoid by identi- fying words in the same position of isomorphic connected components, and induces the definition of an analogue of Kashiwara operators on words over the alpha- bet {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, called quasi-Kashiwara operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, this construction is purely combinatorial and does not have an algebraic foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It cannot be used to construct a monoid starting with a crystal graph of another type [Gui22, Re- mark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is therefore natural to ask whether quasi-Kashiwara operators on words can be defined recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The main goal of this paper is to establish a general theory of quasi-crystals that allows a generalization of the hypoplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It addresses the problems discussed above, while showing that the construction in [CM17] can be placed in the context of this new theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It follows the work in [Gui22] and presents a more consolidated theory with new and improved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 2 introduces notation and discusses preliminaries relating to monoids, root systems, and graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 3 states the definitions of quasi-crystals and homomorphisms between quasi-crystals, which give rise to a category, and is devoted to making an algebraic study of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 4 presents the notion of the quasi-crystal graph associated to a quasi-crystal, leading to a combinatorial study of quasi-crystals, and describes a one-to-one correspon- dence between the class of seminormal quasi-crystals and a class of weighted labelled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 5 states the definition of the quasi-tensor product of quasi-crystals and describes a practical method to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 6 states the definition of the quasi-crystal monoid and is devoted to making an algebraic study of it, concern- ing homomorphisms, congruences and free objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is shown that a free quasi- crystal monoid satisfies a universal property which defines it up to isomorphism, and that congruences on a quasi-crystal monoid form a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Homomorphism theorems for quasi-crystal monoids are also proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 7 shows that identi- fying elements in isomorphic connected components of a free quasi-crystal monoid gives rise to a congruence, called the hypoplactic congruence, which leads to the definition of hypoplactic monoid associated to a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is shown that the central elements of a hypoplactic monoid correspond to the isolated elements of the free quasi-crystal monoid, and the idempotents correspond to isolated elements of weight zero, leading to the conclusion that the idempotents of a hypoplactic monoid commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 8 proves that the hypoplactic monoid associated to the standard quasi-crystal of type An is isomorphic to the classical hypoplactic monoid of rank n, indicating that this approach results in a genuine generalization of the classical hypoplactic monoid, by showing that the construction in [CM17] can be placed in the context of the developed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Section 9 is devoted to the study of the hypoplactic monoid associated to the standard quasi-crystal of type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Highest- weight and isolated words are characterized, allowing an identification of central and idempotent elements of this monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Relations satisfied by the hypoplactic 4 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO monoid of type Cn are then studied, in particular, it is investigated whether this monoid satisfies the Knuth relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' it is shown that the hypoplactic monoid of type Cn satisfies a non-trivial identity if and only if n = 2, in contrast to the classi- cal hypoplactic monoid, which satisfies a non-trivial identity independently of rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is proven that the hypoplactic monoid of type Cn is not finitely presented, for any n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, embeddings of the hypoplactic monoids of types An−1 and Cn−1 into the hypoplactic monoid of type Cn are presented, and it is shown that the ‘obvious’ approach to defining such embeddings does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Preliminaries We assume some familiarity with the basic concepts related with monoids and graphs, so we will not make a proper introduction to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For background on monoids see [How95], on presentations see [Hig92], and on graphs see [Bol98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We will introduce crystals as a subclass of quasi-crystals, and so, we will not need to present a complete introduction to crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We refer to [Kas95] for an intro- duction to crystals as they originally emerged in connection to quantized universal envelopping algebras (also called quantum groups) or [HK02] for a comprehensive background on this approch, to [BS17] for a study of crystals detached from their origin, and to [CGM19] for the relations between crystals and plactic monoids for the infinite Cartan types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this section, we give the essential background on root systems, as these alge- braic structures will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Root systems are commonly found in representation theory, in particular, they arise on the study of Lie groups and Lie algebras, but we will detach them from this context, as our aim is to con- struct an algebraic structure for defining crystals and quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we only introduce the necessary notions needed for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For further context see for example [FH91, Bou02, EW06, Bum13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let V be a Euclidean space, that is, a real vector space with an inner product ⟨ · , · ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For α ∈ V other than 0, denote by rα the reflection in the hyperplane orthogonal to α, which is given by rα(v) = v − � v, α∨� α, where α∨ = 2 ⟨α, α⟩α, for each v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that rα is bijective, as rα � rα(v) � = v, for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, rα preserves the inner product, as � rα(u), rα(v) � = ⟨u, v⟩ for any u, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A root system in V is a subset Φ of V satisfying the following conditions: (RS1) Φ is nonempty, finite, and 0 /∈ Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (RS2) rα(β) ∈ Φ, for all α, β ∈ Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (RS3) � α, β∨� ∈ Z, for all α, β ∈ Φ, (RS4) if α ∈ Φ and kα ∈ Φ, then k = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The elements of Φ are called roots, and the elements α∨, with α ∈ Φ, are called coroots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that the definition of root system may differ in the literature, as some authors omit some of the conditions above and use them to characterize root systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For instance, some authors say that a root system is crystallographic when (RS3) is satisfied, or that it is reduced when (RS4) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, some authors require Φ to span V , we say that a root system is semisimple when this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Together with a root system, we always fix an index set I and simple roots (αi)i∈I, that is, a collection of roots satisfying the following conditions: (SR1) {αi | i ∈ I} is a linearly independent subset of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and (SR2) every root β ∈ Φ can be expressed as β = � i∈I kiαi, where all ki are either nonnegative or nonpositive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 5 For each i ∈ I, the reflection rαi is called a simple reflection and is denoted by si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also fix a weight lattice Λ, that is, a Z-submodule of V satisfying the following conditions: (WL1) Λ spans V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (WL2) Φ ⊆ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (WL3) � λ, α∨� ∈ Z, for any λ ∈ Λ and α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The elements of Λ are called weights and are compared using the following partial order λ ≥ µ ⇐⇒ λ − µ = � i∈I kiαi, for some ki ∈ R≥0, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) Finally, we draw attention to the root systems associated to Cartan types An and Cn, which will be the only non-arbitrary root systems considered in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider V to be the real vector space Rn with the usual inner product, and denote by ei ∈ Rn the n-tuple with 1 in the i-th position, and 0 elsewhere, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The root system associated to Cartan type An based on the general linear Lie algebra gln consists of Φ = {ei − ej | i ̸= j}, the index set for the simple roots is I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}, the simple roots are αi = ei − ei+1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1, and the weight lattice is Λ = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The root system associated to Cartan type Cn based on the symplectic Lie al- gebra sp2n consists of Φ = {±ei ± ej | i < j} ∪ {±2ei | i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, the index set for the simple roots is I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, the simple roots are αi = ei − ei+1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1, and αn = 2en, and the weight lattice is Λ = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For more examples of root systems see [BS17, Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystals and homomorphisms In this section we introduce the notion of quasi-crystals associated to a root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We then study some basic properties satisfied by quasi-crystals, some of which correspond to generalizations of properties verified by crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, we introduce the notion of quasi-crystal homomorphisms and study their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Although we rely on root systems (Section 2) to define quasi-crystals, we only make use of properties that are also satisfied by other algebraic structures com- monly used to define crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, all subsequent definitions and results can be reinterpreted using the algebraic data in [Kas95] or a Cartan datum as in [HK02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider Z ∪ {−∞, +∞} to be the usual set of integers where we add a minimal element −∞ and a maximal element +∞, that is, −∞ < m < +∞ for all m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, set m + (−∞) = (−∞) + m = −∞ and m + (+∞) = (+∞) + m = +∞, for all m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Φ be a root system with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal Q of type Φ consists of a set Q together with maps wt : Q → Λ, ¨ei, ¨fi : Q → Q ⊔ {⊥} and ¨εi, ¨ϕi : Q → Z ∪ {−∞, +∞}, for each i ∈ I, satisfying the following conditions: (1) ¨ϕi(x) = ¨εi(x) + � wt(x), α∨ i � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if ¨ei(x) ∈ Q, then wt � ¨ei(x) � = wt(x) + αi, ¨εi � ¨ei(x) � = ¨εi(x) − 1, and ¨ϕi � ¨ei(x) � = ¨ϕi(x) + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) if ¨fi(x) ∈ Q, then wt � ¨fi(x) � = wt(x) − αi, ¨εi � ¨fi(x) � = ¨εi(x) + 1, and ¨ϕi � ¨fi(x) � = ¨ϕi(x) − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) ¨ei(x) = y if and only if x = ¨fi(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (5) if ¨εi(x) = −∞ then ¨ei(x) = ¨fi(x) = ⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (6) if ¨εi(x) = +∞ then ¨ei(x) = ¨fi(x) = ⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 6 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO for x, y ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The set Q is called the underlying set of Q, and the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) form the quasi-crystal structure of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, the map wt is called the weight map, where wt(x) is said to be the weight of x ∈ Q, and the maps ¨ei and ¨fi (i ∈ I) are called the raising and lowering quasi-Kashiwara operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this definition, ⊥ is an auxilary symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the definition of crystals, 0 is oftenly used instead of ⊥, but since some well-known crystals have 0 as an element, we have adopted this notation for quasi-crystals to avoid ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ Q, by ¨ei(x) = ⊥ (or ¨fi(x) = ⊥) we mean that ¨ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', ¨fi) is undefined on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, we say that ¨ei (or ¨fi) is defined on x whenever ¨ei(x) ∈ Q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', ¨fi(x) ∈ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, alternatively one can consider the quasi-Kashiwara operators ¨ei and ¨fi (i ∈ I) to be partial maps from Q to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' When this point of view is more suitable to describe quasi-Kashiwara operators, we will make use of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In comparison with the definition of crystal [BS17, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12], we have that ¨εi and ¨ϕi (i ∈ I) can also take the value +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to the addition of condition (6), as conditions (1) to (5) coincide in both definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we can take the following as the definition of crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A crystal is a quasi-crystal B where ¨εi(x) ̸= +∞ and ¨ϕi(x) ̸= +∞, for all x ∈ B and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From condition (1) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, we get that ¨εi(x) = ±∞ if and only if ¨ϕi(x) = ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, ¨εi(x) = ¨ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, conditions (5) and (6) could have been stated replacing ¨εi by ¨ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, we could have only stated one of conditions (2) and (3) as justified by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Φ be a root system with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider a set Q and maps wt : Q → Λ, ¨ei, ¨fi : Q → Q⊔{⊥} and ¨εi, ¨ϕi : Q → Z∪{−∞, +∞}, for each i ∈ I, satisfying Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(2) holds if and only if Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q and i ∈ I such that ¨fi(x) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4), we have that ¨ei � ¨fi(x) � = x, and so, wt(x) = wt � ¨ei( ¨fi(x)) � = wt � ¨fi(x) � − αi, ¨εi(x) = ¨εi � ¨ei( ¨fi(x)) � = ¨εi � ¨fi(x) � − 1, and ¨ϕi(x) = ¨ϕi � ¨ei( ¨fi(x)) � = ¨ϕi � ¨fi(x) � + 1, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The converse implication is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ In the same way, by conditions (1) and (4) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 we have that a quasi- crystal is determined by a set Q and the weight map wt together with either ¨ei or ¨fi, and either ¨εi or ¨ϕi, for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, for a purpose of clarity, we usually give explicit definitions for each map when defining a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Consider the root system of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, the standard crystal of type An gives rise to the quasi-crystal An defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The underlying set is the ordered set An = {1 < 2 < · · · < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ An, the weight of x is wt(x) = ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1, the quasi-Kashiwara operators ¨ei and ¨fi are only defined on i + 1 and i, respectively, where ¨ei(i + 1) = i and ¨fi(i) = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, ¨εi(x) = δx,i+1 and ¨ϕi(x) = δx,i, where δk,l = 1 if k = l, and δk,l = 0 whenever k ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We call An the standard quasi-crystal of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Consider the root system of type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, the standard crystal of type Cn gives rise to the quasi-crystal Cn defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The underlying set QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 7 is Cn = {1 < 2 < · · · < n < n < n − 1 < · · · < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, the weight of x is wt(x) = ex, and the weight of x is wt(x) = −ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1, the quasi-Kashiwara operators ¨ei and ¨fi are only defined on the following cases: ¨ei(i + 1) = i, ¨ei(i) = i + 1, ¨fi(i) = i + 1, and ¨fi(i + 1) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quasi-Kashiwara operators ¨en and ¨fn are only defined in n and n, respectively, where ¨en(n) = n and ¨fn(n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, for y ∈ Cn, ¨εi(y) = δy,i+1 + δy,i, ¨εn(y) = δy,n, ¨ϕi(y) = δy,i + δy,i+1, and ¨ϕn(y) = δy,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We call Cn the standard quasi-crystal of type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) Consider the root system of type A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have a quasi-crystal A2 3 of type A3 whose underlying set is A2 3 = A3 × A3 and whose quasi-crystal structure is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' x wt(x) ¨e1(x) ¨e2(x) ¨f1(x) ¨f2(x) ¨ε1(x) ¨ε2(x) ¨ϕ1(x) ¨ϕ2(x) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) 2e1 ⊥ ⊥ (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) ⊥ 0 0 2 0 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) e1 + e2 ⊥ ⊥ ⊥ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) +∞ 0 +∞ 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) e1 + e3 ⊥ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) ⊥ 0 1 1 0 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) e1 + e2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) ⊥ (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) 1 0 1 1 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) 2e2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) ⊥ ⊥ (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) 2 0 0 2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) e2 + e3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) ⊥ ⊥ ⊥ 1 +∞ 0 +∞ (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) e1 + e3 ⊥ (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) ⊥ 0 1 1 0 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) e2 + e3 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) ⊥ (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) 1 1 0 1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 3) 2e3 ⊥ (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 2) ⊥ ⊥ 0 2 0 0 (4) Consider the root system of type A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have a quasi-crystal Q of type A2 consisting of a set Q = {a, b} and maps defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' x wt(x) ¨e(x) ¨f(x) ¨ε(x) ¨ϕ(x) a e1 ⊥ ⊥ 0 1 b e2 ⊥ ⊥ 1 0 Since the root system of type A2 has exactly one simple root, we omit the subscript index in the maps, for instance ¨e instead of ¨e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the previous example we only introduce quasi-crystals that will be relevant below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As crystals are quasi-crystals (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2), more examples can be found in [BS17, Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='25], where the standard crystals for types Bn and Dn are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Recall the partial order defined on a weight lattice Λ described in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result justifies the terminology of raising and lowering used to characterize the quasi-Kashiwara operators ¨ei and ¨fi (i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal, and let x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(x) ∈ Q, then wt � ¨ei(x) � > wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨fi(x) ∈ Q, then wt(x) > wt � ¨fi(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(x) ∈ Q, then wt � ˜ei(x) � − wt(x) = wt(x) + αi − wt(x) = αi, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(2), and so, wt � ¨ei(x) � ≥ wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since αi is a root, we have that αi ̸= 0, which implies that wt � ¨ei(x) � ̸= wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, wt � ¨ei(x) � > wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨fi(x) ∈ Q, then x = ¨ei � ¨fi(x) � , by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As proved above, we have that wt(x) = wt � ¨ei( ¨fi(x)) � > wt � ¨fi(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From the previous result, we have that, like the Kashiwara operators in crystals, the raising quasi-Kashiwara operators ¨ei (i ∈ I) increase the weight of elements, whenever defined, and the lowering quasi-Kashiwara operators ¨fi (i ∈ I) decrease the weight of elements, whenever they are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the notions of highest- and lowest-weight elements from crystals can be generalized in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q be an element of a quasi-crystal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 8 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO (1) x is said to be of highest weight if ¨ei(x) = ⊥, for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) x is said to be of lowest weight if ¨fi(x) = ⊥, for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similar to crystals, notice that a quasi-crystal may have a highest-weight element whose weight is less than or equal to the weight of an element that is not of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For instance, consider the quasi-crystal A2 3 described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(3), take x = (1, 1), y = (1, 2) and z = (2, 1), then x and y are of highest weight, z is not of highest weight, wt(x) > wt(y) and wt(y) = wt(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, if a quasi-crystal Q has an element x ∈ Q such that ¨εi(x) ∈ {−∞, +∞}, for all i ∈ I, then we can change the weight wt(x) of x to any weight in Λ, and the resulting structure is still a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, if we extend this definition to the weights as follows, we get some more natural results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal, and let λ ∈ Λ be a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) λ is called a highest weight in Q if there exists a highest-weight element x ∈ Q such that λ = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) λ is called a lowest weight in Q if there exists a lowest-weight element x ∈ Q such that λ = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal, and let λ be a weight in wt(Q) = {wt(x) | x ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) If λ is maximal among weights in wt(Q), then λ is a highest weight, and any element x ∈ Q such that wt(x) = λ is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) If λ is minimal among weights in wt(Q), then λ is a lowest weight, and any element x ∈ Q such that wt(x) = λ is of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Let x ∈ Q be such that wt(x) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x is not of highest weight, then ¨ei(x) ∈ Q, for some i ∈ I, and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, wt � ¨ei(x) � > wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, λ is not maximal among weights in wt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Let x ∈ Q be such that wt(x) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x is not of lowest weight, then ¨fi(x) ∈ Q, for some i ∈ I, and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, wt(x) > wt � ¨fi(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, λ is not minimal among weights in wt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Since the quasi-Kashiwara operators of a quasi-crystal Q can be regarded as partial maps from Q to Q, we can compose them in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As usual, for i ∈ I, set ¨e0 i and ¨f 0 i to be the identity map on Q, and recursively, define ¨ek+1 i = ¨ei¨ek i and ¨f k+1 i = ¨fi ¨f k i , for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal Q is said to be seminormal if for any x ∈ Q and i ∈ I, ¨εi(x) = max � k ∈ Z≥0 �� ¨ek i (x) ∈ Q � and ¨ϕi(x) = max � k ∈ Z≥0 �� ¨f k i (x) ∈ Q � , whenever ¨εi(x) ̸= +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quasi-crystals described in items (1) to (3) of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 are seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, the quasi-crystal described in item (4) is not seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As pointed out in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, a crystal B satisfies ¨εi(x) ̸= +∞, for all x ∈ B and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If B is seminormal, then the equalities in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 are verified for any x ∈ B and i ∈ I, and so, B is seminormal as a crystal [BS17, formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the seminormal property for quasi-crystals generalize the one for crystals in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For a crystal B, we have that B is seminormal as a crystal if and only if it is seminormal as a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 9 We have just seen that the seminormal property for quasi-crystals is consistent with the corresponding property for crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The exception when ¨εi(x) takes the value +∞ is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Without this exception, in the case ¨εi(x) = +∞ we would have that max � k ∈ Z≥0 �� ¨ek i (x) ∈ Q � = 0, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(6), and hence the class of seminormal quasi-crystals would coincide with the class of seminormal crystals, and we would not have a proper generalization of the seminormal property as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus this exception is vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, it has deep implications, as some common results for seminormal crystals are not satisfied by seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For example, we can no longer guarantee the weight of a highest-weight element to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Instead we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q be an element of a seminormal quasi-crystal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) If x is of highest weight and � wt(x), α∨ i � < 0, for some i ∈ I, then ¨εi(x) = ¨ϕi(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) If x is of lowest weight and � wt(x), α∨ i � > 0, for some i ∈ I, then ¨εi(x) = ¨ϕi(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Suppose that ¨εi(x) ̸= +∞ (or equivalently, ¨ϕi(x) ̸= +∞), for some i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As Q is seminormal, we have that ¨εi(x), ¨ϕi(x) ∈ Z≥0 and by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(1), � wt(x), α∨ i � = ¨ϕi(x) − ¨εi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if � wt(x), α∨ i � < 0, then ¨εi(x) > 0, which implies that ¨ei(x) ∈ Q, because Q is seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, x is not of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) As above, if ¨εi(x) ̸= +∞ and � wt(x), α∨ i � > 0, then ¨ϕi(x) ∈ Z>0, which implies that ¨fi(x) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And therefore, x is not of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Now, we introduce the definition of a homomorphism between quasi-crystals, which is analogous to the one for crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be quasi-crystals of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal homomorphism ψ from Q to Q′, denoted by ψ : Q → Q′, is a map ψ : Q ⊔ {⊥} → Q′ ⊔ {⊥} that satisfies the following conditions: (1) ψ(⊥) = ⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if ψ(x) ∈ Q′, then wt � ψ(x) � = wt(x), ¨εi � ψ(x) � = ¨εi(x), and ¨ϕi � ψ(x) � = ¨ϕi(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) if ¨ei(x) ∈ Q and ψ(x), ψ � ¨ei(x) � ∈ Q′, then ψ � ¨ei(x) � = ¨ei � ψ(x) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) if ¨fi(x) ∈ Q and ψ(x), ψ � ¨fi(x) � ∈ Q′, then ψ � ¨fi(x) � = ¨fi � ψ(x) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' for x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal isomorphism ψ between Q and Q′ is a bijection ψ : Q ⊔ {⊥} → Q′⊔{⊥} such that ψ : Q → Q′ and ψ−1 : Q′ → Q are quasi-crystal homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We say that Q and Q′ are isomorphic if there exists a quasi-crystal isomorphism between Q and Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Due to condition (1), when defining a quasi-crystal homomorphism ψ, we omit the explicit mention to ψ(⊥) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, as ⊥ is an auxilary symbol which stands for undefinition, alternatively a quasi-crystal homomorphism ψ : Q → Q′ can be regarded as a partial map ψ from Q to Q′ satisfying conditions (2) to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, when defining a quasi-crystal homomorphism, we usually only give the images for the elements x ∈ Q such that ψ(x) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For the sake of simplicity, by saying that a map ψ : Q → Q′ is a quasi-crystal homomorphism from Q to Q′, we mean that the map ψ′ : Q ⊔ {⊥} → Q′ ⊔ {⊥}, given by ψ′(⊥) = ⊥ and ψ′(x) = ψ(x), for each x ∈ Q, is a quasi-crystal homomorphism from Q to Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The notion of crystal homomorphism can be placed in the context of quasi- crystals in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 10 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A crystal homomorphism is a quasi-crystal homomorphism between two crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' At this point we defined quasi-crystals and homomorphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is immediate from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12 that given a quasi-crystal Q, the identity map on Q is a quasi-crystal homomorphism from Q to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result follows by a straightforward application of the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q1, Q2 and Q3 be quasi-crystals of the same type, and let ψ1 : Q1 → Q2 and ψ2 : Q2 → Q3 be quasi-crystal homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ2 ◦ ψ1 is a quasi-crystal homomorphism from Q1 to Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus we obtain a category whose objects are quasi-crystals of the same type and morphisms are quasi-crystal homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We say that a quasi-crystal homomorphism ψ : Q → Q′ is injective, surjective or bijective if the map ψ : Q ⊔ {⊥} → Q′ ⊔ {⊥} is injective, surjective or bijective, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As the following example shows, a bijective quasi-crystal homomor- phosm is not necessarily a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let A2 and Q be the quasi-crystals of type A2 described respec- tively in items (1) and (4) of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a map ψ : Q → A2 by ψ(a) = 1 and ψ(b) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a quasi-crystal homomorphism from Q to A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' But ψ is not a quasi-crystal isomorphism as ψ−1 does not verify conditions (3) and (4) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, we have that ψ is a bijective crystal homomorphism that is not a crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, notice that Q is not seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, in the following results we present an alternative characterization of quasi-crystal isomorphisms for seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → Q′ be a bijective quasi-crystal homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following conditions are equivalent (1) ψ � ¨ei(x) � = ¨ei � ψ(x) � for all x ∈ Q and i ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) ψ � ¨fi(x) � = ¨fi � ψ(x) � for all x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that ψ satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is bijective and ψ(⊥) = ⊥ by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(1), then ψ(y) ∈ Q′ for all y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if ¨fi(x) ∈ Q, then ψ � ¨fi(x) � ∈ Q′ which implies ψ � ¨fi(x) � = ¨fi � ψ(x) � by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨fi � ψ(x) � ∈ Q′, or equivalently, ψ−1� ¨fi(ψ(x)) � ∈ Q, then ψ � ¨ei � ψ−1� ¨fi(ψ(x)) ��� = ¨ei � ψψ−1� ¨fi(ψ(x)) �� = ¨ei ¨fi � ψ(x) � = ψ(x), as we assumed that ψ satisfies (1), and so, x = ¨ei � ψ−1� ¨fi(ψ(x)) �� which implies ψ � ¨fi(x) � = ψ � ¨fi¨ei � ψ−1� ¨fi(ψ(x)) ��� = ψψ−1� ¨fi(ψ(x)) � = ¨fi � ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ satisfies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The fact that (2) implies (1) follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → Q′ be a quasi-crystal homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a quasi-crystal isomorphism if and only if ψ is bijective and satisfies (1) or (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that ψ is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12, ψ is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(x) ∈ Q, we also have that ψ(x), ψ � ¨ei(x) � ∈ Q′ as ψ is bijective and ψ(⊥) = ⊥, and so, ψ � ¨ei(x) � = ¨ei � ψ(x) � by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, since ψ−1 is also a quasi-crystal isomorphism, if ¨ei � ψ(x) � ∈ Q′, then ψ−1� ¨ei(ψ(x)) � = ¨ei � ψ−1ψ(x) � = ¨ei(x), QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 11 which implies ¨ei � ψ(x) � = ψ � ¨ei(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ satisfies Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, we can assume that ψ is bijective and satisfies con- ditions (1) and (2) of that lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Clearly, ψ−1(⊥) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x′ ∈ Q′ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a quasi-crystal homomorphism, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2) we have that wt � ψ−1(x′) � = wt � ψ � ψ−1(x′) �� = wt(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, we get ¨εi � ψ−1(x′) � = ¨εi(x′) and ¨ϕi � ψ−1(x′) � = ¨ϕi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ satisfies Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16(1), then ¨ei � ψ−1(x′) � = ψ−1ψ � ¨ei � ψ−1(x′) �� = ψ−1� ¨ei � ψψ−1(x′) �� = ψ−1� ¨ei(x′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since ψ satisfies Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16(2), then ¨fi � ψ−1(x′) � = ψ−1ψ � ¨fi � ψ−1(x′) �� = ψ−1� ¨fi � ψψ−1(x′) �� = ψ−1� ¨fi(x′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ−1 is a quasi-crystal homomorphism from Q′ to Q, and therefore, ψ is a quasi-crystal isomorphism between Q and Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type, and let ψ : Q → Q′ be a bijective quasi-crystal homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ψ is bijective, we get that ψ(x) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q and Q′ are seminormal and ¨εi � ψ(x) � = ¨εi(x), we have that ¨ei(x) ∈ Q if and only if ¨ei � ψ(x) � ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, if ¨ei(x) ∈ Q, then ψ � ¨ei(x) � ∈ Q, as ψ is bijective, and ψ � ¨ei(x) � = ¨ei � ψ(x) � by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, ¨ei(x) = ⊥ = ¨ei � ψ(x) � , which implies that ψ � ¨ei(x) � = ⊥ = ¨ei � ψ(x) � , by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ satisfies Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16(1), and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17, ψ is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal graphs In this section we present a combinatorial approach to quasi-crystals, which results in a generalization of the notion of crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this framework we are able to characterize some substructures of quasi-crystals, generalizing similar structures described for crystals based on crystal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, as a crystal graph of a seminormal crystal completely determines its crystal structure, we show a similar connection between quasi-crystal graphs and seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Λ be a weight lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A weight map on a graph Γ with vertex set X is a map wt : X → Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For a vertex x ∈ X of Γ, wt(x) is called the weight of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this case, we say the graph Γ is Λ-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Φ be a root system with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quasi-crystal graph ΓQ of a quasi-crystal Q of type Φ is a Λ-weighted I-labelled directed graph with vertex set Q and an edge x i −−−→ y from x ∈ Q to y ∈ Q labelled by i ∈ I whenever ¨fi(x) = y, and a loop on x ∈ Q labelled by i ∈ I whenever ¨εi(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ Q, let ΓQ(x) denote the connected component of ΓQ containing the vertex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In comparison with crystal graphs, by requiring quasi-crystal graphs to be Λ- weighted, we accommodate the weight map wt of a quasi-crystal directly in the definition of its quasi-crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, we have that a quasi-crystal graph may not be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, a quasi-crystal graph is simple if the maps ¨εi (i ∈ I) do not take the value +∞, and from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For a quasi-crystal Q, the quasi-crystal graph ΓQ is simple if and only if Q is a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, the quasi-crystal graph of Q coincides with its crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 12 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) The quasi-crystal graph ΓAn of the standard quasi-crystal An of type An, described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1), is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 1 −−−→ 2 2 −−−→ 3 3 −−−→ · · · n−1 −−−→ n (2) The quasi-crystal graph ΓCn of the standard quasi-crystal Cn of type Cn, described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2), is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 1 −−−→ 2 2 −−−→ · · · n−1 −−−→ n n −−−→ n n−1 −−−→ n − 1 n−2 −−−→ · · · 1 −−−→ 1 (3) The quasi-crystal graph ΓA2 3 of the quasi-crystal A2 3 of type A3, described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(3), is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1,1) (2,1) (3,1) (1,2) (2,2) (3,2) (1,3) (2,3) (3,3) 1 1 2 1 1 2 2 2 1 2 Note that (1) and (2) of the previous example are crystal graphs (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For the crystal graphs associated to the standard crystals of type Bn and Dn, see for example [CGM19, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x i −−−→ y be an edge of a quasi-crystal graph ΓQ of a quasi-crystal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x ̸= y, then y = ¨fi(x) by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, or equivalently, x = ¨ei(y) due to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, we have that x = y, that is, x has a loop labelled by i, and so, ¨εi(x) = +∞ by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, which implies that ¨ei and ¨fi are undefined on x, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In either case, we have that if x i −−−→ y′ is an edge of ΓQ, then y = y′, and similarly, if x′ i −−−→ y is an edge of ΓQ, then x = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, for i ∈ I, a vertex of ΓQ is the start of at most one edge, and is the end of at most one edge labelled by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We will show that quasi-crystal graphs provide a combinatorial framework to study quasi-crystals, analogous to the tools that crystal graphs provide for crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For instance, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 is equivalent to state that for a quasi-crystal Q, if x i −−−→ y is an edge of ΓQ with x ̸= y, then wt(x) > wt(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6 is equivalent to stating that an element x ∈ Q is of highest (or lowest) weight if the only edges of ΓQ ending (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', starting) at x are loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, the combinatorial framework formed by quasi-crystal graphs is a genuine generalization of the framework formed by crystal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This allows a natural generalization of structures such as connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A connected component of Q is a subset Q′ of Q that satisfies the following conditions: (1) for each x, y ∈ Q′ there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gm(x) = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) ¨ei(x), ¨fi(x) ∈ Q′ ⊔ {⊥} for all x ∈ Q′ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also use the term connected component to refer to the quasi-crystal Q′ consisting of Q′ together with the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) of Q restricted to Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each x ∈ Q, the connected component of Q containing x is denoted by Q(x), and the associated quasi-crystal is denoted by Q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As a justification for this terminology, we check that connected components of a quasi-crystal Q and the vertex sets of connected components of the quasi-crystal graph ΓQ identify the same subsets of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 13 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal, and let Q′ ⊆ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, Q′ is a con- nected component of Q if and only if Q′ is the vertex set of a connected component of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that Q′ is a connected component of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5(1), there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gm(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set x0 = x, and xk+1 = gm−k(xk), for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that xk+1 = gm−k · · · gm(x) ∈ Q′, by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, xm = g1 · · · gm(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If gm−k = ¨fi, for some i ∈ I, then xk i −−−→ xk+1 is an edge of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, gm−k = ¨ei, for some i ∈ I, and so, xk+1 i −−−→ xk is an edge of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For z ∈ Q, if x i −−−→ z or z i −−−→ x is an edge of ΓQ, then z = ¨fi(x) or z = ¨ei(x), respectively, which implies that z ∈ Q′, by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the subgraph of ΓQ induced by Q′ is a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, assume that Q′ is a vertex set of a connected component of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then there exist x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Q′ such that x = x0, y = xm, and for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m − 1, xk i −−−→ xk+1 or xk+1 i −−−→ xk is an edge of ΓQ, for some i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If xk i −−−→ xk+1 is an edge of ΓQ, for some i ∈ I, set gm−k = ¨fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, xk+1 i −−−→ xk is an edge of ΓQ, for some i ∈ I, and so, set gm−k = ¨ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In any case we have that xk+1 = gm−k(xk), which implies that g1 · · · gm(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ I, if ¨fi(x) ∈ Q, then x i −−−→ ¨fi(x) is an edge of ΓQ, and so, ¨fi(x) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, if ¨ei(x) ∈ Q, then ¨ei(x) i −−−→ x is an edge of ΓQ, which implies that ¨ei(x) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, Q′ is a connected component of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Given Λ-weighted I-labelled directed graphs Γ1 and Γ2 with vertex sets X1 and X2, respectively, a homomorphism ψ from Γ1 to Γ2 is a map ψ : X1 → X2 such that wt � ψ(x) � = wt(x), for all x ∈ X1, and ψ(x) i −−−→ ψ(y) is an edge of Γ2, whenever x i −−−→ y is an edge of Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ψ is also bijective and ψ−1 is a homomorphism from Γ2 to Γ1, then ψ is said to be an isomorphism between Γ1 and Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also have the following relation between quasi-crystal homomorphisms and graph homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be quasi-crystals of the same type, and let ψ : Q → Q′ be a quasi-crystal homomorphism such that ψ(Q) ⊆ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a graph homomorphism from ΓQ to ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), we have that wt � ψ(x) � = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that x i −−−→ y is an edge of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x = y, that is, x has a loop labelled by i, then ¨εi(x) = +∞, and by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), ¨εi � ψ(x) � = +∞, which implies that ψ(x) has a loop labelled by i in ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, x ̸= y, we have by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 that ¨fi(x) = y, and since ψ(x), ψ(y) ∈ ψ(Q) ⊆ Q′, we get by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(4) that ¨fi � ψ(x) � = ψ(y), which implies that ψ(x) i −−−→ ψ(y) is an edge of ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ is a graph homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Notice that the converse of the previous result does not hold, as ψ may be a graph homomorphism from ΓQ to ΓQ′ and not be a quasi-crystal homomorphism from Q to Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider the root system of type A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take Q consisting of the set Q = {x}, where wt(x) = 0, ¨e(x) = ¨f(x) = ⊥ and ¨ε(x) = ¨ϕ(x) = 0, and take 14 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Q′ consisting of the set Q′ = {x′}, where wt(x′) = 0, ¨e(x′) = ¨f(x′) = ⊥ and ¨ε(x′) = ¨ϕ(x′) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quasi-crystal graphs of Q and Q′ are respectively x and x′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The map ψ : Q → Q′, defined by ψ(x) = x′, is a graph homomorphism, but not a quasi-crystal homomorphism, because ¨ε � ψ(x) � = +∞ ̸= 0 = ¨ε(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal isomorphism ψ between quasi-crystals Q and Q′ satisfies the property ψ(Q) = Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, it is immediate from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7 that ψ and ψ−1 are graph homomorphisms, which implies that ψ is a graph isomorphism between ΓQ and ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be two quasi-crystals of the same type, and let ψ : Q → Q′ be a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, Q0 is a connected component of Q if and only if ψ(Q0) is a connected component of Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Furthermore, for each x ∈ Q, the restriction of ψ to Q(x) is a quasi-crystal isomorphism between Q(x) and Q′� ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose Q0 is a connected component of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x′, y′ ∈ ψ(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set x = ψ−1(x′) and y = ψ−1(y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As x, y ∈ Q0, there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gm(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17, we have that g1 · · · gm(x′) = g1 · · · gm � ψ(x) � = ψ � g1 · · · gm(x) � = ψ(y) = y′, which implies that ψ(Q0) satisfies Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By the same results, for i ∈ I, we have that ψ � ¨ei(x) � = ¨ei � ψ(x) � = ¨ei(x′) and ψ � ¨fi(x) � = ¨fi � ψ(x) � = ¨fi(x′), and since ¨ei(x), ¨fi(x) ∈ Q0 ⊔ {⊥}, we get that ¨ei(x′), ¨fi(x′) ∈ ψ(Q0) ⊔ {⊥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ(Q0) is a connected component of Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ−1 is a quasi-crystal isomorphism between Q′ and Q, by the previous implication, if ψ(Q0) is a connected component of Q′, then ψ−1� ψ(Q0) � = Q0 is a connected component of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, let z ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q(z) is a connected component of Q, then ψ � Q(z) � is a connected component of Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ψ(z) ∈ ψ � Q(z) � , we get that ψ � Q(z) � = Q′� ψ(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The restriction of ψ to Q(z) is a bijective quasi-crystal homomorphism from Q(z) to Q′� ψ(z) � , because ψ is a quasi-crystal homomorphism from Q to Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, the restriction of ψ−1 to Q′� ψ(z) � is a bijective quasi-crystal homomorphism from Q′� ψ(z) � to Q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And therefore, the restriction of ψ to Q(z) is a quasi-crystal isomorphism between Q(z) and Q′� ψ(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ For a quasi-crystal Q, it is immediate that the quasi-Kashiwara operators ¨ei and ¨fi (i ∈ I) are completely determined by the quasi-crystal graph ΓQ, because given x, y ∈ Q with x ̸= y, we have that x i −−−→ y is an edge of ΓQ if and only if ¨fi(x) = y and ¨ei(y) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Now, we show that if Q is seminormal, then also the maps ¨εi and ¨ϕi (i ∈ I) are completely determined by ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal, and let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Given an i-labelled walk x0 i −−−→ x1 i −−−→ · · · i −−−→ xm on ΓQ, then either (1) x0 = x1 = · · · = xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' or (2) xk = ¨f k i (x0), for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m, and thus, x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm form the unique i-labelled path on ΓQ starting at x0 and ending at xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x0 = x1, then x0 has an i-labelled loop, and so, ¨εi(x0) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, ¨fi(x1) = ¨fi(x0) = ⊥, which implies that x1 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And recursively, we obtain that x0 = x1 = · · · = xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 15 Otherwise, we have that x0 ̸= x1, which implies that ¨fi(x0) = x1 (or equivalently, ¨ei(x1) = x0), because x0 i −−−→ x1 is an edge of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ¨ei is defined on x1, then ¨εi(x1) ̸= +∞, and so, x1 does not have an i-labelled loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, x1 ̸= x2, and so, ¨fi(x1) = x2, because x1 i −−−→ x2 is an edge of ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Recursively, we get that ¨fi(xk) = xk+1, for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, we have that wt(x0) > wt(x1) > · · · > wt(xm), which implies that x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm are pairwise distinct, and thus, form an i-labelled path on ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is the unique i-labelled path on ΓQ starting at x0 and ending at xm, because in a quasi-crystal graph every vertex is the start of at most an edge and is the end of at most an edge labelled by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ Q and i ∈ I, we have that (1) ¨ϕi(x) is the supremum among nonnegative integers m ∈ Z≥0 such that there exists an i-labelled walk on ΓQ starting at x with length m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) ¨εi(x) is the supremum among nonnegative integers m ∈ Z≥0 such that there exists an i-labelled walk on ΓQ ending at x with length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Let z ∈ Z≥0 ∪ {+∞} be the supremum among nonnegative integers m ∈ Z≥0 such that there exists an i-labelled walk on ΓQ starting on x with length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕi(x) = +∞, then x has an i-labelled loop on ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so, for any m ∈ Z≥0, the sequence x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm, where x0 = · · · = xm = x, is an i-labelled walk starting on x with length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, z = +∞ = ¨ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, we have that ¨ϕi(x) = max � k ∈ Z≥0 �� ¨f k i (x) ∈ Q � , because Q is seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since x i −−−→ ¨fi(x) i −−−→ · · · i −−−→ ¨f ¨ϕi(x) i (x) is an i-labelled path on ΓQ starting on x, we have that ¨ϕi(x) ≤ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since x has no i-labelled loops, if x0 i −−−→ x1 i −−−→ · · · i −−−→ xm is a walk on ΓQ such that x0 = x, then xk = ¨f k i (x), for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since ¨f ¨ϕi(x)+1 i (x) = ⊥, we get that m ≤ ¨ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, z ≤ ¨ϕi(x), and therefore, ¨ϕi(x) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Analogously to (1), we have that if ¨εi(x) = +∞, then the lengths of i-labelled walks on ΓQ ending on x are unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And otherwise, ¨e¨εi(x) i (x) i −−−→ ¨e¨εi(x)−1 i (x) i −−−→ · · · i −−−→ x is the longest i-labelled walk on ΓQ ending on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We have shown how the maps ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) of a seminormal quasi- crystal can be described by an I-labelled directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so a seminor- mal quasi-crystal can be completely described by a Λ-weighted I-labelled directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From this correspondence between seminormal quasi-crystals and weighted I-labelled directed graphs, we can identify a subclass of graphs which leads to a purely combinatorial description of seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By translating Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 for weighted labelled directed graphs we obtain a subclass of graphs, whose elements are call seminormal quasi- crystal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider a root system Φ with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A Λ-weighted I-labelled directed graph Γ is a seminormal 16 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO quasi-crystal graph if for any vertices x and y, and any i ∈ I, the following conditions are satisfied: (1) x is the start of at most one edge labelled by i, and is the end of at most one edge labelled by i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) any i-labelled path of Γ is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) if x i −−−→ y is an edge of Γ with x ̸= y, then wt(y) = wt(x) − αi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) ¨ϕi(x) = ¨εi(x)+ � wt(x), α∨ i � , where ¨ϕi(x) is the supremum among nonnega- tive integers k ∈ Z≥0 such that there exists an i-labelled walk on Γ starting on x with length k, and ¨εi(x) is the supremum among nonnegative integers l ∈ Z≥0 such that there exists an i-labelled walk on Γ ending on x with length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that if Γ is a seminormal quasi-crystal with vertex set Q, we can define partial maps ¨ei and ¨fi (i ∈ I) on Q, by setting ¨ei(y) = x and ¨fi(x) = y, whenever x i −−−→ y is an edge of Γ with x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, we get a seminormal quasi-crystal Q, and Γ coincides with ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Due to this relation between seminormal quasi-crystals and weighted labelled directed graphs we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type, and let ψ : Q → Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a quasi-crystal isomorphism between Q and Q′ if and only if ψ is a graph isomorphism between the weighted labelled directed graphs ΓQ and ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ψ is a quasi-crystal isomorphism between Q and Q′, then ψ and ψ−1 are graph homomorphisms, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ is a graph isomorphism between ΓQ and ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, suppose that ψ is a graph isomorphism between ΓQ and ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By definition, ψ is weight-preserving, that is, wt � ψ(x) � = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨fi(x) ∈ Q, then x i −−−→ ¨fi(x) is an edge of ΓQ, which implies that ψ(x) i −−−→ ψ � ¨fi(x) � is an edge of ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since x ̸= ¨fi(x) and ψ is bijective, then ψ(x) ̸= ψ � ¨fi(x) � , which implies that ¨fi � ψ(x) � = ψ � ¨fi(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if ¨ei(x) ∈ Q, then ψ � ¨ei(x) � = ¨ei � ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a graph isomorphism, we have that x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Q form an i-labelled walk on ΓQ if and only if ψ(x0), ψ(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , ψ(xm) form an i-labelled walk on ΓQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11, ¨εi � ψ(x) � = ¨εi(x) and ¨ϕi � ψ(x) � = ¨ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ is a bijective quasi-crystal homomorphism from Q to Q′, and by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18, ψ is a quasi-crystal isomorphism between Q and Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ In contrast to the relation between graph homomorphisms and crystal homo- morphisms of seminormal crystals associated to semisimple root systems, we can- not replace the word isomorphism by homomorphism in the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8, we have two seminormal quasi-crystals Q and Q′, and a map ψ which is a graph homomorphism from ΓQ to ΓQ′, but not a quasi-crystal homomorphism from Q to Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the converse of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7 does not hold even for seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, we can give a stronger version of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 in the particular case of seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type, and let ψ : Q → Q′ be a quasi-crystal homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each x ∈ Q, if ψ � Q(x) � ⊆ Q′, then the restriction of ψ to Q(x) is a surjective quasi-crystal homo- morphism from Q(x) to Q′� ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q be such that ψ � Q(x) � ⊆ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For any y ∈ Q(x) and i ∈ I, we have that ¨ϕi(y) = ¨ϕi � ψ(y) � by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), and as Q and Q′ are seminormal, QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 17 ¨fi(y) ∈ Q if and only if ¨fi � ψ(y) � ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, we get that ¨fi(y) ∈ Q(x) by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5(2), and ψ(y), ψ � ¨fi(y) � ∈ Q′ as ψ � Q(x) � ⊆ Q′, which implies by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(4) that ψ � ¨fi(y) � = ¨fi � ψ(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, ¨ei(y) ∈ Q if and only if ¨ei � ψ(y) � ∈ Q′, and if so, ¨ei(y) ∈ Q(x) and ψ � ¨ei(y) � = ¨ei � ψ(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, given g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ej, ¨fj | j ∈ I}, we have that g1 · · · gm(x) is defined if and only if g1 · · · gm � ψ(x) � is defined, in which case ψ � g1 · · · gm(x) � = g1 · · · gm � ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that ψ � Q(x) � = Q′� ψ(x) � , and thus, the restriction of ψ to Q(x) induces a well-defined surjective map from Q(x) to Q′� ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a quasi-crystal homomorphism, we obtain that the restriction of ψ to Q(x) is a surjective quasi- crystal homomorphism from Q(x) to Q′� ψ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Due to the connection between quasi-crystals and graphs, an element of a quasi- crystal Q is also a vertex of the graph ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to characterizations based on either perspective and justifies terminology as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' An element x ∈ Q is said to be isolated if Q(x) = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' An isolated element of a quasi-crystal Q is an isolated vertex of the quasi-crystal graph ΓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, an element x ∈ Q is isolated if and only if ¨ei(x) = ¨fi(x) = ⊥, for all i ∈ I, or equivalently, x is isolated if and only if x is of highest and lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Furthermore, if Q is seminormal, then x is isolated if and only if for each i ∈ I, either ¨εi(x) = ¨ϕi(x) = 0 or ¨εi(x) = ¨ϕi(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-tensor product of quasi-crystals A definition of tensor product for quasi-crystals can be given in a similar way as it was originally done for crystals (see [Kas90, Kas91, KN94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Such a definition would lead to a generalization to quasi-crystals of the construction of a plactic monoid from a crystal as in [Gui22, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since we are interested in a general construction of the hypoplactic monoid from a quasi-crystal, in this section we introduce a slightly different definition: the quasi-tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We then study its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, as this notion will be used in the subsequent sections to relate quasi-crystals and monoids, we describe a combinatorial method to compute the quasi-crystal structure of a quasi-tensor product of quasi-crystals, which is analogous to the signature rule for the tensor product of crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following theorem we establish the founda- tions to introduce the notion of quasi-tensor product of quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider a root system Φ with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of type Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set Q ¨⊗Q′ to be the Cartesian product Q×Q′ whose ordered pairs are denoted by x ¨⊗x′ with x ∈ Q and x′ ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a map wt : Q ¨⊗ Q′ → Λ by wt(x ¨⊗ x′) = wt(x) + wt(x′), for x ∈ Q and x′ ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And for each i ∈ I, define maps ¨ei, ¨fi : Q ¨⊗ Q′ → (Q ¨⊗ Q′) ⊔ {⊥} and ¨εi, ¨ϕi : Q ¨⊗ Q′ → Z ∪ {−∞, +∞} as follows: (1) if ¨ϕi(x) > 0 and ¨εi(x′) > 0, set ¨ei(x ¨⊗ x′) = ¨fi(x ¨⊗ x′) = ⊥ and ¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) otherwise, set ¨ei(x ¨⊗ x′) = � ¨ei(x) ¨⊗ x′ if ¨ϕi(x) ≥ ¨εi(x′) x ¨⊗ ¨ei(x′) if ¨ϕi(x) < ¨εi(x′), 18 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO ¨fi(x ¨⊗ x′) = � ¨fi(x) ¨⊗ x′ if ¨ϕi(x) > ¨εi(x′) x ¨⊗ ¨fi(x′) if ¨ϕi(x) ≤ ¨εi(x′), ¨εi(x ¨⊗ x′) = max � ¨εi(x), ¨εi(x′) − � wt(x), α∨ i �� , and ¨ϕi(x ¨⊗ x′) = max � ¨ϕi(x) + � wt(x′), α∨ i � , ¨ϕi(x′) � , where x ¨⊗ ⊥ = ⊥ ¨⊗ x′ = ⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' for x ∈ Q and x′ ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, Q ¨⊗ Q′ together with the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) forms a seminormal quasi-crystal of type Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q, x′ ∈ Q′ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨εi(x), ¨ϕi(x), ¨εi(x′), ¨ϕi(x′) are all non-negative as Q and Q′ are seminormal (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕi(x) > 0 and ¨εi(x′) > 0, it is immediate that x ¨⊗ x′ satisfies all conditions of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, namely, conditions (1) and (6) which are the ones that apply to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, assume that ¨ϕi(x) = 0 or ¨εi(x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨ϕi(x ¨⊗ x′) = max � ¨ϕi(x) + � wt(x′), α∨ i � , ¨ϕi(x′) � = max � ¨εi(x) + � wt(x), α∨ i � + � wt(x′), α∨ i � , ¨εi(x′) + � wt(x′), α∨ i �� = max � ¨εi(x), ¨εi(x′) − � wt(x), α∨ i �� + � wt(x), α∨ i � + � wt(x′), α∨ i � = ¨εi(x ¨⊗ x′) + � wt(x ¨⊗ x′), α∨ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, condition (1) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕi(x) = +∞ and ¨εi(x′) = 0, , then ¨εi(x) = +∞ and ¨ei(x) = ¨fi(x) = ⊥, by (1) and (6) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, which implies that ¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞, ¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ = ⊥, and ¨fi(x ¨⊗ x′) = ¨fi(x) ¨⊗ x′ = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if ¨ϕi(x) = 0 and ¨εi(x′) = +∞, we have that ¨ei(x ¨⊗ x′) = ¨fi(x ¨⊗ x′) = ⊥ and ¨εi(x ¨⊗ x′) = ¨ϕi(x ¨⊗ x′) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, besides ¨ϕi(x) = 0 or ¨εi(x′) = 0, we may further assume that ¨ϕi(x) ̸= +∞ ̸= ¨εi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get that � wt(x), α∨ i � = ¨ϕi(x) − ¨εi(x) and � wt(x′), α∨ i � = ¨ϕi(x′) − ¨εi(x′) by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(1), implying that ¨εi(x ¨⊗x′) = max � ¨εi(x), ¨εi(x′)− ¨ϕi(x)+ ¨εi(x) � and ¨ϕi(x ¨⊗ x′) = max � ¨ϕi(x) + ¨ϕi(x′) − ¨εi(x′), ¨ϕi(x′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ¨εi(x), ¨εi(x′), ¨ϕi(x), ¨ϕi(x′) are all non-negative where ¨ϕi(x) = 0 or ¨εi(x) = 0, we obtain that ¨εi(x ¨⊗ x′) = ¨εi(x) + ¨εi(x′) and ¨ϕi(x ¨⊗ x′) = ¨ϕi(x) + ¨ϕi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) Now we consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: ¨ϕi(x) = ¨εi(x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ and ¨fi(x ¨⊗ x′) = x ¨⊗ ¨fi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, ¨ei(x ¨⊗ x′) is defined if and only if ¨ei(x) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If so, then we have that wt � ¨ei(x ¨⊗ x′) � = wt � ¨ei(x) � + wt(x′) = wt(x) + αi + wt(x′) = wt(x ¨⊗ x′) + αi, and since ¨ei(x) ¨⊗x′ satisfies the conditions leading to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1), we deduce that ¨εi � ¨ei(x ¨⊗ x′) � = ¨εi � ¨ei(x) � + ¨εi(x′) = ¨εi(x) − 1 + ¨εi(x′) = ¨εi(x ¨⊗ x′) − 1 and ¨ϕi � ¨ei(x ¨⊗ x′) � = ¨ϕi � ¨ei(x) � + ¨ϕi(x′) = ¨ϕi(x) + 1 + ¨ϕi(x′) = ¨ϕi(x ¨⊗ x′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, as ¨ϕi � ¨ei(x) � = ¨ϕi(x) + 1 = 1 and ¨εi(x′) = 0, we get that ¨fi � ¨ei(x ¨⊗ x′) � = ¨fi � ¨ei(x) ¨⊗ x′� = ¨fi � ¨ei(x) � ¨⊗ x′ = x ¨⊗ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 19 On the other hand, ¨fi(x ¨⊗ x′) is defined if and only if ¨fi(x′) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If so, we have that ¨ϕi(x) = 0 and ¨εi � ¨fi(x′) � = ¨εi(x′) + 1 = 1, which implies that ¨ei � ¨fi(x ¨⊗ x′) � = ¨ei � x ¨⊗ ¨fi(x′) � = x ¨⊗ ¨ei � ¨fi(x′) � = x ¨⊗ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: ¨ϕi(x) > 0 and ¨εi(x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨ei(x ¨⊗ x′) = ¨ei(x) ¨⊗ x′ and ¨fi(x ¨⊗ x′) = ¨fi(x) ¨⊗ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, ¨ei(x ¨⊗ x′) is defined if and only if ¨ei(x) ∈ Q, and if so, the facts that condition (3) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 holds and ¨fi � ¨ei(x ¨⊗ x′) � = x ¨⊗ x′ follow as in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q is seminormal and ¨ϕi(x) > 0, we get that ¨fi(x) ∈ Q, which implies that ¨fi(x ¨⊗ x′) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ¨ϕi � ¨fi(x) � = ¨ϕi(x) − 1 ≥ 0 and ¨εi(x′) = 0, we obtain that ¨ei � ¨fi(x ¨⊗ x′) � = ¨ei � ¨fi(x) ¨⊗ x′� = ¨ei � ¨fi(x) � ¨⊗ x′ = x ¨⊗ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 3: ¨ϕi(x) = 0 and ¨εi(x′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨ei(x ¨⊗ x′) = x ¨⊗ ¨ei(x′) and ¨fi(x ¨⊗ x′) = x ¨⊗ ¨fi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q′ is seminormal and ¨εi(x′) > 0, then ¨ei(x′) ∈ Q′, which implies that ¨ei(x ¨⊗ x′) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that wt � ¨ei(x ¨⊗ x′) � = wt(x) + wt(x′) + αi = wt(x ¨⊗ x′) + αi, and since ¨ei(x) ¨⊗ x′ satisfies the conditions leading to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1), we get that ¨εi � ¨ei(x ¨⊗ x′) � = ¨εi(x) + ¨εi(x′) − 1 = ¨εi(x ¨⊗ x′) − 1 and ¨ϕi � ¨ei(x ¨⊗ x′) � = ¨ϕi(x) + ¨ϕi(x′) + 1 = ¨ϕi(x ¨⊗ x′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, as ¨ϕi(x) = 0 and ¨εi � ¨ei(x′) � = ¨εi(x′) − 1 ≥ 0, we obtain that ¨fi � ¨ei(x ¨⊗ x′) � = ¨fi � x ¨⊗ ¨ei(x′) � = x ¨⊗ ¨fi � ¨ei(x′) � = x ¨⊗ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The fact that ¨ei � ¨fi(x ¨⊗x′) � = x ¨⊗x′, whenever ¨fi(x ¨⊗x′) is defined, follows as in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In each case we showed that conditions (2) and (4) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also showed that if x ¨⊗ x′ lies in one of these cases, so does ¨fi(x ¨⊗ x′) when defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, condition (3) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, Q ¨⊗Q′ together with wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) forms a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It remains to prove that this quasi-crystal is seminormal (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that ¨εi(x ¨⊗ x′) ̸= +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨εi(x), ¨ϕi(x), ¨εi(x′), ¨ϕi(x′) ∈ Z≥0 where ¨ϕi(x) = 0 or ¨εi(x′) = 0, and so, we have one of the three cases above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q and Q′ are seminormal, then ¨e¨εi(x) i (x) ∈ Q, ¨e¨εi(x)+1 i (x) = ⊥, ¨e¨εi(x′) i (x′) ∈ Q′, and ¨εi � ¨e¨εi(x′) i (x′) � = 0 ≤ ¨ϕi � ¨e¨εi(x) i (x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) and cases 1 and 3 above, we get that ¨e¨εi(x ¨⊗x′) i (x ¨⊗ x′) = ¨e ¨ϕi(x)+¨εi(x′) i (x ¨⊗ x′) = ¨e¨εi(x) i � x ¨⊗ ¨e¨εi(x′) i (x′) � = ¨e¨εi(x) i (x) ¨⊗ ¨e¨εi(x′) i (x′) is defined, and ¨e¨εi(x ¨⊗x′)+1 i (x ¨⊗ x′) = ¨ei � ¨e¨εi(x) i (x) ¨⊗ ¨e¨εi(x′) i (x′) � = ¨e¨εi(x)+1 i (x) ¨⊗ ¨e¨εi(x′) i (x′) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, we have that ¨f ¨ϕi(x ¨⊗x′) i (x ¨⊗ x′) = ¨f ¨ϕi(x) i (x) ¨⊗ ¨f ¨ϕi(x′) i (x′) is defined, and ¨f ¨ϕi(x ¨⊗x′)+1 i (x ¨⊗ x′) = ¨f ¨ϕi(x) i (x) ¨⊗ ¨f ¨ϕi(x′)+1 i (x′) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, Q ¨⊗ Q′ together with the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) is a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Note that the quasi-crystal structure on Q ¨⊗ Q′ given in (2) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 is similar to the original definition of the crystal structure for the tensor product of crystals [Kas90, Kas91, KN94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if we omitted (1) and applied (2) to all elements, we would have obtained a generalization to quasi-crystals of the tensor product of crystals, as remarked in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note also that 20 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO if we apply the maps ¨ei, ¨fi, ¨εi and ¨ϕi as defined in (2) to elements of the form x ¨⊗ x′ with ¨ϕi(x) = +∞ or ¨εi(x′) = +∞, then we get the same images as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q and Q′ are seminormal, we have that ¨ϕi(x), ¨εi(x′) ∈ Z>0 if and only if ¨fi(x) ∈ Q and ¨ei(x′) ∈ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, condition (1) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 is specifying values for the crystal structure on elements of the form x ¨⊗ x′, where ¨fi(x) and ¨ei(x′) are defined, different from what they would be if the definitions in (2) would apply to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This is a quasi-crystal interpretation of the notion of an i-inversion in a word, introduced in [CM17, § 5], which justifies the following terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The inverse-free quasi-tensor product of Q and Q′, or simply the quasi-tensor product of Q and Q′, is the seminormal quasi-crystal defined in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and is denoted by Q ¨⊗ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We chose to give definitions of the maps of the quasi-crystal structure of a quasi- tensor product in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 to emphasize their resemblance with the maps of the crystal structure of a tensor product of crystals, although in the proof we deduced alternative definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result is an immediate consequence of the arguments that led to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) and the cases that followed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x ∈ Q, x′ ∈ Q′ and i ∈ I with ¨ϕi(x) = 0 or ¨εi(x′) = 0, we have that ¨ei(x ¨⊗ x′) = � ¨ei(x) ¨⊗ x′ if ¨εi(x′) = 0 x ¨⊗ ¨ei(x′) if ¨εi(x′) > 0, ¨fi(x ¨⊗ x′) = � ¨fi(x) ¨⊗ x′ if ¨ϕi(x) > 0 x ¨⊗ ¨fi(x′) if ¨ϕi(x) = 0, ¨εi(x ¨⊗ x′) = ¨εi(x) + ¨εi(x′), ¨ϕi(x ¨⊗ x′) = ¨ϕi(x) + ¨ϕi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) The quasi-crystal A2 3, described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(3) is isomorphic to A3 ¨⊗ A3 as the map A3 × A3 → A3 ¨⊗ A3, given by (x, y) �→ x ¨⊗ y for each x, y ∈ A3, is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13 the quasi-crystal graph ΓA3 ¨⊗A3 is isomorphic to the quasi-crystal graph ΓA2 3, which is drawn in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) The quasi-crystal graph ΓC2 ¨⊗C2 of the quasi-tensor product C2 ¨⊗ C2 (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2)) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨⊗11 2 ¨⊗1 2 ¨⊗1 1 ¨⊗1 1 ¨⊗2 2 ¨⊗2 2 ¨⊗2 1 ¨⊗2 1 ¨⊗2 2 ¨⊗2 2 ¨⊗2 1 ¨⊗2 1 ¨⊗1 2 ¨⊗1 2 ¨⊗1 1 ¨⊗1 1 1 2 1 1 2 2 2 2 1 1 1 1 2 1 1 2 1 1 where wt(x ¨⊗ y) = wt(x) + wt(y), for x, y ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From the previous example, we can see that the quasi-tensor product of semi- normal crystals may not be a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Indeed, if B and B′ are seminormal crystals with elements x ∈ B and x′ ∈ B′ such that ¨fi(x) ∈ B and ¨ei(x′) ∈ B′, for some i ∈ I, then ¨εi(x ¨⊗ x′) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, apart from trivial cases, the quasi-tensor product of seminormal crystals is not a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We only defined quasi-tensor product between quasi-crystals that are seminor- mal, although if we did not require Q and Q′ in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 to be seminormal, the resulting structure would still be a quasi-crystal, eventually not seminormal too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following example shows that this condition is essential to model inversions in words by quasi-crystals, that is, we need both ¨ei and ¨fi to be undefined on x ¨⊗ x′, whenever ¨fi(x) and ¨ei(x′) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 21 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider the standard quasi-crystal A2 of type A2, described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a quasi-crystal of type A2 consisting of a set Q = {−1, −2} and maps given as follows: x wt(x) ¨e(x) ¨f(x) ¨ε(x) ¨ϕ(x) −1 −e1 −2 ⊥ 0 −1 −2 −e2 ⊥ −1 −1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Clearly, A2 is seminormal, but Q is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Nonetheless, set a quasi-crystal structure on A2 ¨⊗Q as defined in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ¨f � 1 ¨⊗(−1) � = 2 ¨⊗(−1), which implies that ¨f is defined on an element of the form x ¨⊗ y where ¨f(x) and ¨e(y) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Alternatively, let ¨e′, ¨f ′ : A2 ¨⊗ Q → (A2 ¨⊗ Q) ⊔ {⊥} be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' x ¨⊗ y ¨e′(x ¨⊗ y) ¨f ′(x ¨⊗ y) 1 ¨⊗ (−1) ⊥ ⊥ 2 ¨⊗ (−1) 1 ¨⊗ (−1) ⊥ 1 ¨⊗ (−2) ⊥ 2 ¨⊗ (−2) 2 ¨⊗ (−2) 1 ¨⊗ (−2) ⊥ So, ¨e′ and ¨f ′ are undefined on x ¨⊗ y whenever ¨f(x) ∈ A2 and ¨e(y) ∈ Q, otherwise ¨e′ and ¨f ′ follow the rule in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, there is no quasi-crystal of type A2 whose quasi-Kashiwara operators are ¨e′ and ¨f ′, because Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4) is not satisfied, as ¨e′� 2 ¨⊗ (−1) � = 1 ¨⊗ (−1) and ¨f ′� 1 ¨⊗ (−1) � = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This illustrates that requiring quasi-crystals to be seminormal is essential to give an interpretation of an inversion on a word by the quasi-tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following result we show that the quasi-tensor product ¨⊗ of quasi-crystals is an associative operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q1, Q2 and Q3 be seminormal quasi-crystals of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The map (Q1 ¨⊗Q2) ¨⊗Q3 → Q1 ¨⊗(Q2 ¨⊗Q3), given by (x1 ¨⊗x2) ¨⊗x3 �→ x1 ¨⊗(x2 ¨⊗x3), is a quasi-crystal isomorphism between (Q1 ¨⊗ Q2) ¨⊗ Q3 and Q1 ¨⊗ (Q2 ¨⊗ Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define ψ : (Q1 ¨⊗ Q2) ¨⊗ Q3 → Q1 ¨⊗ (Q2 ¨⊗ Q3) by ψ((x1 ¨⊗ x2) ¨⊗ x3) = x1 ¨⊗(x2 ¨⊗x3) for x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is immediate that ψ is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since (Q1 ¨⊗Q2) ¨⊗Q3 and Q1 ¨⊗(Q2 ¨⊗Q3) are seminormal by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, to prove that ψ is a quasi-crystal isomorphism, it suffices to show that ψ is a quasi-crystal homomorphism by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, wt � (x1 ¨⊗ x2) ¨⊗ x3 � = wt(x1) + wt(x2) + wt(x3) = wt � x1 ¨⊗ (x2 ¨⊗ x3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For k = 1, 2, 3, we have that ¨εi(xk), ¨ϕi(xk) ≥ 0 as Qk is seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨εi(xk) = +∞, for some k, then ¨εi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨εi � x1 ¨⊗ (x2 ¨⊗ x3) � = ¨εi(xk) = +∞, which implies by (1) and (6) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 that ¨ϕi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨ϕi � x1 ¨⊗ (x2 ¨⊗ x3) � = +∞, ¨ei � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨ei � x1 ¨⊗ (x2 ¨⊗ x3) � = ⊥, and ¨fi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨fi � x1 ¨⊗ (x2 ¨⊗ x3) � = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So assume that ¨εi(xk) ∈ Z≥0 for all k = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following cases we show that if ¨ϕi(xk), ¨εi(xl) > 0 for some 1 ≤ k < l ≤ 3, then ¨εi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨εi � x1 ¨⊗ (x2 ¨⊗ x3) � = +∞, and thus, it follows as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: ¨ϕi(x1), ¨εi(x2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then ¨εi(x1 ¨⊗ x2) = +∞ and ¨εi(x2 ¨⊗ x3) ≥ ¨εi(x2) > 0, which imply that ¨εi � (x1 ¨⊗x2) ¨⊗x3 � = ¨εi � x1 ¨⊗(x2 ¨⊗x3) � = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: ¨ϕi(x1), ¨εi(x3) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then ¨ϕi(x1 ¨⊗x2) ≥ ¨ϕi(x1) > 0 and ¨εi(x2 ¨⊗x3) ≥ ¨εi(x3) > 0, which imply that ¨εi � (x1 ¨⊗x2) ¨⊗x3 � = ¨εi � x1 ¨⊗(x2 ¨⊗x3) � = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 3: ¨ϕi(x2), ¨εi(x3) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then ¨ϕi(x1 ¨⊗x2) ≥ ¨ϕi(x2) > 0 and ¨εi(x2 ¨⊗x3) = +∞, which imply that ¨εi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨εi � x1 ¨⊗ (x2 ¨⊗ x3) � = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we further assume that ¨εi(xk), ¨ϕi(xl) > 0 implies k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 22 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we get that ¨εi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨εi(x1) + ¨εi(x2) + ¨εi(x3) = ¨εi � x1 ¨⊗ (x2 ¨⊗ x3) � and ¨ϕi � (x1 ¨⊗ x2) ¨⊗ x3 � = ¨ϕi(x1) + ¨ϕi(x2) + ¨ϕi(x3) = ¨ϕi � x1 ¨⊗ (x2 ¨⊗ x3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also have that ¨ei � (x1 ¨⊗ x2) ¨⊗ x3 � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � ¨ei(x1) ¨⊗ x2 � ¨⊗ x3 if ¨εi(x2) = ¨εi(x3) = 0 � x1 ¨⊗ ¨ei(x2) � ¨⊗ x3 if ¨εi(x2) > 0 = ¨εi(x3) (x1 ¨⊗ x2) ¨⊗ ¨ei(x3) if ¨εi(x3) > 0 and ¨ei � x1 ¨⊗ (x2 ¨⊗ x3) � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ¨ei(x1) ¨⊗ (x2 ¨⊗ x3) if ¨εi(x2) = ¨εi(x3) = 0 x1 ¨⊗ � ¨ei(x2) ¨⊗ x3 � if ¨εi(x2) > 0 = ¨εi(x3) x1 ¨⊗ � x2 ¨⊗ ¨ei(x3) � if ¨εi(x3) > 0, implying that ψ � ¨ei((x1 ¨⊗ x2) ¨⊗ x3) � = ¨ei � ψ(x1 ¨⊗ (x2 ¨⊗ x3)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, ¨fi � (x1 ¨⊗ x2) ¨⊗ x3 � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � ¨fi(x1) ¨⊗ x2 � ¨⊗ x3 if ¨ϕi(x1) > 0 � x1 ¨⊗ ¨fi(x2) � ¨⊗ x3 if ¨ϕi(x1) = 0 < ¨ϕi(x2) (x1 ¨⊗ x2) ¨⊗ ¨fi(x3) if ¨ϕi(x1) = ¨ϕi(x2) = 0 and ¨fi � x1 ¨⊗ (x2 ¨⊗ x3) � = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ¨fi(x1) ¨⊗ (x2 ¨⊗ x3) if ¨ϕi(x1) > 0 x1 ¨⊗ � ¨fi(x2) ¨⊗ x3 � if ¨ϕi(x1) = 0 < ¨ϕi(x2) x1 ¨⊗ � x2 ¨⊗ ¨fi(x3) � if ¨ϕi(x1) = ¨ϕi(x2) = 0, implying that ψ � ¨fi((x1 ¨⊗ x2) ¨⊗ x3) � = ¨fi � ψ(x1 ¨⊗ (x2 ¨⊗ x3)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Due to the previous result, we may omit parenthesis for the quasi-tensor prod- uct of seminormal quasi-crystals and simply write Q1 ¨⊗ Q2 ¨⊗ Q3, whose elements are denoted by x1 ¨⊗ x2 ¨⊗ x3, for x1 ∈ Q1, x2 ∈ Q2 and x3 ∈ Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From the proofs of Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, we deduce the following result, which generalizes Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3 and describes the quasi-crystal structure of a quasi-tensor product of an arbitrary number of seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , Qm be seminormal quasi-crystals of the same type, and let x1 ∈ Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, wt(x1 ¨⊗ · · · ¨⊗ xm) = wt(x1) + · · · + wt(xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, for i ∈ I, by setting p = max � 1 ≤ k ≤ m �� ¨εi(xk) > 0 � and q = min � 1 ≤ l ≤ m �� ¨ϕi(xl) > 0 � , we have that (1) if p > q or ¨εi(xk) = +∞ for some 1 ≤ k ≤ m, then ¨ei(x1 ¨⊗ · · · ¨⊗ xm) = ¨fi(x1 ¨⊗ · · · ¨⊗ xm) = ⊥ and ¨εi(x1 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(x1 ¨⊗ · · · ¨⊗ xm) = +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 23 (2) otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨ei(x1 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xp−1 ¨⊗ ¨ei(xp) ¨⊗ xp+1 ¨⊗ · · · ¨⊗ xm ¨fi(x1 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xq−1 ¨⊗ ¨fi(xq) ¨⊗ xq+1 ¨⊗ · · · ¨⊗ xm ¨εi(x1 ¨⊗ · · · ¨⊗ xm) = ¨εi(x1) + · · · + ¨εi(xp) and ¨ϕi(x1 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(xq) + · · · + ¨ϕi(xm) In the following result we show that quasi-crystal homomorphisms between semi- normal quasi-crystals give rise to homomorphisms between quasi-tensor products of their domains and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q1, Q2, Q′ 1 and Q′ 2 be seminormal quasi-crystals of the same type, and let ψ1 : Q1 → Q′ 1 and ψ2 : Q2 → Q′ 2 be quasi-crystal homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The partial map ψ1 ¨⊗ψ2 : Q1 ¨⊗Q2 → Q′ 1 ¨⊗Q′ 2, given by x1 ¨⊗x2 �→ ψ1(x1) ¨⊗ψ2(x2) for each x1 ∈ Q1 and x2 ∈ Q2 such that ψ1(x1) ∈ Q′ 1 and ψ2(x2) ∈ Q′ 2, is a quasi-crystal homomorphism from Q1 ¨⊗ Q2 to Q′ 1 ¨⊗ Q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, if ψ1 and ψ2 are quasi-crystal isomorphisms, then ψ1 ¨⊗ ψ2 is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x1 ∈ Q1 and x2 ∈ Q2 be such that ψ1(x1) ∈ Q′ 1 and ψ2(x2) ∈ Q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get that wt � ψ1(x1) ¨⊗ψ2(x2) � = wt � ψ1(x1) � +wt � ψ2(x2) � = wt(x1)+wt(x2) = wt(x1 ¨⊗x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I and k ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), We have that ¨εi � ψk(xk) � = ¨εi(xk) and ¨ϕi � ψk(xk) � = ¨ϕi(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if ¨ϕi(x1) > 0 and ¨εi(x2) > 0, then ¨ei � ψ1(x1) ¨⊗ ψ2(x2) � = ¨fi � ψ1(x1) ¨⊗ ψ2(x2) � = ¨ei(x1 ¨⊗ x2) = ¨fi(x1 ¨⊗ x2) = ⊥ and ¨εi � ψ1(x1) ¨⊗ ψ2(x2) � = ¨ϕi � ψ1(x1) ¨⊗ ψ2(x2) � = ¨εi(x1 ¨⊗ x2) = ¨ϕi(x1 ¨⊗ x2) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, we get that ¨εi � ψ1(x1) ¨⊗ ψ2(x2) � = max � ¨εi � ψ1(x1) � , ¨εi � ψ2(x2) � − � wt � ψ1(x1) � , α∨ i �� = max � ¨εi(x1), ¨εi(x2) − � wt(x1), α∨ i �� = ¨εi(x1 ¨⊗ x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, ¨ϕi � ψ1(x1) ¨⊗ ψ2(x2) � = ¨ϕi(x1 ¨⊗ x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(3), if ¨ei(xk) ∈ Qk and ψk � ¨ei(xk) � ∈ Q′ k, then ψk � ¨ei(xk) � = ¨ei � ψk(xk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if ¨ei(x1 ¨⊗ x2) ∈ Q1 ¨⊗ Q2 and (ψ1 ¨⊗ ψ2) � ¨ei(x1 ¨⊗ x2) � ∈ Q′ 1 ¨⊗ Q′ 2, then (ψ1 ¨⊗ ψ2) � ¨ei(x1 ¨⊗ x2) � = � ψ1 � ¨ei(x1) � ¨⊗ ψ2(x2) if ¨ϕi(x1) ≥ ¨εi(x2) ψ1(x1) ¨⊗ ψ2 � ¨ei(x2) � if ¨ϕi(x1) < ¨εi(x2) = � ¨ei � ψ1(x1) � ¨⊗ ψ2(x2) if ¨ϕi � ψ1(x1) � ≥ ¨εi � ψ2(x2) � ψ1(x1) ¨⊗ ¨ei � ψ2(x2) � if ¨ϕi � ψ1(x1) � < ¨εi � ψ2(x2) � = ¨ei � ψ1(x1) ¨⊗ ψ2(x2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, if ¨fi(x1 ¨⊗ x2) ∈ Q1 ¨⊗ Q2 and (ψ1 ¨⊗ ψ2) � ¨fi(x1 ¨⊗ x2) � ∈ Q′ 1 ¨⊗ Q′ 2, then (ψ1 ¨⊗ψ2) � ¨fi(x1 ¨⊗x2) � = ¨fi � ψ1(x1) ¨⊗ψ2(x2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ1 ¨⊗ψ2 is a quasi-crystal homomorphism from Q1 ¨⊗ Q2 to Q′ 1 ¨⊗ Q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ψ1 and ψ2 are quasi-crystal isomorphisms, then ψ1 ¨⊗ ψ2 is a bijective quasi- crystal homomorphism between seminormal quasi-crystals, as proved above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18, ψ1 ¨⊗ ψ2 is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 24 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The signature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now describe a practical method to compute the quasicrystal structure of the quasi-tensor product of seminormal quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This method is essentially a combinatorial interpretation of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7, and has a procedure similar to the signature rule for the tensor product of seminormal crystals [HK02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Z0 be the monoid with zero defined by the following presentation ⟨−, + | (+−, 0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, an element of Z0, other than 0, has the form −a+b with a, b ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ I define sgni : Q → Z0 by sgni(x) = � 0 if ¨εi(x) = +∞ −¨εi(x)+ ¨ϕi(x) otherwise, for each x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The map sgni is called the i-signature map for the quasi-tensor product ¨⊗, and sgni(x) is called the i-signature of x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In comparison with the signature map for the tensor product of crystals, we have that the bicyclic monoid ⟨−+ | (+−, ǫ)⟩ (where ǫ denotes the empty word) has been replaced by the monoid Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This allows sgni to interact with the quasi- tensor product of seminormal quasi-crystals in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q and Q′ be seminormal quasi-crystals of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, sgni(x ¨⊗ x′) = sgni(x) sgni(x′), for all x ∈ Q, x′ ∈ Q′ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Q, x′ ∈ Q′ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7, if ¨εi(x) = +∞ (or equivalently, ¨ϕi(x) = +∞), then sgni(x) = 0 and ¨εi(x ¨⊗ x′) = +∞, which implies that sgni(x ¨⊗ x′) = 0 = sgni(x) sgni(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, if ¨εi(x′) = +∞, we have that sgni(x ¨⊗ x′) = 0 = sgni(x) sgni(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, assume that ¨ϕi(x), ¨εi(x′) ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕi(x), ¨εi(x′) > 0, then ¨εi(x ¨⊗ x′) = +∞, and sgni(x) sgni(x′) = −¨εi(x)+ ¨ϕi(x)−1+−−¨εi(x′)−1+ ¨ϕi(x′) = 0 = sgni(x ¨⊗ x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, assume that ¨ϕi(x) = 0 or ¨εi(x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕi(x) = 0, then sgni(x) sgni(x′) = −¨εi(x)+¨εi(x′)+ ¨ϕi(x′) = −¨εi(x ¨⊗x′)+ ¨ϕi(x ¨⊗x′) = sgni(x ¨⊗ x′), by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨εi(x′) = 0, then sgni(x) sgni(x′) = −¨εi(x)+ ¨ϕi(x)+ ¨ϕi(x′) = −¨εi(x ¨⊗x′)+ ¨ϕi(x ¨⊗x′) = sgni(x ¨⊗ x′), by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From the previous result, given seminormal quasi-crystals Q1, Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , Qm of the same type and elements x1 ∈ Q1, x2 ∈ Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Qm, we can easily compute the i-signature of x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm as sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = sgni(x1) sgni(x2) · · · sgni(xm) (i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = 0, then ¨εi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = ¨ϕi(x1 ¨⊗ x2 ¨⊗ · · ¨⊗ xm) = +∞ which implies ¨ei(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = ¨fi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = −a+b, for some a, b ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ¨εi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = a and ¨ϕi(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7, if a ≥ 1, then ¨ei(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm) = x1 ¨⊗ · · · ¨⊗ xp−1 ¨⊗ ¨ei(xp) ¨⊗ xp+1 ¨⊗ · · · ¨⊗ xm, where xp originates the right-most symbol − in sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, if b ≥ 1, then ¨fi(x1 ¨⊗x2 ¨⊗· · · ¨⊗xm) = x1 ¨⊗· · · ¨⊗xq−1 ¨⊗ ¨fi(xq) ¨⊗xq+1 ¨⊗· · · ¨⊗xm, where xq originates the left-most symbol + in sgni(x1 ¨⊗ x2 ¨⊗ · · · ¨⊗ xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This process is called the signature rule for the quasi-tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 25 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider the quasi-crystal A4 ¨⊗A4 ¨⊗A4 ¨⊗A4 ¨⊗A4, where A4 is the standard quasi-crystal of type A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We compute ¨e2, ¨f2, ¨ε2 and ¨ϕ2 on 3 ¨⊗1 ¨⊗2 ¨⊗2 ¨⊗3 using the signature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To keep track to which element originates each − and + we write a subscript with the position of the element, this is just an auxiliary notation and the binary operation of Z0 should be applied ignoring the subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So we have that sgn2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = −1+3+4−5 = −1+30 = 0, and therefore, ¨e2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = ⊥, ¨ε2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = +∞, ¨f2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = ⊥, ¨ϕ2(3 ¨⊗ 1 ¨⊗ 2 ¨⊗ 2 ¨⊗ 3) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Now we compute ¨e1, ¨f1, ¨ε1 and ¨ϕ1 on 2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Using the same notation as above, we obtain that sgn1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = −1−3+5, and therefore, ¨e1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2 ¨⊗ 3 ¨⊗ 1 ¨⊗ 3 ¨⊗ 1, ¨ε1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2, ¨f1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 2, ¨ϕ1(2 ¨⊗ 3 ¨⊗ 2 ¨⊗ 3 ¨⊗ 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal monoids In this section we study the algebraic framework relating quasi-crystals and monoids, which will be used to give a general definition of hypoplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, we present the definition of quasi-crystal monoid, which is the basic concept for relating quasi-crystals and monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we introduce the definition of free quasi-crystal monoid over a seminormal quasi- crystal, and show that free quasi-crystal monoids satisfy a universal property that defines them up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we present the notion of congruences on a quasi-crystal monoid, which form a lattice and allow to consider quotients of quasi-crystal monoids, leading to the homomorphism theorems for quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Quasi-crystal monoids and homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first introduce the fun- damental concept relating quasi-crystals and monoids with respect to the quasi- tensor product ¨⊗, studied in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Φ be a root system with weight lattice Λ and index set I for the simple roots (αi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A ¨⊗-quasi-crystal monoid M of type Φ consists of a set M together with maps wt : M → Λ, ¨ei, ¨fi : M → M⊔{⊥}, ¨εi, ¨ϕi : M → Z∪{−∞, +∞} (i ∈ I) and a binary operation · : M × M → M satisfying the following conditions: (1) M together with wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) forms a seminormal quasi- crystal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) M together with · forms a monoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) the map M ¨⊗ M → M, given by x ¨⊗ y �→ x · y for x, y ∈ M, induces a quasi-crystal homomorphism from M ¨⊗ M to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We stated a definition of quasi-crystal monoid with respect to the quasi-tensor product ¨⊗, because we shall see that it models the binary operation of the hy- poplactic monoid, which we want to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A similar definition can be given by replacing the quasi-tensor product by other operation on quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For instance, if we considered the tensor product instead, the subsequent would lead to a notion of plactic monoid over a quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since our goal is to introduce the notion of hypoplactic monoid associated to a quasi-crystal, we will only consider 26 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO quasi-crystal monoids with respect to the quasi-tensor product ¨⊗, and thus, we will omit ¨⊗ and just say that M is a quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In a quasi-crystal monoid the interaction between the quasi-crystal structure and the binary operation satisfies rules similar to those satisfied by the quasi- crystal structure of a quasi-tensor product (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3), as shown in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x, y ∈ M, we have that wt(xy) = wt(x) + wt(y), and for i ∈ I, if ¨ϕi(x) > 0 and ¨εi(y) > 0, then ¨ei(xy) = ¨fi(xy) = ⊥ and ¨εi(xy) = ¨ϕi(xy) = +∞, otherwise, ¨ei(xy) = � ¨ei(x) · y if ¨εi(y) = 0 x · ¨ei(y) if ¨εi(y) > 0, ¨fi(xy) = � ¨fi(x) · y if ¨ϕi(x) > 0 x · ¨fi(y) if ¨ϕi(x) = 0, ¨εi(xy) = ¨εi(x) + ¨εi(y), and ¨ϕi(xy) = ¨ϕi(x) + ¨ϕi(y), where x⊥ = ⊥y = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(3), let ψ : M ¨⊗ M → M be the quasi-crystal homo- morphism given by ψ(x ¨⊗ y) = xy, for x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ M and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), we get that wt(xy) = wt � ψ(x ¨⊗ y) � = wt(x ¨⊗ y) = wt(x) + wt(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and similarly, ¨εi(xy) = ¨εi(x ¨⊗ y) and ¨ϕi(xy) = ¨ϕi(x ¨⊗ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(1), M is seminormal (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9), and by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, M ¨⊗ M is also seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ¨εi(xy) = ¨εi(x ¨⊗y), we have that ¨ei is defined on xy if and only if ¨ei is defined on x ¨⊗y, and since ¨ϕi(xy) = ¨ϕi(x ¨⊗y), ¨fi is defined on xy if and only if ¨fi is defined on x ¨⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, as ψ(M ¨⊗ M) ⊆ M, we obtain that ¨ei(xy) = ψ � ¨ei(x ¨⊗ y) � and ¨fi(xy) = ψ � ¨fi(x ¨⊗ y) � , by conditions (3) and (4) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the result follows directly from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The previous result can be generalized to get the values of the quasi-crystal structure on an element of the form x1 · · · xm based only on their values on each xk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to an analogue of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid, and let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, wt(x1 · · · xm) = wt(x1) + · · · + wt(xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, for i ∈ I, by setting p = max � 1 ≤ k ≤ m �� ¨εi(xk) > 0 � and q = min � 1 ≤ l ≤ m �� ¨ϕi(xl) > 0 � , we have that (1) if p > q or ¨εi(xk) = +∞ for some 1 ≤ k ≤ m, then ¨ei(x1 · · · xm) = ¨fi(x1 · · · xm) = ⊥ and ¨εi(x1 · · · xm) = ¨ϕi(x1 · · · xm) = +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) otherwise, ¨ei(x1 · · · xm) = x1 · · · xp−1 · ¨ei(xp) · xp+1 · · · xm, ¨fi(x1 · · · xm) = x1 · · · xq−1 · ¨fi(xq) · xq+1 · · · xm, ¨εi(x1 · · · xm) = ¨εi(x1) + · · · + ¨εi(xp), QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 27 and ¨ϕi(x1 · · · xm) = ¨ϕi(xq) + · · · + ¨ϕi(xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We proceed by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If m = 1, then the result is trivial, and if m = 2, then it coincides with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume as induction hypothesis (IH) that the result holds for any x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xk ∈ M with k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , ym, ym+1 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since · is associative, we have that y1 · · · ymym+1 = (y1 · · · ym)ym+1, where wt � (y1 · · · ym)ym+1 � = wt(y1 · · · ym)+wt(ym+1) = wt(y1)+· · ·+wt(ym)+wt(ym+1), by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 and (IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨εi(yk) = +∞, for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', m + 1}, then we have when k ≤ m that ¨εi(y1 · · · ym) = +∞, by (IH), which implies that ¨εi(y1 · · · ymym+1) = ¨εi((y1 · · · ym) ¨⊗ ym+1) = +∞, and by conditions (1) and (6) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, ¨ei(y1 · · · ym+1) = ¨fi(y1 · · · ym+1) = ⊥ and ¨ϕi(y1 · · · ym+1) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, assume that ¨εi(yk) ∈ Z≥0, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set p = max � 1 ≤ k ≤ m + 1 �� ¨εi(xk) > 0 � and q = min � 1 ≤ l ≤ m + 1 �� ¨ϕi(xl) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that p > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, p > 1 and q < m+1 implying that the sets where the maximum and minimum are taken are nonempty, and thus, ¨εi(yp), ¨ϕi(yq) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By (IH), ¨ϕi(y1 · · · yp−1) ≥ ¨ϕi(yq) > 0, and ¨εi(yp · · · ym+1) ≥ ¨εi(yp) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 that ¨εi(y1 · · · ym+1) = ¨εi � (y1 · · · yp−1)(yp · · · ym+1) � = +∞, and by conditions (1) and (6) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, ¨ei(y1 · · · ym+1) = ¨fi(y1 · · · ym+1) = ⊥ and ¨ϕi(y1 · · · ym+1) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, suppose that p ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ¨ϕi(yk) = 0, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , q − 1, we have by (IH) that ¨εi(y1 · · · yp−1) = ¨εi(y1)+· · ·+¨εi(yp−1) and ¨ϕi(y1 · · · yp−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we get that ¨ei � (y1 · · · yp−1)yp � = y1 · · · yp−1 · ¨ei(yp) (note that if ¨εi(yp) = 0, then p = 1 and ¨ei(yp) = ⊥, as M is seminormal), and by (IH), ¨εi � (y1 · · · yp−1)yp � = ¨εi(y1 · · · yp−1) + ¨εi(yp) = ¨εi(y1) + · · · + ¨εi(yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, since ¨εi(yk) = 0, for k = p + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m + 1, we have by (IH) that ¨εi(yp+1 · · · ym+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, we obtain that ¨ei � (y1 · · · yp)(yp+1 · · · ym+1) � = ¨ei(y1 · · · yp) · yp+1 · · · ym+1 = y1 · · · yp−1 · ¨ei(yp) · yp+1 · · · ym+1 and ¨εi � (y1 · · · yp)(yp+1 · · · ym+1) � = ¨εi(y1 · · · yp) + ¨εi(yp+1 · · · ym+1) = ¨εi(y1) + · · · + ¨εi(yp), by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, we have that ¨fi � (y1 · · · yq−1)(yq · · · ym+1) � = y1 · · · yq−1 · ¨fi(yq) · yq+1 · · · ym+1 and ¨ϕi � (y1 · · · yq−1)(yq · · · ym+1) � = ¨ϕi(yq) + · · · + ¨ϕi(ym+1), by (IH) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ In the previous result we saw how the monoid binary operation · interacts with the quasi-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that this allows us to relate some prop- erties of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' First, we recall that an element x of a monoid M is called a commutative element, also known as central element, if x commutes with every ele- ment, that is, xy = yx, for any y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also recall that x is called an idempotent element if x2 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 28 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid and let x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) If x is a commutative element, then x is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) If x is an idempotent element, then x is isolated and wt(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Suppose that x is not an isolated element of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, take i ∈ I such that ¨ei or ¨fi is defined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As M is seminormal, if ¨ei is defined on x, then ¨εi(x), ¨ϕi(x) ∈ Z≥0, where ¨εi(x) > 0, and the element y = ¨e¨εi(x) i (x) satisfies ¨εi(y) = 0 and ¨ϕi(y) ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, ¨εi(xy) = ¨εi(x) ∈ Z>0 and ¨εi(yx) = +∞, which implies that xy ̸= yx, and thus, x is not commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, ¨fi is defined on x, and since M is seminormal, we have that ¨εi(x), ¨ϕi(x) ∈ Z≥0 where ¨ϕi(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The element z = ¨f ¨ϕi(x) i (x) is such that ¨εi(z) ∈ Z>0 and ¨ϕi(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, ¨ϕi(xz) = +∞ and ¨ϕi(zx) = ¨ϕi(x) ∈ Z>0, which implies that xz ̸= zx, and therefore, x is not commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Assume that x is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we have that wt(x) = wt � x2� = wt(x) + wt(x), which implies that wt(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨εi(x) ̸= +∞ (or equivalently, ¨ϕi(x) ̸= +∞), for some i ∈ I, then ¨εi(x) = ¨εi � x2� = ¨εi(x) + ¨εi(x) and ¨ϕi(x) = ¨ϕi � x2� = ¨ϕi(x) + ¨ϕi(x), impliying that ¨εi(x) = ¨ϕi(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ¨εi(x), ¨ϕi(x) ∈ {0, +∞}, for any i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As M is seminormal, we obtain that x is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The monoid identity is in particular both a commutative and an idempotent element, but as we show in the following result, its properties may affect the whole quasi-crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid where the monoid identity is denoted by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, 1 is isolated and wt(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, for each i ∈ I, either ¨εi(1) = ¨ϕi(1) = 0 or ¨εi(x) = +∞, for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 1 is an idempotent element, then 1 is isolated and wt(1) = 0, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As M is seminormal and 1 is isolated, we get for each i ∈ I that either ¨εi(1) = ¨ϕi(1) = 0 or ¨εi(1) = ¨ϕi(1) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose there exists x ∈ M such that ¨εi(x) ̸= +∞, for some i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since M is seminormal, we get that ¨εi � ¨e¨εi(x) i (x) � = ¨ϕi � ¨f ¨ϕi(x) i (x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set y = ¨e¨εi(x) i (x) and z = ¨f ¨ϕi(x) i (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, 0 ≤ ¨εi(1) ≤ ¨εi(1y) = ¨εi(y) = 0 and 0 ≤ ¨ϕi(1) ≤ ¨ϕi(1z) = ¨ϕi(z) = 0, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Note that in the case where for some i ∈ I we have ¨εi(x) = +∞, for all x ∈ M, the quasi-Kashiwara operators ¨ei and ¨fi are undefined on every element in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Such a case has little interest to study in the context of this paper, and we say that such a quasi-crystal monoid is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, we get the following characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal monoid M is said to be nondegenerate if ¨εi(1) = ¨ϕi(1) = 0, for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 29 Due to the interaction between the binary operation · of a quasi-crystal monoid M and the quasi-crystal structure of the quasi-tensor product M ¨⊗ M required by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(3), we can extend the signature rule described in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 to quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ M and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ¨εi(xy) = ¨εi(x ¨⊗ y) and ¨ϕi(xy) = ¨ϕi(x ¨⊗ y), we have that the i-signature of xy and x ¨⊗ y coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, sgni(xy) = sgni(x) sgni(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, either sgni(1) = ǫ or sgni(z) = 0, for any z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we obtained the following result, which can be seen as an improvement of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 for nondegenerate quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid, and let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, sgni(xy) = sgni(x) sgni(y), for any x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, sgni is a monoid homomorphism from M to Z0 if and only if ¨εi(x) ∈ Z≥0, for some x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The signature rule for quasi-crystal monoids follows directly from the previous result and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider a quasi-crystal monoid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ M and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, we can compute the i-signature of x1 · · · xm based only in the i-signature of each xk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m, because sgni(x1 · · · xm) = sgni(x1) · · · sgni(xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If sgni(x1 · · · xm) = 0, then ¨εi(x1 · · · xm) = ¨ϕi(x1 · · · xm) = +∞ which implies that ¨ei(x1 · · · xm) = ¨fi(x1 · · · xm) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, sgni(x1 · · · xm) = −a+b, for some a, b ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ¨εi(x1 · · · xm) = a and ¨ϕi(x1 · · · xm) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The raising quasi- Kashiwara operator ¨ei is defined on x1 · · · xm if and only if a ≥ 1, in which case ¨ei(x1 · · · xm) = x1 · · · xp−1 · ¨ei(xp) · xp+1 · · · xm, where xp originates the right-most symbol − in sgni(x1 · · · xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, the lowering quasi-Kashiwara operator ¨fi is defined on x1 · · · xm if and only if b ≥ 1, in which case ¨fi(x1 · · · xm) = x1 · · · xq−1 · ¨fi(xq) · xq+1 · · · xm, where xq originates the left-most symbol + in sgni(x1 · · · xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now introduce the notion of a homomorphism between quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M and M′ be quasi-crystal monoids of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal monoid homomorphism ψ from M to M′, denoted by ψ : M → M′, is a map ψ : M → M ′ that satisfies the following conditions: (1) ψ is a quasi-crystal homomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) ψ is a monoid homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ψ is also bijective, it is called a quasi-crystal monoid isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that in the previous definition we only consider maps from M to M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' But as we observed after Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12, when we state that ψ is a quasi-crystal homomorphism in condition (1) above, we mean that the map ψ′ : M ⊔ {⊥} → M ′ ⊔ {⊥}, defined by ψ′(⊥) = ⊥ and ψ′(x) = ψ(x), for each x ∈ M, is a quasi- crystal homomorphism from M to M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, if ψ : M → M′ is a quasi-crystal monoid isomorphism, then ψ is both a quasi-crystal isomorphism (by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18) and a monoid isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The converse is immediate, because a monoid isomorphism is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, a map ψ : M → M ′ is a quasi-crystal monoid isomorphism if and only if ψ is a quasi-crystal isomorphism and a monoid isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that if ψ : M → M′ is a quasi-crystal monoid isomorphism, then ψ−1 is a quasi-crystal monoid isomorphism between M′ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The free quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For k ≥ 1, set Q ¨⊗k = Q ¨⊗ · · · ¨⊗ Q � �� � k times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 30 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18 and Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, for k, l ≥ 1 the map Q ¨⊗k ¨⊗ Q ¨⊗l → Q ¨⊗(k+l), given by (x1 ¨⊗ · · · ¨⊗ xk) ¨⊗ (y1 ¨⊗ · · · ¨⊗ yl) �→ x1 ¨⊗ · · · ¨⊗ xk ¨⊗ y1 ¨⊗ · · · ¨⊗ yl, for x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xk, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , yl ∈ Q, is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let ζ be an element that does not lie in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set Q ¨⊗0 to be the seminormal quasi- crystal of the same type as Q formed by the set Q ¨⊗0 = {ζ} and maps given by wt(ζ) = 0, ¨ei(ζ) = ¨fi(ζ) = ⊥ and ¨εi(ζ) = ¨ϕi(ζ) = 0 (i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that Q ¨⊗0 ¨⊗Q ¨⊗0 is quasi-crystal isomorphic to Q ¨⊗0, as both quasi-crystals consist of a single element where the quasi-crystal structure maps coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For any x ∈ Q and i ∈ I, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we have that wt(x ¨⊗ζ) = wt(x), ¨εi(x ¨⊗ζ) = ¨εi(x) and ¨ϕi(x ¨⊗ζ) = ¨ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, by the signature rule, since sgni(ζ) = ǫ, it is immediate that ¨ei (or ¨fi) is defined on x ¨⊗ ζ if and only if ¨ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', ¨fi) is defined on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, ¨ei(x ¨⊗ ζ) = ¨ei(x) ¨⊗ ζ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', ¨fi(x ¨⊗ ζ) = ¨fi(x) ¨⊗ ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the map Q ¨⊗ Q ¨⊗0 → Q, given by y ¨⊗ ζ �→ y for each y ∈ Q, is a quasi-crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, the map Q ¨⊗0 ¨⊗Q → Q, given by ζ ¨⊗y �→ y for each y ∈ Q, is a quasi- crystal isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since the quasi-tensor product of quasi-crystals is associative (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6), we get that Q ¨⊗k ¨⊗ Q ¨⊗0 and Q ¨⊗0 ¨⊗Q ¨⊗k are isomorphic to Q ¨⊗k, for any k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The sets Q ¨⊗k and Q ¨⊗l are disjoint, whenever k ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we can extend the maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I) defined on each Q ¨⊗k to the set M = � k≥0 Q ¨⊗k obtaining a seminormal quasi-crystal M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By the quasi-crystal isomorphisms Q ¨⊗k ¨⊗ Q ¨⊗l → Q ¨⊗(k+l) (k, l ≥ 0) defined above, we get that M becomes a quasi-crystal monoid with the binary operation · : M × M → M given by (x1 ¨⊗ · · · ¨⊗ xk) · (y1 ¨⊗ · · · ¨⊗ yl) = x1 ¨⊗ · · · ¨⊗ xk ¨⊗ y1 ¨⊗ · · · ¨⊗ yl and (x1 ¨⊗ · · · ¨⊗ xk) · ζ = ζ · (x1 ¨⊗ · · · ¨⊗ xk) = x1 ¨⊗ · · · ¨⊗ xk, for x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xk, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , yl ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If we identify ζ with the empty word ǫ and each element of the form x1 ¨⊗ · · · ¨⊗ xk in M with the word x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xk over the alphabet Q, we obtain a monoid isomorphism between M and the free monoid Q∗ over Q, and through this identification we can also define a quasi-crystal structure on Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we have constructed a quasi-crystal monoid that leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The free quasi-crystal monoid Q¨∗ over Q is a quasi-crystal monoid of the same type as Q consisting of the set Q∗ of all words over Q, the usual concatenation of words, and quasi-crystal structure maps defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' set wt(ǫ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨ei(ǫ) = ¨fi(ǫ) = ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and ¨εi(ǫ) = ¨ϕi(ǫ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and for u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' v ∈ Q∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' set wt(uv) = wt(u) + wt(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' if ¨ϕi(u) > 0 and ¨εi(v) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' set ¨ei(uv) = ¨fi(uv) = ⊥ and ¨εi(uv) = ¨ϕi(uv) = +∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' set ¨ei(uv) = � ¨ei(u)v if ¨ϕi(u) ≥ ¨εi(v) u¨ei(v) if ¨ϕi(u) < ¨εi(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 31 ¨fi(uv) = � ¨fi(u)v if ¨ϕi(u) > ¨εi(v) u ¨fi(v) if ¨ϕi(u) ≤ ¨εi(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨εi(uv) = max � ¨εi(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨εi(v) − � wt(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' α∨ i �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and ¨ϕi(uv) = max � ¨ϕi(u) + � wt(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' α∨ i � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' ¨ϕi(v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' where u⊥ = ⊥v = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As we constructed the free quasi-crystal monoid Q¨∗ based on the quasi-tensor product ¨⊗ of quasi-crystals, we described its quasi-crystal structure based on the definition of the quasi-crystal structure of a quasi-tensor product (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Notice that we explicitly gave the values of the quasi-crystal structure maps of Q¨∗ on ζ (which we identify with the empty word ǫ), on letters the values follow from the quasi-crystal structure maps of Q, and on a word of the form uv they depend only on their values on u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the definition of the quasi-crystal structure above is not circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we can obtain the values of the quasi-crystal structure maps on a word based only on their values on its letters, which implies the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For any w ∈ Q∗ and i ∈ I, if ¨ei(w) ∈ Q∗ then ��¨ei(w) �� = |w|, and if ¨fi(w) ∈ Q∗ then �� ¨fi(w) �� = |w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, |u| = |v|, whenever u and v lie in the same connected component of Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ Q∗ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨fi is defined on w, then w ̸= ǫ, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, and so, w = x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm, for some x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Q and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, there exists q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m} such that ¨fi(w) = x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xq−1 ¨fi(xq)xq+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since xq ∈ Q, then ¨fi(xq) ∈ Q, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, which implies that �� ¨fi(w) �� = |w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, for any w ∈ Q∗ and i ∈ I, �� ¨fi(w) �� = |w|, whenever ¨fi(w) ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, for any w ∈ Q∗ and i ∈ I, if ¨ei is defined on w, then ��¨ei(w) �� = |w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, let u and v be words lying in the same connected component of Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gk ∈ {¨ei, ¨fi | i ∈ I} such that g1 · · · gk(u) = v, and by applying recursively what we have proven above, |g1 · · · gk(u)| = |u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From the previous result, we can deduce the following properties of the connected components of Q¨∗ (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal whose underlying set Q is finite, and let Q′ ⊆ Q∗ be a connected component of Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, (1) Q′ is finite, (2) Q′ has at least a highest-weight element, and (3) Q′ has at least a lowest-weight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10, we have that words in Q′ have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q is finite, there are finitely many words of a given length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, Q′ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Since Q′ is finite, we can take a word w ∈ Q′ whose weight wt(x) is maximal among weights of words in Q′ with respect to the partial order given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8(1), w is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) Analogously to (2), we can take a word w′ ∈ Q′ whose weight wt(w′) is minimal among weights of words in Q′, and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8(2), we get that w′ is of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We observed before Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 that our intention with the (inverse-free) quasi- tensor product was to allow an interpretation in terms of quasi-crystals of the notion 32 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO of i-inversion in a word, introduced in [CM17, § 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the context of quasi-crystals of type An (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1)), for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}, a word w ∈ A∗ n has an i-inversion if it admits a decomposition of the form w = w1iw2(i + 1)w3, for some w1, w2, w3 ∈ A∗ n, and to accomplish our intention, the quasi-Kashiwara operators ¨ei and ¨fi should be undefined on such a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following example, we check that indeed this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider the standard quasi-crystal An of type An as described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following is a direct consequence of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The weight of w is given by wt(w) = |w|1e1 + |w|2e2 + · · · + |w|nen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If w has a decomposition of the form w = w1iw2(i+1)w3, for some w1, w2, w3 ∈ A∗ n, then ¨εi(w) = ¨ϕi(w) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, ¨εi(w) = |w|i+1 and ¨ϕi(w) = |w|i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The raising quasi-Kashiwara operator ¨ei is defined on w if and only if w has a decomposition of the form w = u1(i+1)u2, for some u1, u2 ∈ A∗ n with |u1|i = |u2|i+1 = 0, and if so, ¨ei(w) = u1iu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The lowering quasi-Kashiwara operator ¨fi is defined on w if and only if w has a decomposition of the form w = v1iv2, for some v1, v2 ∈ A∗ n with |v1|i = |v2|i+1 = 0, and if so, ¨fi(w) = v1(i + 1)v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Informally, provided that w does not have a decomposition of the form w = w1iw2(i + 1)w3, we have that ¨ei(w) is obtained by replacing the right-most symbol i + 1 in w by i, and ¨fi(w) is obtained by replacing the left-most symbol i in w by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(3) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1), We have that the words of length 2 form the following subgraph of the quasi-crystal graph of A¨∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 11 21 31 12 22 32 13 23 33 1 2 1 1 1 2 2 2 2 2 The term free, used in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 to characterize the quasi-crystal monoid Q¨∗ over a seminormal quasi-crystal Q, is justified by the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal and M be a nondegenerate quasi-crystal monoid of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, for each quasi-crystal homomorphism ψ : Q → M satisfying ψ(Q) ⊆ M, there exists a unique quasi-crystal monoid homomorphism ˆψ : Q¨∗ → M such that ˆψ(x) = ψ(x), for all x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let ψ : Q → M be a quasi-crystal homomorphism such that ψ(Q) ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we can consider ψ as a map from Q to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is well-known that there exists a unique monoid homomorphism ˆψ : Q∗ → M such that ˆψ(x) = ψ(x), for all x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also have that ˆψ is given by ˆψ(ǫ) = 1 and ˆψ(x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xm) = ψ(x1) · · · ψ(xm), for x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It remains to show that ˆψ is a quasi-crystal homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 and Definitions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, we have that wt(ǫ) = 0 = wt(1), ¨ei(ǫ) = ¨fi(ǫ) = ⊥ = ¨ei(1) = ¨fi(1), and ¨εi(ǫ) = ¨ϕi(ǫ) = 0 = ¨εi(1) = ¨ϕi(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Q, m ≥ 1, and set w = x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(2), we have that wt � ψ(xk) � = wt(xk), ¨εi � ψ(xk) � = ¨εi(xk) and ¨ϕi � ψ(xk) � = ¨ϕi(xk) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we obtain that wt � ˆψ(w) � = wt � ψ(x1) � + · · · + wt � ψ(xm) � = wt(x1) + · · · + wt(xm) = wt(w), and since p = max � 1 ≤ k ≤ m �� ¨εi(xk) > 0 � = max � 1 ≤ k ≤ m �� ¨εi � ψ(xk) � > 0 � QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 33 and q = min � 1 ≤ l ≤ m �� ¨ϕi(xl) > 0 � = min � 1 ≤ l ≤ m �� ¨ϕi � ψ(xl) � > 0 � , we also get that ¨εi � ˆψ(w) � = ¨εi(w) and ¨ϕi � ˆψ(w) � = ¨ϕi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(w) ∈ Q∗, then ¨ei(xp) ∈ Q∗, more precisely ¨ei(xp) ∈ Q as xp ∈ Q, and since ψ(Q) ⊆ M we have by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12(3) that ψ � ¨ei(xp) � = ¨ei � ψ(xp) � , which implies that ˆψ � ¨ei(w) � = ψ(x1) · · · ψ(xp−1) · ψ � ¨ei(xp) � ψ(xp+1) · · · ψ(xm) = ψ(x1) · · · ψ(xp−1) · ¨ei � ψ(xp) � ψ(xp+1) · · · ψ(xm) = ¨ei � ˆψ(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogous reasoning applies if ¨fi(w) ∈ Q∗, leading to ˆψ � ¨fi(w) � = ¨fi � ˆψ(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The property described in the previous result can be used to define the free quasi- crystal monoid up to isomorphism among nondegenerate quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal, M be a quasi-crystal monoid of the same type, and ι : Q → M be an injective quasi-crystal homomorphism such that for each nondegenerate quasi-crystal monoid M′ and each quasi-crystal homomorphism ψ : Q → M′ satisfying ψ(Q) ⊆ M ′, there exists a unique quasi- crystal monoid homomorphism ˆψ : M → M′ for which ψ = ˆψι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, there exists a quasi-crystal monoid isomorphism between M and Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define ψ : Q → Q∗ by ψ(x) = x, for each x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As the quasi-crystal structure maps of Q¨∗ agree on words of length 1 with the quasi-crystal structure maps of Q, we have that ψ is a quasi-crystal homomorphism from Q to Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, there exists a unique quasi-crystal monoid homomorphism ˆψ : M → Q¨∗ such that ˆψι(x) = x, for all x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, M is nondegenerate, because ¨εi(1) = ¨εi � ˆψ(1) � = ¨εi(ǫ) = 0 and ¨ϕi(1) = ¨ϕi � ˆψ(1) � = ¨ϕi(ǫ) = 0, for any i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, there exists a unique quasi-crystal monoid homomorphism ˆι : Q¨∗ → M such that ˆι(x) = ι(x), for all x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ˆψˆι is a quasi-crystal monoid homomorphism from Q¨∗ to Q¨∗ such that ˆψˆι(x) = x, for any x ∈ Q, and by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, we obtain that ˆψˆι must be the identity map on Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, ˆι ˆψ is a quasi-crystal monoid homomorphism from M to M such that ˆι ˆψι(x) = ι(x), for any x ∈ Q, and by uniqueness, we get that ˆι ˆψ must be the identity map on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ˆψ is a quasi- crystal monoid isomorphism between M and Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Congruences and quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now study the notion of congruence on a quasi-crystal monoid which leads to the definition of quotient quasi-crystal monoid and to the proof of homomorphism theorems for quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A quasi-crystal monoid con- gruence on M is an equivalence relation θ ⊆ M × M satisfying the conditions: (1) if (x, y) ∈ θ, then wt(x) = wt(y), ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y) for all i ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if (x, y) ∈ θ and ¨ei(x) ∈ M, then � ¨ei(x), ¨ei(y) � ∈ θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) if (x, y) ∈ θ and ¨fi(x) ∈ M, then � ¨fi(x), ¨fi(y) � ∈ θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) if (x1, y1), (x2, y2) ∈ θ, then (x1x2, y1y2) ∈ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is immediate from the definition that the equality relation ∆ = � (x, x) �� x ∈ M � is a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ∆ ⊆ θ, for any quasi-crystal monoid congruence θ on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, given a nonempty family Θ of quasi-crystal monoid congruences on M, it is straightforward to show that � Θ is a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From this and the following result, we are able to show that the quasi-crystal monoid congruences on a quasi-crystal monoid form a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 34 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid, and let R ⊆ M ×M be a relation on M such that for any (x, y) ∈ R and i ∈ I, the following conditions are satisfied: (1) wt(x) = wt(y), ¨εi(x) = ¨εi(y), and ¨ϕi(x) = ¨ϕi(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if ¨ei(x) ∈ M, then � ¨ei(x), ¨ei(y) � ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and (3) if ¨fi(x) ∈ M, then � ¨fi(x), ¨fi(y) � ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, the monoid congruence θR generated by R is a quasi-crystal monoid congru- ence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We check that in every step of constructing θR from R, properties (1) to (3) are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set R1 = � (ux, uy) �� (x, y) ∈ R, u ∈ M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For (x, y) ∈ R, u ∈ M and i ∈ I, since wt(x) = wt(y), ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y), we get that wt(ux) = wt(uy), ¨εi(ux) = ¨εi(uy) and ¨ϕi(ux) = ¨ϕi(uy), by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As M is seminormal, we have that ¨ei(ux) ∈ M if and only if ¨ei(uy) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, we obtain when ¨εi(x) = 0 that � ¨ei(ux), ¨ei(uy) � = � ¨ei(u) · x, ¨ei(u) · y � ∈ R1, and when ¨εi(x) > 0 that � ¨ei(ux), ¨ei(uy) � = � u · ¨ei(x), u · ¨ei(y) � ∈ R1, because � ¨ei(x), ¨ei(y) � ∈ R, by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, if ¨fi(ux) ∈ M, we get when ¨ϕi(u) > 0 that � ¨fi(ux), ¨fi(uy) � = � ¨fi(u) · x, ¨fi(u) · y � ∈ R1, and when ¨ϕi(u) = 0 that � ¨fi(ux), ¨fi(uy) � = � u · ¨fi(x), u · ¨fi(y) � ∈ R1, because � ¨fi(x), ¨fi(y) � ∈ R, by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, R1 satisfies conditions (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We can analogously deduce that R2 = � (xv, yv) �� (x, y) ∈ R1, v ∈ M � satisfies conditions (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is immediate that the reflexive closure R3 = R2 ∪ � (x, x) �� x ∈ M � of R2 satisfies conditions (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set R4 = R3 ∪ � (x, y) �� (y, x) ∈ R3 � , which correspondes to the symmetric closure of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since R3 satisfies condition (1), so it does R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For (x, y) ∈ R3, if ¨ei(y) ∈ M, then ¨ei(x) ∈ M, because ¨εi(x) = ¨εi(y) and M is seminormal, and so, � ¨ei(x), ¨ei(y) � ∈ R3, by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, if ¨fi(y) ∈ M, then � ¨fi(x), ¨fi(y) � ∈ R3, by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, R4 also satisfies conditions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, θR corresponds to the transitive closure of R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For (x, y) ∈ θR, there exist x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ M such that x = x0, y = xm and (xk−1, xk) ∈ R4, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ I, since R4 satisfies condition (1), we get that wt(x) = wt(x0) = wt(x1) = · · · = wt(xm) = wt(y), and similarly, ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As R4 satisfies condition (2), if ¨ei(x) ∈ M, we get that � ¨ei(x), ¨ei(x1) � ∈ R4, and recursively, � ¨ei(xk−1), ¨ei(xk) � ∈ R4, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m, which implies that � ¨ei(x), ¨ei(y) � ∈ θR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, as R4 satisfies condition (3), if ¨fi(x) ∈ M, then � ¨fi(x), ¨fi(y) � ∈ θR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, θR satisfies conditions (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, θR satisfies Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(4), as by construction θR is a monoid congruence on M, and thus, θR is a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, the quasi-crystal monoid congruences on M form a lattice with respect to the partial order ⊆ of inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Θ be the set of all quasi-crystal monoid congruences on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since the equality relation ∆ lies in Θ, we have that Θ is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Clearly, ⊆ is a partial order on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ, σ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since θ ∩ σ is a quasi-crystal monoid congruence and the largest set contained in θ and σ, we have that θ ∩ σ is the infimum of θ and σ in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, the relation R = θ ∪ σ satisfies the conditions of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, and so, the monoid congruence θR generated by R is a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, θR is the smallest equivalence relation on M satisfying Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(4) and containing R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, θR is the supremum of θ and σ in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 35 Let θ be a quasi-crystal monoid congruence on a quasi-crystal monoid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15, it follows that the quasi-crystal structure maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi (i ∈ I), and the monoid binary operation · of M give rise in a natural way to a quasi-crystal monoid whose underlying set is the set of all θ-equivalence classes M/θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each x ∈ M, denote the θ-equivalence class of x by [x]θ, or simply, [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x, y ∈ M are such that [x] = [y], then wt(x) = wt(y), ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y), by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As M is seminormal and ¨εi(x) = ¨εi(y), we have that ¨ei is defined on x if and only if ¨ei is defined on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, we have by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(2) that [¨ei(x)] = [¨ei(y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(3), if ¨fi is defined on x or y, then [ ¨fi(x)] = [ ¨fi(y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, if x1, x2, y1, y2 ∈ M are such that [x1] = [y1] and [x2] = [y2], then [x1x2] = [y1y2], by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we obtain the following construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ be a congruence on a quasi-crystal monoid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quotient quasi-crystal monoid of M by θ is a quasi-crystal monoid M/θ of the same type as M consisting of the set M/θ and maps given by wt([x]) = wt(x) ¨ei([x]) = [¨ei(x)] ¨fi([x]) = [ ¨fi(x)] ¨εi([x]) = ¨εi(x) ¨ϕi([x]) = ¨ϕi(x) [x] · [y] = [x · y], where [⊥] = ⊥, for x, y ∈ M and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result follows directly from the previous definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ be a congruence on a quasi-crystal monoid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, the map π : M → M/θ, given by π(x) = [x] for each x ∈ M, is a surjective quasi-crystal monoid homomorphism from M to M/θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to the following result that relates congruences and homomorphisms on quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M be a quasi-crystal monoid, and let θ ⊆ M × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, θ is a congruence on M if and only if there exist a quasi-crystal monoid M′ and a quasi-crystal monoid homomorphism ψ : M → M′ such that θ = ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ be a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set π : M → M/θ to be the quasi-crystal monoid homomorphism defined in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, for x, y ∈ M, we have that π(x) = π(y) if and only if (x, y) ∈ θ, which implies that θ = ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, let ψ : M → M′ be a quasi-crystal monoid homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is immediate that ker ψ is an equivalence relation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let (x, y) ∈ ker ψ and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that wt(x) = wt � ψ(x) � = wt � ψ(y) � = wt(y), and similarly, ¨εi(x) = ¨εi(y) and ¨ϕi(x) = ¨ϕi(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(x) ∈ M, then ¨ei(y) ∈ M, because M is seminormal and ¨εi(x) = ¨εi(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, since ψ(M) ⊆ M ′, we get that ψ � ¨ei(x) � , ψ � ¨ei(y) � ∈ M ′, and so, ψ � ¨ei(x) � = ¨ei � ψ(x) � = ¨ei � ψ(y) � = ψ � ¨ei(y) � , which implies that � ¨ei(x), ¨ei(y) � ∈ ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if ¨fi(x) ∈ M, then we obtain that � ¨fi(x), ¨fi(y) � ∈ ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, for (x1, y1), (x2, y2) ∈ ker ψ, we have that ψ(x1x2) = ψ(x1)ψ(x2) = ψ(y1)ψ(y2) = ψ(y1y2), which implies that (x1x2, y1y2) ∈ ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ker ψ is a quasi-crystal monoid congruence on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Now, we introduce the homomorphism theorems for quasi-crystal monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 36 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let M and M′ be quasi-crystal monoids of the same type, and let ψ : M → M′ be a quasi-crystal monoid homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, for each quasi- crystal monoid congruence θ on M satisfying θ ⊆ ker ψ, there exists a unique quasi-crystal monoid homomorphism ˆψ : M/θ → M′ such that ˆψ([x]θ) = ψ(x) for any x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Furthermore, if ψ is surjective, then there exists a unique quasi-crystal monoid isomorphism ˆψ : M/ ker ψ → M′ such that ˆψ([x]) = ψ(x), for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ be a quasi-crystal monoid congruence on M such that θ ⊆ ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x, y ∈ M, if [x] = [y], then ψ(x) = ψ(y), because θ ⊆ ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we can define a map ˆψ : M/θ → M ′ by ˆψ([x]) = ψ(x), for each x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a quasi-crystal monoid homomorphism from M to M′, it is immediate from Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18 that ˆψ is a quasi-crystal monoid homomorphism from M/θ to M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that ψ is surjective and take θ = ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, given x′ ∈ M ′, there exists x ∈ M such that ψ(x) = x′, and so, ˆψ([x]) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, if y, z ∈ M are such that ˆψ([y]) = ˆψ([z]), then ψ(y) = ψ(z), or equivalently, (y, z) ∈ ker ψ, which implies that [y] = [z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ˆψ is bijective, and therefore, a quasi-crystal monoid isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let θ and σ be congruences on a quasi-crystal monoid M such that θ ⊆ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a relation σ/θ on M/θ by σ/θ = � ([x]θ, [y]θ) �� (x, y) ∈ σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, σ/θ is a quasi-crystal monoid congruence on M/θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, the map (M/θ)/(σ/θ) → M/σ, given by [[x]θ]σ/θ �→ [x]σ for each x ∈ M, is a quasi-crystal monoid isomorphism between (M/θ)/(σ/θ) and M/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a map π : M → M/σ by π(x) = [x]σ, for each x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='19, π is a quasi-crystal monoid homomorphism from M to M/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since θ ⊆ σ = ker π, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21, we have a quasi-crystal monoid homomorphism ˆπ : M/θ → M/σ given by ˆπ([x]θ) = [x]σ, for each x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As π is surjective, then ˆπ is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For x, y ∈ M, we have that ˆπ([x]θ) = ˆπ([y]θ) if and only if (x, y) ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ker ˆπ = θ/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21, we obtain that the map (M/θ)/(σ/θ) → M/σ, given by [[x]θ]σ/θ �→ [x]σ for each x ∈ M, is a quasi-crystal monoid isomorphism between (M/θ)/(σ/θ) and M/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic congruence This section is devoted to study the hypoplactic congruence on a free quasi- crystal monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We start by proving that it results in a quasi-crystal monoid congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Based on this, we give the definition of hypoplactic monoid associated to a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We then characterize the commutative elements of such a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic congruence on Q¨∗ is a relation ¨∼ on Q∗ given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For u, v ∈ Q∗, u ¨∼ v if and only if there exists a quasi-crystal isomorphism ψ : Q¨∗(u) → Q¨∗(v) such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To prove that ¨∼ is a quasi-crystal monoid congruence on Q¨∗ we first show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each u, v ∈ Q∗, the map � Q∗(u) ¨⊗ Q∗(v) � (u ¨⊗ v) → Q∗(uv), given by x ¨⊗ y �→ xy, is a quasi-crystal isomor- phism between � Q¨∗(u) ¨⊗ Q¨∗(v) � (u ¨⊗ v) and Q¨∗(uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since Q¨∗ is a quasi-crystal monoid, the map Q∗ ¨⊗ Q∗ → Q∗, defined by x ¨⊗ y �→ xy for each x, y ∈ Q∗, is a quasi-crystal homomorphism, by Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='14 we have a surjective quasi-crystal homomorphism ψ : � Q¨∗ ¨⊗ Q¨∗� (u ¨⊗ v) → Q¨∗(uv) given by ψ(x ¨⊗ y) = xy, for each x ¨⊗ y ∈ (Q∗ ¨⊗ Q∗)(u ¨⊗ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that � Q¨∗ ¨⊗Q¨∗� (u ¨⊗v) and � Q¨∗(u) ¨⊗Q¨∗(v) � (u ¨⊗v) correspond to the same quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since they are formed by connected components of Q¨∗ ¨⊗Q¨∗ and Q¨∗, it suffices to prove that their underlying sets coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As Q∗(u), Q∗(v) ⊆ Q∗, we get that � Q∗(u) ¨⊗Q∗(v) � (u ¨⊗v) ⊆ (Q∗ ¨⊗Q∗)(u ¨⊗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ Q∗ be such that x ¨⊗ y ∈ (Q∗ ¨⊗ Q∗)(u ¨⊗ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ei, ¨fi | i ∈ I} such that x ¨⊗ y = g1 · · · gm(u ¨⊗ v), and by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, x = g′ 1 · · · g′ k(u) and y = g′′ 1 · · · g′′ l (v) for some g′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , g′ k, g′′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , g′′ l ∈ {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, x ∈ Q∗(u) and y ∈ Q∗(v), and so, (Q∗ ¨⊗ Q∗)(u ¨⊗ v) = � Q∗(u) ¨⊗ Q∗(v) � (u ¨⊗ v) Therefore, ψ is a surjective quasi-crystal homomorphism from � Q¨∗(u) ¨⊗ Q¨∗(v) � (u ¨⊗ v) to Q¨∗(uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, we show that ψ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x1 ¨⊗y1, x2 ¨⊗y2 ∈ � Q∗(u) ¨⊗Q∗(v) � (u ¨⊗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 that |x1| = |x2| and |y1| = |y2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if x1y1 = x2y2, then x1 = x2 and y1 = y2 implying x1 ¨⊗ y1 = x2 ¨⊗ y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ψ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18, ψ is a quasi-crystal isomorphism between � Q¨∗(u) ¨⊗ Q¨∗(v) � (u ¨⊗ v) and Q¨∗(uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, the hypoplactic con- gruence ¨∼ on Q¨∗ is a quasi-crystal monoid congruence on Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It is straightforward to see that ¨∼ is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ Q∗ with u ¨∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then there exists a quasi-crystal isomorphism ψ : Q¨∗(u) → Q¨∗(v) such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a quasi-crystal isomorphism, we get that wt(u) = wt � ψ(u) � = wt(v), and similarly, ¨εi(u) = ¨εi(v) and ¨ϕi(u) = ¨ϕi(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ei(u) ∈ Q∗, then ψ � ¨ei(u) � = ¨ei � ψ(u) � = ¨ei(v), and since Q∗� ¨ei(u) � = Q∗(u) and Q∗� ¨ei(v) � = Q∗(v), we obtain ¨ei(u) ¨∼ ¨ei(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if ¨fi(u) ∈ Q∗, then ¨fi(u) ¨∼ ¨fi(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Additionally, let u′, v′ ∈ Q∗ with u′ ¨∼ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then we also have a quasi-crystal isomorphism ψ′ : Q¨∗(u′) → Q¨∗(v′) such that ψ(u′) = v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we have quasi-crystal isomorphisms ψ1 : Q¨∗(uv) → � Q¨∗(u) ¨⊗ Q¨∗(v) � (u ¨⊗ v) and ψ2 : � Q¨∗(u′) ¨⊗ Q¨∗(v′) � (u′ ¨⊗ v′) → Q¨∗(u′v′) such that ψ1(uv) = u ¨⊗ v and ψ2(u′ ¨⊗ v′) = u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8, we have a quasi-crystal isomorphism ψ ¨⊗ ψ′ between Q¨∗(u) ¨⊗Q¨∗(v) and Q¨∗(u′) ¨⊗Q¨∗(v′) satisfying (ψ ¨⊗ψ′)(u ¨⊗v) = u′ ¨⊗v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set ψ3 to be the restriction of ψ ¨⊗ψ′ to � Q∗(u) ¨⊗Q∗(v) � (u ¨⊗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, ψ3 is a quasi- crystal isomorphism between � Q¨∗(u) ¨⊗Q¨∗(v) � (u ¨⊗v) and � Q¨∗(u′) ¨⊗Q¨∗(v′) � (u′ ¨⊗v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ2ψ3ψ1 is a quasi-crystal isomorphism between Q¨∗(uv) and Q¨∗(u′v′) that satisfies ψ2ψ3ψ1(uv) = u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, uv ¨∼ u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ¨∼ is a quasi-crystal monoid congruence on Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We have now set up the framework to present the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal, and let ¨∼ be the hypoplactic congruence on Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The quotient quasi-crystal monoid Q¨∗/ ¨∼ is called the hypoplac- tic quasi-crystal monoid, or simply the hypoplactic monoid, associated to Q, and is denoted by hypo(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 38 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Although hypo(Q) is a quasi-crystal monoid, we are interested in studying its properties as a monoid, and thus, we just refer it as the hypoplactic monoid asso- ciated to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' However, we will be constantly considering its quasi-crystal structure, as it plays a fundamental role in the construction of hypo(Q), and consequently, in its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This terminology will make more sense in the following section, where we see how the classical hypoplactic monoid can be placed in context as the hypoplactic monoid associated to the standard quasi-crystal of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In a hypoplactic monoid the converse of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(1) also holds, because the isolated elements (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15) are commutative, which is a consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal, and let u, v ∈ Q∗ be such that uv is an isolated element of Q¨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, uvw ¨∼ uwv ¨∼ wuv, for any w ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we have that wt(uvw) = wt(u) + wt(v) + wt(w) = wt(uwv), for any w ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since uv is isolated, we have that ¨ei(uv) = ¨fi(uv) = ⊥, for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, for each i ∈ I, either ¨εi(uv) = ¨ϕi(uv) = 0 or ¨εi(uv) = ¨ϕi(uv) = +∞, because Q¨∗ is seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set J = � i ∈ I �� ¨εi(uv) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, for any i ∈ I \\ J, since ¨εi(uv) = +∞, we have that ¨εi(u) = +∞, ¨εi(v) = +∞, or ¨ϕi(u), ¨εi(v) ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3 that, for any w ∈ Q∗, ¨εi(uvw) = ¨ϕi(uvw) = ¨εi(uwv) = ¨ϕi(uwv) = +∞, and so, ¨ei and ¨fi are undefined on uvw and on uwv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, for any j ∈ J, we have that ¨εj(u) = ¨εj(v) = ¨ϕj(u) = ¨ϕj(v) = 0, because ¨εj(uv) = ¨ϕj(uv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, for any w ∈ Q∗, we get that ¨ej(uvw) = uv¨ej(w), ¨fj(uvw) = uv ¨fj(w), ¨εj(uvw) = ¨εj(w), ¨ϕj(uvw) = ¨ϕj(w), ¨ej(uwv) = u ¨fj(w)v, ¨fj(uwv) = u ¨fj(w)v, ¨εj(uwv) = ¨εj(w), ¨ϕj(uwv) = ¨ϕj(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, given w ∈ Q∗ and g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ei, ¨fi | i ∈ I}, we have that g1 · · · gm is defined on uvw if and only if g1 · · · gm is defined on uwv if and only if g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ej, ¨fj | j ∈ J} and g1 · · · gm is defined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define X = � w′ ∈ Q∗ �� w′ = g1 · · · gm(w), for some g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , gm ∈ {¨ej, ¨fj | j ∈ J} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, Q∗(uvw) = {uvw′ | w′ ∈ X} and Q∗(uwv) = {uw′v | w′ ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, the map ψ : Q∗(uvw) → Q∗(uwv) given by ψ(uvw′) = uw′v, for each w′ ∈ X, is a bijective quasi-crystal homomorphism from Q¨∗(uvw) to Q¨∗(uwv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18, ψ is a quasi-crystal isomorphism between Q¨∗(uvw) and Q¨∗(uwv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ψ(uvw) = uwv, we obtain that uvw ¨∼ uwv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The fact that uwv ¨∼ wuv follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We now show that the converse of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2) holds for hypoplactic monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to a characterization of the idempotent elements of a hy- poplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 39 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal, and let w ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w ¨∼ w2 if and only if w is an isolated element of Q¨∗ and wt(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The direct implication follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, assume that w is an isolated element of Q¨∗ and wt(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get that wt � w2� = wt(w) + wt(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As Q¨∗ is seminormal and w is isolated, we have that either ¨εi(w) = ¨ϕi(w) = 0 or ¨εi(w) = ¨ϕi(w) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨εi(w) = ¨ϕi(w) = 0, then ¨εi � w2� = ¨εi(w) + ¨εi(w) = 0 and ¨ϕi � w2� = ¨ϕi(w) + ¨ϕi(w) = 0, implying that ¨ei and ¨fi are undefined on w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, ¨εi(w) = ¨ϕi(w) = +∞, we get by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 that ¨εi � w2� = ¨ϕi � w2� = +∞, which implies that ¨ei and ¨fi are undefined on w2, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, w2 is isolated and the map ψ : q∗(w) → Q∗� w2� , defined by ψ(w) = w2, is a quasi-crystal isomorphism between Q¨∗(w) and Q¨∗� w2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, w ¨∼ w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The following result is a direct consequence of Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Q be a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the hypoplactic monoid hypo(Q), the idempotent elements commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Crystallizing the classical hypoplactic monoid In this section we prove that the classical hypoplactic monoid hypon of rank n arises as the hypoplactic monoid hypo(An) associated to the standard quasi-crystal An of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This is accomplished by showing that the direct approach in [CM17] can be placed in the context developped in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Recall that Kashiwara crystals [Kas90, Kas91] give rise to a plactic monoid anti-isomorphic to the original one [LS81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since we introduced the quasi-tensor product of quasi-crystals (Section 5) based on the tensor product of crystals defined by Kashiwara, it is natural to expect the hypoplactic monoid obtained from quasi- crystals to be anti-isomorphic to the original one [KT97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the results in this section concerning the classical hypoplactic monoid are adaptations of the original results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The classical hypoplactic monoid hypon of rank n is given by the presentation ⟨An | R1 ∪ R2 ∪ R3 ∪ R4⟩ where R1 = � (yzx, yxz), (xzy, zxy) �� x < y < z � , R2 = � (xyx, xxy), (xyy, yxy) �� x < y � , R3 = � (xzty, zxyt) �� t ≤ x < y ≤ z � , and R4 = � (ytzx, tyxz) �� t < x ≤ y < z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The Knuth relations consist of R1 ∪ R2, and the quartic relations consist of R3 ∪ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The classical hypoplactic congruence ∼hypon is the monoid congruence on A∗ n generated by R1 ∪ R2 ∪ R3 ∪ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The Knuth and quartic relations given above are respectively the reverse of the ones given in [Knu70] and [KT97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This is part of the adaptations we pointed out in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the rest of this section, fix the root system associated to Cartan type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The maps wt, ¨ei, ¨fi, ¨εi and ¨ϕi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1, always refer to the quasi-crystal structure of A¨∗ n, and ¨∼ always denote the hypoplactic congruence on A¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 40 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO As we saw in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12, for w ∈ A∗ n and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}, ¨ei is defined on w if and only if w does not have an i-inversion and |w|i+1 > 0, and if so, ¨ei(w) is obtained from w by replacing the right-most symbol i + 1 by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, ¨fi is defined on w if and only if w does not have an i-inversion and |w|i > 0, and if so, ¨fi(w) is obtained from w by replacing the left-most symbol i by i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we can use the quasi-crystal structure of A¨∗ n to construct a graph similar to the one in [CM17, § 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let Γ′ n denote the Λ-weighted {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}-labelled directed graph consisting of the vertex set A∗ n, the weight map of A¨∗ n, and for each u, v ∈ A∗ n and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}, an edge u i −−−→ v whenever ¨fi(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that Γ′ n can be obtained from the graph constructed in [CM17, § 5] by reversing the words on each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This is one of the adaptations pointed out in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3 ([CM17, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ∼hypon v if and only if there exists a (weight-preserving labelled directed) graph isomorphism ψ between Γ′ n(u) and Γ′ n(v) such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now proceed to prove that ¨∼ and ∼hypon are the same relation on A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The graph Γ′ n coincides with the graph that results from ΓA¨∗n by removing all loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we have that the vertex sets and weight maps of Γ′ n and ΓA¨∗n coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For u, v ∈ A∗ n and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}, if ¨fi(u) = v, then wt(u) > wt(v), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, which implies that u ̸= v, and so, Γ′ n is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, we have that u i −−−→ v is an edge of Γ′ n if and only if ¨fi(u) = v if and only if u i −−−→ v is an edge of ΓA¨∗n and u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ A∗ n be such that u ∼hypon v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}, u has an i-inversion if and only if v has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that u has an i-inversion and v does not have an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, there exists a graph isomorphism ψ between Γ′ n(u) and Γ′ n(v) such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u has an i-inversion, we have that |u|i ≥ 1, and since ψ preserves weights, wt(u) = wt(v), which implies that i occurs in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As v does not have an i-inversion, ¨fi is defined on v, and so, v i −−−→ ¨fi(v) is an edge of Γ′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a graph isomorphism and ψ(u) = v, we get that u i −−−→ ψ−1� ¨fi(v) � is an edge of Γ′ n, which is a contradiction, because ¨fi is undefined on u, as u has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The other direction is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ¨∼ v if and only if u ∼hypon v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, hypo(An) and hypon are isomorphic monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that u ∼hypon v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definitions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1, there exists a quasi-crystal isomorphism ψ : A¨∗ n(u) → A¨∗ n(v) such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, ψ is a graph isomorphism between ΓA¨∗ n(u) and ΓA¨∗ n(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4, Γ′ n(u) and Γ′ n(v) can be obtained from ΓA¨∗n(u) and ΓA¨∗n(v), respectively, by removing all loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ is a graph isomorphism between Γ′ n(u) and Γ′ n(v) satisfying ψ(u) = v, which implies by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3 that u ∼hypon v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, assume that u ∼hypon v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, there exists a graph isomorphism ψ′ between Γ′ n(u) and Γ′ n(v) such that ψ′(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To prove that ψ′ is a graph isomorphism between ΓA¨∗n(u) and ΓA¨∗n(v), by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 it just remains QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 41 to show that ψ′ and (ψ′)−1 preserve loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ A∗ n and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1} such that w has an i-labelled loop in ΓA¨∗ n(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ¨εi(w) = +∞, which implies that u has an i-inversion (see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4, w lies in Γ′ n(u), and since w ∼hypon ψ′(w) by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we get by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that ψ′(w) also has an i- inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ¨εi � ψ′(w) � = +∞, which implies that ψ′(w) has an i-labelled loop in ΓA¨∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if w′ ∈ A∗ n has an i-labelled loop in ΓA¨∗n(v), then (ψ′)−1(w′) also has an i-labelled loop in ΓA¨∗n, because (ψ′)−1 is a graph isomorphism between Γ′ n(v) and Γ′ n(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, ψ′ is a graph isomorphism between ΓA¨∗n(u) and ΓA¨∗n(v) We have by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13 that ψ′ is a quasi-crystal isomorphism between A¨∗ n(u) and A¨∗ n(v) satisfying ψ′(u) = v, which implies that u ¨∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The previous result justifies the term hypoplactic used in Definitions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4, since the classical hypoplactic monoid can be obtained as the hypoplactic monoid associated to the standard quasi-crystal An of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, it shows that the theory of quasi-crystals presented in Sections 3 to 7 gives rise to a genuine general- ization of the classical hypoplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now have a process of crystallizing the hypoplactic monoid which allows the construction of the classical hypoplactic monoid from the standard quasi-crystal An of type An, and allows the analogous construction of a monoid based on any other seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic monoid of type Cn We described in Section 7 a method of obtaining a monoid from a seminormal quasi-crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We then showed in Section 8 that for the standard quasi-crystal An of type An it results in the classical hypoplactic monoid of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A natural way of proceeding is to study the monoids that are obtained for other quasi-crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this section, we make a detailed study of the hypoplactic monoid hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We start in Subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 by presenting a description of the free quasi-crystal monoid C¨∗ n over Cn, from which hypo(Cn) emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Subsections 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we characterize the highest-weight and isolated words of C¨∗ n, which allows us to identify the commutative and idempotent elements of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4, we investigate whether hypo(Cn) satisfies some well-known relations, such as the Knuth relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, we show that hypo(C2) satisfies nontrivial identities, and describe some of the properties any such identity must have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We also show that hypo(Cn) does not satisfy nontrivial identities, for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, we prove that hypo(Cn) does not admit a finite presentation, but we identify the connected components of C¨∗ 2 up to isomorphism, leading to a class of representatives for the elements of hypo(C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, in Subsections 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8 we describe monoid embeddings of hypo(An−1) and hypo(Cn−1) into hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The definition of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2), we have that the standard quasi-crystal Cn of type Cn is a seminormal quasi-crystal consisting of an ordered set Cn = � 1 < 2 < · · · < n < n < n − 1 < · · · < 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Its quasi-crystal graph is 1 1 −−−→ 2 2 −−−→ · · · n−1 −−−→ n n −−−→ n n−1 −−−→ n − 1 n−2 −−−→ · · · 1 −−−→ 1, where the weight map wt : Cn → Zn is defined by wt(x) = ex and wt(x) = −ex, for x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 42 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Notice that for x, y ∈ Cn and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}, if ¨ϕi(x) > 0 and ¨εi(y) > 0, then x ∈ � i, i + 1 � and y ∈ � i + 1, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If ¨ϕn(x) > 0 and ¨εn(y) > 0, then x = n and y = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To avoid constant division into cases where i ̸= n and i = n, for brevity, in the rest of this section we formally consider n + 1 and n + 1 to be symbols that never appear in any word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the observation in the previous paragraph can be simply re-stated as follows: for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if ¨ϕi(x) > 0 and ¨εi(y) > 0, then x ∈ � i, i + 1 � and y ∈ � i + 1, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This leads to the concept of an i-inversion for words over the alphabet Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A word w ∈ C∗ n is said to have an i-inversion if w admits a decomposition of the form w = w1xw2yw3, for some w1, w2, w3 ∈ C∗ n, x ∈ � i, i + 1 � and y ∈ � i + 1, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' A word w ∈ C∗ n is said to be i-inversion-free if w does not have an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3, we obtain the following description of the free quasi-crystal monoid C¨∗ n over Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The free quasi-crystal monoid C¨∗ n over Cn consists of the set C∗ n of all words over Cn, under the operation of concatenation of words, and a quasi- crystal structure given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For w ∈ C∗ n, the weight of w is wt(w) = � |w|1 − |w|1, |w|2 − |w|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , |w|n − |w|n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if w has an i-inversion, then ¨εi(w) = ¨ϕi(w) = +∞, otherwise, ¨εi(w) = |w|i+1 + |w|i and ¨ϕi(w) = |w|i + |w|i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The raising quasi-Kashiwara operator ¨ei is defined on w if and only if ¨εi(w) ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The lowering quasi-Kashiwara operator ¨fi is defined on w if and only if ¨ϕi(w) ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' When they are defined, the quasi-Kashiwara operators can be com- puted (as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3) as follows: Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, and let w ∈ C∗ n be i-inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If i + 1 or i occurs in w, let x be the right-most i + 1 or i occurring in w, then ¨ei(w) is obtained from w by replacing x by ¨ei(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If i or i + 1 occurs in w, let y be the left-most i or i + 1 occurring in w, then ¨fi(w) is obtained from w by replacing y by ¨fi(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take w = 253553234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that wt(w) = (0, 2, −1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since w admits a decomposition of the form w = w12w23w3, we get that w has a 2-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It has a 3-inversion, as it admits a decomposition of the form w = w13w23w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It has a 4-inversion, as it admits a decomposition of the form w = w15w25w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, for i ∈ {2, 3, 4}, the quasi-Kashiwara operators ¨ei and ¨fi are undefined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, w is 1-inversion-free and 5- inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that ¨f1 is undefined on w, as neither 1 nor 2 occurs in w, ¨e1(w) = 253553134, ¨e5(w) = 253553234, and ¨f5(w) = 253553234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In Section 4, we showed that a seminormal quasi-crystal can be described by its quasi-crystal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, to study the hypoplactic monoid hypo(Cn), we will frequently resort to the quasi-crystal graph ΓC¨∗ n of C¨∗ n Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The empty word ǫ is an isolated vertex in ΓC¨∗ n without loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The set of letters Cn forms a connected component which is described in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now turn our attention to the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Words of length 2 form a subgraph of ΓC¨∗ 2 that is isomorphic to the quasi-crystal graph of C2 ¨⊗ C2, which is described in QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 43 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The connected component ΓC¨∗ 2 (121) of ΓC¨∗ 2 containing 121 is the following: 121 121 221 211 2 11 212 2 12 2 1 2 2 1 1 1 2 2 2 1 2 1 1 1 The connected component ΓC¨∗ 2 (212) of ΓC¨∗ 2 containing 212 is the following: 212 212 212 112 112 122 121 1 2 1 2 2 1 2 1 1 2 1 1 1 2 1 By Definitions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4, the hypoplactic monoid hypo(Cn) is the quotient monoid of C∗ n by the hypoplactic congruence ¨∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Although we omitted the weight map wt in the example above, recall that ΓC¨∗ n is a weighted labelled graph, from which ¨∼ can be obtained, since, for two words u, v ∈ C∗ n, we have by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13 that u ¨∼ v if and only if there exists a graph isomorphism ψ between the connected components ΓC¨∗ n(u) and ΓC¨∗ n(v) of ΓC¨∗ n such that ψ(u) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result shows that the hypoplactic congruence ¨∼ respects inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ n with u ¨∼ v, and let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u has an i-inversion if and only if v has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If u has an i-inversion, then ¨εi(u) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ v, then ¨εi(v) = ¨εi(u) = +∞, which implies that v has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The converse follows from the fact that u ¨∼ v implies v ¨∼ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ This result is analogous to one obtained for the classical hypoplactic monoid in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, where, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n−1}, either all words in a congruence class of the hypoplactic congruence have an i-inversion, or all of them are i-inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 and [CM17, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2], we can deduce a construction of the quasi-crystal graph ΓA¨∗n from the crystal graph over A∗ n of type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' An analogous construction of the quasi-crystal graph ΓC¨∗ n from the crystal graph over C∗ n of type Cn [KN94, Lec02] can also be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' First, note that the weight maps coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9, we have that when the quasi-Kashiwara operators are defined, they coincide with the Kashiwara operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the quasi- crystal graph ΓC¨∗ n is obtained from the crystal graph over C∗ n of type Cn by deleting all i-labelled edges starting or ending on a word with an i-inversion, and then, adding i-labelled loops on all words with an i-inversion, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The empty word ǫ and the word 11 are related by the plactic congruence on C∗ n [Lec02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 11 has a 1-inversion, we get by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that 11 ̸¨∼ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, the plactic congruence on C∗ n is not contained in the hypoplactic congruence on C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This contrasts with the well-known result for type An, where the plactic congruence on A∗ n is contained in the hypoplactic congruence on A∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, notice that a word w in C∗ n may have unbarred symbols and barred symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For the sake of simplicity, we introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set ǫ = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, set x = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Given a word w = x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xm, with x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xm ∈ Cn, set w = xm xm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 44 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO The following result shows that this notation preserves inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w has an i-inversion if and only if w has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If w has an i-inversion, then w = w1xw2yw3, for some w1, w2, w3 ∈ C∗ n, x ∈ � i, i + 1 � and y ∈ � i + 1, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, w = w3 y w2 x w1, where y ∈ � i + 1, i � and x ∈ � i, i + 1 � , and so, w has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The converse is immediate since w = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ In the quasi-crystal monoid C¨∗ n we have the following relation between a word w ∈ C∗ n and its barred version w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n, and let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, wt(w) = − wt(w), ¨εi(w) = ¨ϕi(w), and ¨ϕi(w) = ¨εi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Furthermore, if ¨ei(w) or ¨fi(w) are defined, then ¨ei(w) = ¨fi(w), and if ¨fi(w) or ¨ei(w) are defined, then ¨fi(w) = ¨ei(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we have that wt(w) = � |w|1 − |w|1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , |w|n − |w|n � = � |w|1 − |w|1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , |w|n − |w|n � = − wt(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7, we get that ¨εi(w) = ¨ϕi(w) = +∞ if and only if ¨εi(w) = ¨ϕi(w) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, the result follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, assume that neither w nor w has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ̸= n, we have that ¨εi(w) = |w|i+1 + |w|i = |w|i+1 + |w|i = ¨ϕi(w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' furthermore, ¨εn(w) = |w|n = |w|n = ¨ϕn(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As C¨∗ n is seminormal, ¨ei(w) is defined if and only if ¨fi(w) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If so, then there exist w1, w2 ∈ C∗ n and x ∈ � i, i + 1 � such that |w1|i = |w1|i+1 = 0, w = w1xw2, and ¨fi(w) = w1 ¨fi(x)w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since w = w2 x w1, where x ∈ � i + 1, i � and |w1|i+1 = |w1|i = 0, we get that ¨ei(w) = w2¨ei(x)w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that if x = i, then ¨ei(x) = i + 1 = ¨fi(x), and otherwise, x = i + 1 and ¨ei(x) = i = ¨fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ¨ei(w) = ¨fi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, ¨fi(w) = ¨ei(w), whenever ¨fi(w) or ¨ei(w) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The following results are straightforward consequences of the previous result: Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Given u, v ∈ C∗ n, there is an edge u i −−−→ v in the quasi-crystal graph ΓC¨∗ n if and only if there is an edge v i −−−→ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, for any w ∈ C∗ n, C∗ n(w) = � u �� u ∈ Cn(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ¨∼ v if and only if u ¨∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Highest-weight words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the study of plactic monoids for the infinite Car- tan types [Lec02, Lec03], words of highest weight are extremely relevant as they are used to index connected components of crystal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' An analogous relation was proven in [CM17] for the classical hypoplactic monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we now characterize the highest-weight words of C¨∗ n, and check whether they satisfy properties similar to highest-weight words in the mentioned contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6(1), we have that a word w ∈ C∗ n is of highest weight if ¨ei is undefined on w, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Equivalently, w is of highest weight if the only edges in ΓC¨∗ n ending on w are loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w is of highest weight if and only if for each letter x ∈ Cn occurring in w, the following conditions are satisfied: QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 45 (1) if x ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, then w has an (x − 1)-inversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if x ∈ � n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 � , then w has an x-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that w is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ Cn be a letter occurring in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, then ¨εx−1(w) > 0, and since C¨∗ n is seminormal and ¨ex−1 is undefined on w, we get that ¨εx−1(w) = +∞, or equivalently, w has an (x − 1)- inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x ∈ � n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 � , then ¨εx(w) > 0, which implies as in the previous case that w has an x-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, suppose that w is not of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, take i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n} such that ¨ei is defined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, we have that ¨ei(w) = w1¨ei(x)w2, for some w1, w2 ∈ C∗ n and x � i + 1, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the letter i + 1 or the letter i occurs in w, and w does not have an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In C¨∗ 4, the following words are of highest weight: 1, 12, 11, 332, 333, and 123443 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the crystal graphs studied in [KN94], which led to the construction of the plac- tic monoids for the infinite Cartan types [Lec02, Lec03], each connected component has exactly one highest-weight element and exactly one lowest-weight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Due to the results in [CM17] and in Section 8, we also have that the connected compo- nents of the free quasi-crystal monoid A¨∗ n have exactly one highest-weight word and exactly one lowest-weight word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The free quasi-crystal monoid C¨∗ n does not have this property, as we can see in Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 that 212 and 112 are highest-weight words of C¨∗ 2 which belong to the same connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The same happens in C¨∗ n for any n ≥ 2, because ¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1 ¨fn ¨fn−1 · · · ¨f2(212) = ¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1 ¨fn(n12) = ¨e2 ¨f1 ¨f2 ¨f2 ¨f3 · · · ¨fn−1(n12) = ¨e2 ¨f1 ¨f2 � 212 � = ¨e2 ¨f1 � 21 ¨f2(2) � = ¨e2 � 11 ¨f2(2) � = 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We can also see that 2 11 and 2 1 2 are lowest-weight words of C¨∗ n which belong to the same connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Although we have that a connected component of C¨∗ n may have more than one highest-weight word or more then one lowest-weight word, we can guarantee by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11 that it has at least one of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, in the following result we describe a one-to-one correspondence between highest-weight words and lowest-weight words of C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w is of highest weight if and only if w is of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, w is of lowest weight if and only if w is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, we have by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='9 that ¨ei (or ¨fi) is defined on w if and only if ¨fi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', ¨ei) is defined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that w is of highest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', lowest) weight if and only if w is of lowest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', highest) weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12, we have that 123443 2 1 is a highest-weight word in C¨∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, wt � 12344 3 2 1 � = 0 and ¨εi � 12344 3 2 1 � = +∞, for all i ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, for any w ∈ C∗ 4, we get that 12344 3 2 1w is a highest-weight word with weight wt(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following result, we generalize this reasoning, from which we can see that highest weights (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7) in C¨∗ n do not identify a relevant subset of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 46 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Any element λ ∈ Zn is a highest weight in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , λn) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if λi ≥ 0, set ai = λi + 1 and bi = 1, otherwise, set ai = 1 and bi = −λi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The word w = 1a12a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' nannbnn − 1 bn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 b1 is such that wt(w) = (a1 − b1, a2 − b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , an − bn) = λ and ¨εi(w) = +∞, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, because w has a decomposition of the form w = w1iw2iw3, for some w1, w2, w3 ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, w is a highest-weight word with weight λ, which implies that λ is a highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ The previous result together with Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13 implies that any element λ ∈ Zn is a lowest weight in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Isolated words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, we have that the com- mutative elements of the hypoplactic monoid hypo(Cn) correspond to the hypoplac- tic congruence classes of isolated words of C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, we also have that the idempotent elements of the hypoplactic monoid hypo(Cn) correspond to the hypoplactic congruence classes of isolated words of C¨∗ n with weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we now turn our attention to characterizing the isolated words in C¨∗ n, and consequently, obtain some relations in hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15, a word w ∈ C∗ n is isolated if it is an isolated vertex in the quasi-crystal graph ΓC¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In other words, w is isolated if and only if it is both of highest and of lowest weight in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w is an isolated word if and only if w is an isolated word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, w is an isolated word if and only if both w and w are of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, w is of highest and of lowest weight if and only if w is of highest and of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, w and w are of highest weight if and only if w and w are of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w is an isolated word if and only if both of the following conditions hold: (1) w has a 1-inversion if 1 or 1 occurs in w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) w has an (i − 1)-inversion and an i-inversion, for all i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} such that i or i occurs in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that w is an isolated word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='15, w and w are of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If i occurs in w, or equivalently, i occurs in w, then we have by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11 that w has an (i − 1)-inversion when i ≥ 2, and that w has an i-inversion which implies by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7 that w has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, if i occurs in w, or equivalently, i occurs in w, then w has an (i − 1)- inversion and an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Conversely, suppose that w is not an isolated word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} such that ¨ei or ¨fi is defined on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, w does not have an i-inversion, and some letter among i, i + 1, i + 1 and i occurs in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In C¨∗ 4, the following are isolated words: 11, 122, 22 1, 333, and 123443 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following result, we show how to obtain commutative and idempotent elements of hypo(Cn) from each word in C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 47 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, www and wwww are isolated words in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, www is a commutative element of hypo(Cn), and wwww is a commuta- tive and idempotent element of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if i occurs in w, then www and wwww have decom- positions of the form w1iw2iw3iw4, for some w1, w2, w3, w4 ∈ C∗ n, which implies that www and wwww have an (i − 1)-inversion and an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If i occurs in w, then www and wwww have decompositions of the form w1iw2iw3iw4, for some w1, w2, w3, w4 ∈ C∗ n, which implies that www and wwww have an (i − 1)-inversion and an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, www and wwww are isolated words, and by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, they are commutative elements of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='3 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8, we have that wt(wwww) = wt(w) − wt(w) + wt(w) − wt(w) = 0, and by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, we get that wwww is an idempotent element of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ To give a complete characterization of the commutative and idempotent elements of hypo(Cn), we first introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each word w ∈ C∗ n, define an n-tuple inv(w) = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , δn) ∈ {0, 1}n, where, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, δi = 1 if and only if w has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ n be isolated words in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ¨∼ v if and only if wt(u) = wt(v) and inv(u) = inv(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u and v are isolated words, we get that C∗ n(u) = {u} and C∗ n(v) = {v}, which implies that ¨εi(u), ¨ϕi(u), ¨εi(v), ¨ϕi(v) ∈ {0, +∞}, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, because C¨∗ n is seminormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, we have that ¨εi(u) = ¨ϕi(u) = +∞ (or ¨εi(u) = ¨ϕi(v) = +∞) if and only if u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', v) has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the map ψ : C∗ n(u) → C∗ n(v), given by ψ(u) = v, is a quasi-crystal isomorphism between C¨∗ n(u) and C¨∗ n(v) if and only if wt(u) = wt(v) and inv(u) = inv(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The map that sends each isolated word w ∈ C∗ n to � wt(w), inv(w) � induces a bijection between the set of commutative elements of hypo(Cn) and the set of pairs (λ, δ) with λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , λn) ∈ Zn and δ = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , δn) ∈ {0, 1}n satisfying the following conditions: (1) if λi ̸= 0, for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, then δi = 1, and δi−1 = 1 when i ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) if δi = 1, for some i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, then δi−1 = 1, or δi+1 = 1 when i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, we have that the commutative ele- ments of hypo(Cn) correspond to the hypoplactic congruence classes of isolated words of C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20, the map that sends each isolated word w ∈ C∗ n to � wt(w), inv(w) � induces a well-defined injective map from the commutative ele- ments of hypo(Cn) to Zn × {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that the pairs � wt(w), inv(w) � , where w ∈ C∗ n is an isolated word of C¨∗ n, satisfy conditions (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n be an isolated word of C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, set λi = |w|i − |w|i, and if w has an i-inversion, take δi = 1, otherwise, take δi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, wt(w) = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , λn) and inv(w) = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If λi ̸= 0, for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, then i or i occurs in w implying that w has an (i − 1)-inversion (if i ≥ 2) and an i-inversion, by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, and so, δi−1 = 1 (if i ≥ 2) and δi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If δi = 1, for some i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, then some letter among i, i + 1, i + 1 and i occurs in w implying that w has an (i − 1)-inversion, or when i ≤ n − 1, an (i + 1)-inversion, by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, and thus, δi−1 = 1 or δi+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the pair � wt(w), inv(w) � satisfies conditions (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 48 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Finally, we show that for each pair (λ, δ) ∈ Zn ×{0, 1}n satisfying conditions (1) and (2), there exists an isolated word w ∈ C∗ n such that wt(w) = λ and inv(w) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , λn) ∈ Zn and δ = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , δn) ∈ {0, 1}n satisfying conditions (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If δ1 = 1, set w1 = 11, otherwise, set w1 = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if δi−1 = δi = 1, take wi = iiii, otherwise, take wi = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By (2), if δi = 1, for some i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, then wi ̸= ǫ or wi+1 ̸= ǫ, which implies that wiwi+1 has an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, for i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, we have that ¨εi−1(wi) = ¨εi(wi) = +∞, and ¨εj(wi) = 0, whenever j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n} \\ {i − 1, i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, the word w′ = w1w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' wn has a 1-inversion if and only if δ1 = 1, and for i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, w′ has an i-inversion if and only if δi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, inv(w′) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since wt(wi) = 0, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, we get that wt(w′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, if λi ≥ 0, set ai = λi and bi = 0, otherwise, set ai = 0 and bi = −λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w = w′1a12a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' nannbnn − 1 bn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By (1), if ai ̸= 0 or bi ̸= 0, for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, then δi−1 = 1 when i ≥ 2, and δi = 1, implying that w′ has an (i − 1)-inversion (if i ≥ 2) and an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, inv(w) = inv(w′) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since wt(w′) = 0, we get that wt(w) = (a1 − b1, a2 − b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , an − bn) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, � wt(w), inv(w) � = (λ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The map that sends each isolated word w ∈ C∗ n to inv(w) induces a bijection between the set of idempotent elements of hypo(Cn) and the set of n- tuples δ = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , δn) ∈ {0, 1}n such that for each i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}, if δi = 1, then δi−1 = 1, or δi+1 = 1 when i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6, the idempotent elements of hypo(Cn) correspond to the hypoplactic congruence classes of isolated words of C¨∗ n with weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, the result follows directly from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this subsection, we first prove some results for hypo(C2) that allow a deeper understanding of this monoid, which will be necessary to deduce some properties in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Motivated by the fact that the plactic monoid of type Cn satisfies the Knuth relations (see Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) with the restric- tion that x ̸= z [Lec02, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1], we then study whether the hypoplactic monoid hypo(Cn) satisfies the Knuth relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In fact, we show that the Knuth relations only hold for one choice of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let m1, m2, p1, p2 ∈ Z≥0 and n1, n2 ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, 1m12n11p1 ¨∼ 1m22n21p2 in C¨∗ 2 implies that m1 = m2, n1 = n2 and p1 = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that 1m12n11p1 ¨∼ 1m22n21p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, m1+p1 = m2+p2 and n1 = n2, because wt(1m12n11p1) = wt(1m22n21p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose m1 ̸= m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Without loss of generality, assume m1 < m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set u = ¨f m2+n1−1 1 ¨f n1 2 (1m12n11p1) = ¨f m2+n1−1 1 � 1m12 n11p1� = 2m11 n12m2−m1−11p1−m2+m1+1 and v = ¨f m2+n2−1 1 ¨f n2 2 (1m22n21p2) = ¨f m2+n2−1 1 � 1m22 n21p2� = 2m21 n2−121p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 1m12n11p1 ¨∼ 1m22n21p2 and n1 = n2, we get that u ¨∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, this is a contradiction, because u is 2-inversion-free and v has a 2-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 49 Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n1, n2, p1, p2, ∈ Z≥0 and m1, m2, q1, q2 ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, in C¨∗ 2, 1m12n11p12q1 ¨∼ 1m22n21p22q2 if and only if m1+p1 = m2+p2 and n1+q1 = n2+q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If we first suppose that 1m12n11p12q1 ¨∼ 1m22n21p22q2 then we get that wt(1m12n11p12q1) = wt(1m22n21p22q2), which implies that m1 + p1 = m2 + p2 and n1 + q1 = n2 + q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that 12k1l2 ¨∼ 1l+12k+1, for any k, l ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The aim is to show that each connected component ΓC¨∗ 2 � 12k1l2 � and ΓC¨∗ 2 � 1l+12k+1� is a path with 2k+2l+5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This will allow us to define a bijection ψ : C∗ 2 � 12k1l2 � → C∗ 2 � 1l+12k+1� that maps each word w ∈ C∗ 2 � 12k1l2 � to the word ψ(w) ∈ C∗ 2 � 1l+12k+1� such that the position of w in ΓC¨∗ 2 � 12k1l2 � is the same as ψ(w) in ΓC¨∗ 2 � 1l+12k+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The paths ΓC¨∗ 2 � 12k1l2 � and ΓC¨∗ 2 � 1l+12k+1� start in 12k1l2 and 1l+12k+1, which are of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From these starting-points, there is a sequence of k +1 edges labelled by 2, each of which transforms a symbol 2 to a symbol 2, in order from left to right through the word;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' at each step except the last, there is a 1-inversion in the word and so a loop labelled by 1 at that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' There are then k + l + 2 edges labelled by 1, each of which transforms a symbol 1 to a symbol 2 or a symbol 2 to a symbol 1, in order from left to right through the word;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' again, in each step except the first and the last, there is a 2-inversion in the word so a loop labelled by 2 at that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, there is a sequence of l + 1 edges labelled by 2, each transforming a symbol 2 to a symbol 2, in order from left to right throughout the word;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' again, there is a loop labelled by 1 at each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, u i −−−→ v is an edge in ΓC¨∗ 2 � 12k1l2 � if and only if ψ(u) i −−−→ ψ(v) is an edge of ΓC¨∗ 2 � 1l+12k+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since ψ � 12k1l2 � = 1l+12k+1, where wt � 12k1l2 � = (l + 1, k + 1) = wt � 1l+12k+1� , we have that ψ preserves weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, ψ is a quasi-crystal isomorphism, which implies that 12k1l2 ¨∼ 1l+12k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, as ¨∼ is a monoid congruence, we can iterately apply 12k1l2 ¨∼ 1l+12k+1 to see that 1m12n11p12q1 ¨∼ 1m1+p12n1+q1 and 1m22n21p22q2 ¨∼ 1m2+p22n2+q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If m1 + p1 = m2 + p2 and n1 + q1 = n2 + q2, we then obtain that 1m12n11p12q1 ¨∼ 1m22n21p22q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ {1, 2}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w ¨∼ 2m11m22m31m4 in C¨∗ 2, for some m1, m2, m3, m4 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that w ̸= 2m11m22m31m4, for any m1, m2, m3, m4 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, w = 2q01p12q11p22q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1pk2qk1pk+1, for some pk+1, q0 ∈ Z≥0 and p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , pk, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , qk ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='24, we have that 1p12q11p22q2 ¨∼ 1p1+p22q1+q2, and by iterating this process, we obtain that 1p12q11p22q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1pk2qk ¨∼ 1p1+p2+···+pk2q1+q2+···+qk, which implies that w ¨∼ 2q01p1+p2+···+pk2q1+q2+···+qk1pk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We will see in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='34 that the previous result does not hold in C¨∗ n when n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now study some properties satisfied by words u, v ∈ C∗ n that are hypoplactic congruent u ¨∼ v in C¨∗ n, for any n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ n with u ¨∼ v in C¨∗ n, and let x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) If x ≤ n − 1 and u ∈ {x, x + 1}∗, then v ∈ {x, x + 1}∗, |u|x = |v|x and |u|x+1 = |v|x+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) If x ≤ n − 1 and u ∈ � x + 1, x �∗, then v ∈ � x + 1, x �∗, |u|x+1 = |v|x+1 and |u|x = |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 50 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO (3) If u ∈ {x}∗, then v ∈ {x}∗ and |u|x = |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) If u ∈ {x}∗, then v ∈ {x}∗ and |u|x = |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (5) If x ̸= 2 and u ∈ {x, x}∗, then v ∈ {x, x}∗ and |u|x − |u|x = |v|x − |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Assume x ≤ n − 1 and u ∈ {x, x + 1}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} \\ {x, x + 1}, since neither i nor i + 1 occurs in u, we have that u is i-inversion-free, and as u ¨∼ v, |v|i ≤ ¨ϕi(v) = ¨ϕi(u) = |u|i + |u|i+1 = 0, which implies that i does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since wt(u) = wt(v), we get that −|v|i = |u|i − |u|i = 0, which implies that i does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, v ∈ � x, x + 1, x + 1, x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since neither x + 2 nor x + 1 occurs in u, we have that u is (x + 1)-inversion-free, and so, |v|x+1 ≤ ¨εx+1(v) = ¨εx+1(u) = |u|x+2 + |u|x+1 = 0, implying that x + 1 does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As wt(u) = wt(v), we get that |u|x+1 = |v|x+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u′ ¨∼ v′ where u′ = ¨f |u|x+1 x+2 ¨f |u|x+1 x+3 · · ¨f |u|x+1 n−1 ¨f |u|x+1 n ¨f |u|x+1 n−1 · · ¨f |u|x+1 x+1 (u) and v′ = ¨f |v|x+1 x+2 ¨f |v|x+1 x+3 · · ¨f |u|x+1 n−1 ¨f |v|x+1 n ¨f |v|x+1 n−1 · · ¨f |v|x+1 x+1 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ and v′ are respectively obtained from u and v by replacing each x + 1 by x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, u′ ∈ � x, x + 1 � and v′ ∈ � x, x + 1, x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since neither x + 1 nor x occurs in u′, we get that u′ is x-inversion-free, and since u′ ¨∼ v′, |v|x = |v′|x ≤ ¨εx(v′) = ¨εx(u′) = |u′|x+1 + |u′|x = 0, which implies that x does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, v ∈ {x, x + 1} which implies that |u|x = |v|x and |u|x+1 = |v|x+1, because wt(u) = wt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) If x ≤ n − 1 and u ∈ � x + 1, x �∗, then u ∈ {x, x + 1}∗, and as u ¨∼ v by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10, we get by (1) that v ∈ {x, x + 1}∗, |u|x = |v|x and |u|x+1 = |v|x+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that v ∈ � x + 1, x �∗, |u|x+1 = |v|x+1 and |u|x = |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) Suppose u ∈ {x}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x > 1 then u lies in {x − 1, x}∗, which implies by (1) that v lies in {x−1, x}∗ , where |u|x = |u|x and |v|x−1 = |v|x+1 = 0 and so v ∈ {x}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x = 1, then u ∈ {1, 2}∗ and the result follows similarly from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) If u ∈ {x}∗, then u ∈ {x}∗ and the result follows from Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (5) Assume x ̸= 2 and u ∈ {x, x}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As u ¨∼ v, we have that wt(u) = wt(v), which implies that |u|x − |u|x = |v|x − |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} \\ {x − 1, x}, since neither i nor i + 1 occurs in u, we have that u is i-inversion-free, and as u ¨∼ v, |v|i ≤ ¨ϕi(v) = ¨ϕi(u) = |u|i + |u|i+1 = 0, which implies that i does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since wt(u) = wt(v), we get that −|v|i = |u|i − |u|i = 0, which implies that i does not occur in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x = 1, then the result is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Otherwise, x ≥ 3, we have that |v|x−1 ≤ ¨εx−2(v) = ¨εx−2(u) = |u|x−1 + |u|x−2 = 0, and then, −|v|x−1 = |u|x−1 − |u|x−1 = 0, implying that neither x − 1 nor x − 1 occurs in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, v ∈ {x, x}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} with {x, y} ̸= � 2, 2 � , and let u, v ∈ C∗ n with u ¨∼ v in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If u ∈ {x, y}∗, then v ∈ {x, y}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that u ∈ {x, y}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x = y, the result follows from items (3) and (4) of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If x = y, then x ̸= 2, because {x, y} ̸= � 2, 2 � , and so, the result follows from Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, in the following, we assume that x ̸= y and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 51 Take i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} such that x ∈ � i, i � and y ∈ � j, j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since x ̸= y and x ̸= y, we have that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Without loss of generality, we assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We thus consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: x = i and y = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If j > i + 1, then u is (j − 1)-inversion-free, because neither j − 1 nor j occurs in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this case, as u ¨∼ v, we get that u′ ¨∼ v′ for u′ = ¨e|u|j i+1¨e|u|j i+2 · · · ¨e|u|j j−1(u) and v′ = ¨e|u|j i+1¨e|u|j i+2 · · · ¨e|u|j j−1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ is obtained from u by replacing each j by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If j = i + 1, we take u′ = u and v′ = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' trivially, u′ ¨∼ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In either case, u′ ∈ {i, i + 1}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26(1) that v′ ∈ {i, i + 1}∗ and |v′|i+1 = |u′|i+1 = |u|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the case j = i + 1, this establishes the result immediately since v = v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the case j > i + 1, v = ¨f |u|j j−1 ¨f |u|j j−2 · · · ¨f |u|j i+1 (v′), we have that v is obtained from v′ by replacing each i + 1 by j, as |v′|i+1 = |u|j, and so, v ∈ {i, j}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: x = i and y = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u is j-inversion-free, because neither j nor j + 1 occurs in u, as i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ v, we get that u′ ¨∼ v′ for u′ = ¨e |u|j i+1¨e |u|j i+2 · · · ¨e |u|j n−1¨e |u|j n ¨e |u|j n−1 · · · ¨e |u|j j (u) and v′ = ¨e |u|j i+1¨e |u|j i+2 · · · ¨e |u|j n−1¨e |u|j n ¨e |u|j n−1 · · · ¨e |u|j j (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ is obtained from u by replacing each j by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' With a reasoning analogous to case 1, we obtain that v ∈ � i, j �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 3: x = i and y = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If j > i + 1, then u is (j − 1)-inversion-free, as neither j nor j − 1 occurs in u, and since u ¨∼ v, we get that u′ ¨∼ v′ for u′ = ¨f |u|j i+1 ¨f |u|j i+2 · · · ¨f |u|j j−1(u) and v′ = ¨f |u|j i+1 ¨f |u|j i+2 · · · ¨f |u|j j−1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ is obtained from u by replacing each j by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If j = i + 1, we take u′ = u and v′ = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' trivially, u′ ¨∼ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In either case, u′ ∈ � i + 1, i �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26(2) that v′ ∈ � i + 1, i �∗ and |v′|i+1 = |u′|i+1 = |u|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the case j = i + 1, this establishes the result immediately since v = v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the case j > i + 1, v = ¨e |u|j j−1¨e |u|j j−2 · · · ¨f |u|j i+1 (v′), we have that v is obtained from v′ by replacing each i + 1 by j, as |v′|i+1 = |u|j, and thus, v ∈ � j, i �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 4: x = i and y = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As i < j, note that u is j-inversion-free, because neither j + 1 nor j occurs in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ v, we get that u′ ¨∼ v′ for u′ = ¨f |u|j i+1 ¨f |u|j i+2 · · · ¨f |u|j n−1 ¨f |u|j n ¨f |u|j n−1 · · · ¨f |u|j j (u) and v′ = ¨f |u|j i+1 ¨f |u|j i+2 · · · ¨f |u|j n−1 ¨f |u|j n ¨f |u|j n−1 · · · ¨f |u|j j (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ is obtained from u by replacing each j by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' With a reasoning analogous to case 3, we obtain that v ∈ � j, i �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In either case, we get that v ∈ {x, y}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let m ∈ Z≥0, and let i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n} with i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In C¨∗ n, for any u, v ∈ � 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', i, j, i − 1, i − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 �∗, we have that (1) jmiu ̸¨∼ jm+1v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and 52 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO (2) ujim ̸¨∼ vim+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) Suppose there exist u, v ∈ � 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , i, j, i − 1, i − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 �∗ such that jmiu ¨∼ jm+1v in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, wt(jmiu) = wt � jm+1v � which implies that |v|i = |u|i + 1 and |u|j = |v|j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, j occurs in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since neither j + 1 nor j occurs in u or v, then jmiu and jm+1v are j-inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w1 = ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(jmiu) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='= i + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='mi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 · · · ¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 · · · ¨f |u|j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='= i + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='miu′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w2 = ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='jm+1v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='= i + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='m+1� ¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='= i + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='m+1v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As jmiu ¨∼ jm+1v, we get that w1 ¨∼ w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that u′ is obtained from u by replacing each j by i + 1, and as |v|j = |u|j − 1, v′ is obtained from v by replacing each j by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, i + 1 occurs in u′, as j occurs in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since neither i + 1 nor i occurs in w1 or w2, we have that w1 and w2 are i-inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set w′ 1 = ¨f m+1 i (w1) = i m(i + 1)u′ and w′ 2 = ¨f m+1 i (w2) = i m+1v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As w1 ¨∼ w2, we get that w′ 1 ¨∼ w′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since i + 1 occurs in u′, then w′ 1 has an (i + 1)-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since neither i + 1 nor i + 2 occurs in w′ 2, then w′ 2 is (i + 1)- inversion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, this is a contradiction, because we obtained that w′ 1 has an (i + 1)-inversion, w′ 2 is (i + 1)-inversion-free, and w′ 1 ¨∼ w′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Suppose there exist u, v ∈ � 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , i, j, i − 1, i − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 �∗ such that ujim ¨∼ vim+1 in C¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, wt(ujim) = wt � vim+1� which implies that |u|i = |v|i + 1 and |v|j = |u|j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As justified in (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w1 = ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(ujim) = u′i + 1im ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 = ¨f |u|i+|u|j+m+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(w1) = u′′i(i + 1)m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1 = ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(w′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) = u′′′ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w2 = ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='vim+1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='= v′im+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 = ¨f |u|i+|u|j+m+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(w2) = v′′(i + 1)m+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='w′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2 = ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='j+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='· · ¨f |u|i+m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='(w′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2) = v′′′j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ujim ¨∼ vim+1, we get that w′′ 1 ¨∼ w′′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that w′′ 1 is obtained from ujim by replacing each i by j and each j by i, and since |v|i = |u|i − 1 and |v|j = |u|j + 1, w′′ 2 is obtained from vim+1 by replacing each i by j and each j by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, u′′′, v′′′ ∈ � 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', i − 1, j, i, i − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , 1 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 that jmiu′′′ = w′′ 1 ¨∼ w′′ 2 = jm+1v′′′, which is a contradiction by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='27 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28, we have for words u, v ∈ C∗ n with length at most 2 that u ¨∼ v implies u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following result, we identify which words of length 3 are hypoplactic congruent, and obtain that for distinct words, it comes under the statement of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 53 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y, z, x′, y′, z′ ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, xyz ¨∼ x′y′z′ in C¨∗ n if and only if xyz = x′y′z′ or xyz, x′y′z′ ∈ � a11, 1a1, 11a � , for some a ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As 11 is an isolated word, we have by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that a11 ¨∼ 1a1 ¨∼ 11a, for any a ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so, the converse implication holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Assume that xyz ¨∼ x′y′z′, that is, there exists a quasi-crystal isomorphism between the connected components C¨∗ n(xyz) and C¨∗ n(x′y′z′) mapping xyz to x′y′z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11(2), we have that all words in C¨∗ n(xyz) have exactly three letters, and C¨∗ n(xyz) has at least one highest-weight word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we first suppose that xyz is a highest-weight word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As xyz ¨∼ x′y′z′, we get that if xyz is isolated, so is x′y′z′, and if xyz is of highest weight but not isolated, so is x′y′z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, in the following, we consider this two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, the isolated words in C¨∗ n with three letters are 111, 11 1, 122, 22 1, or of the form ttt, for t ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If xyz and x′y′z′ are among these words and xyz ̸= x′y′z′, then as wt(xyz) = wt(x′y′z′), we must have that xyz and x′y′z′ lie in � 111, 122, 111 � or � 11 1, 22 1, 111 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since both 122 and 221 have a 2-inversion, we get by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 that 111 ̸¨∼ 122 ̸¨∼ 111 and 11 1 ̸¨∼ 22 1 ̸¨∼ 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if xyz and x′y′z′ are isolated and xyz ̸= x′y′z′, then they lie in � 111, 111 � or � 11 1, 111 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='16, the highest-weight words in C¨∗ n, which are not isolated and consist of three letters, are 111, 112, 121, 122, 212, 123 (if n ≥ 3), 112, 121, 211, of the form ii(i + 1), for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}, or of the form (j + 1)j + 1 j, for j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If xyz and x′y′z′ are among these words and xyz ̸= x′y′z′, then as wt(xyz) = wt(x′y′z′), we must have that xyz and x′y′z′ lie in {112, 121}, {122, 212}, � 112, 121, 211, 112 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We get by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28(1) that 212 ̸¨∼ 122 and by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28(2) that 112 ̸¨∼ 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 112 is 1-inversion-free, we have by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10 that 112 ̸¨∼ w, for any w ∈ � 112, 121, 211 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, if xyz and x′y′z′ are of highest weight, but not isolated, and xyz ̸= x′y′z′, then they lie in � 112, 121, 211 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that C∗ n � 112 � = � 11a �� a ∈ Cn, 2 ≤ a ≤ 2 � , C∗ n � 121 � = � 1a1 �� a ∈ Cn, 2 ≤ a ≤ 2 � , and C∗ n � 211 � = � a11 �� a ∈ Cn, 2 ≤ a ≤ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so, if xyz and x′y′z′ lie in some of these connected components, then as wt(xyz) = wt(x′y′z′), we obtain that xyz and x′y′z′ lie in � 11a, 1a1, a11 � , for some a ∈ Cn with 2 ≤ a ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, for any x, y, z, x′, y′, z′ ∈ Cn such that xyz ̸= x′y′z′ and xyz ¨∼ x′y′z′, we have that xyz and x′y′z′ lie in � 11a, 1a1, a11 � , for some a ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From the previous result, we get that the Knuth relations (Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1) only hold in hypo(Cn) for instances that come under the statement of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x, y, z ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, yzx ¨∼ yxz in C¨∗ n if and only if x = y = z or (y, x) = � 1, 1 � ) or (y, z) = � 1, 1 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, xzy ¨∼ zxy in C¨∗ n if and only if x = y = z or (x, y) = � 1, 1 � ) or (z, y) = � 1, 1 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We start by checking some properties of the identities satisfied by hypo(C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let X be a finite alphabet, and let u, v ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If hypo(C2) satisfies the identity u = v, then the following conditions are satisfied: (1) |u|x = |v|x, for all x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 54 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO (2) until the first occurrence of a letter x ∈ X in u and v, each letter of X occurs exactly the same number of times in u and v, that is, if u = u1xu2 and v = v1xv2, where u1, u2, v1, v2 ∈ X∗ are such that |u1|x = |v1|x = 0, then |u1|y = |v1|y, for all y ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) after the last occurrence of a letter x ∈ X in u and v, each letter of X occurs exactly the same number of times in u and v, that is, if u = u1xu2 and v = v1xv2, where u1, u2, v1, v2 ∈ X∗ are such that |u2|x = |v2|x = 0, then |u2|y = |v2|y, for all y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (1) If we consider the map from X to C∗ 2 that sends x to 1 and each other letter of X to ǫ, we obtain that 1|u|x ¨∼ 1|v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, wt � 1|u|x� = wt � 1|v|x� which implies that |u|x = |v|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) Since |u|x = |v|x, we have that x occurs in u if and only if x occurs in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, there exist u1, u2, v1, v2 ∈ X∗ such that u = u1xu2, v = v1xv2, and x does not occur in u1 or v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Given y ∈ X, consider the monoid homomorphism ψ : X∗ → C∗ 2 induced by ψ(x) = 1, ψ(y) = 2 and ψ(z) = ǫ, for each z ∈ X \\ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, 2|u1|y1ψ(u2) = ψ(u) ¨∼ ψ(v) = 2|v1|y1ψ(v2), which implies that |u1|y = |v1|y, by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) Since |u|x = |v|x, we have that x occurs in u if and only if x occurs in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And if so, there exist u1, u2, v1, v2 ∈ X∗ such that u = u1xu2, v = v1xv2, and x does not occur in u2 or v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Given y ∈ X, consider the monoid homomorphism ψ : X∗ → C∗ 2 induced by ψ(x) = 2, ψ(y) = 1 and ψ(z) = ǫ, for each z ∈ X \\ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ(u1)21|u2|y = ψ(u) ¨∼ ψ(v) = ψ(v1)21|v2|y, which implies that |u2|y = |v2|y, by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic monoid hypo(C2) satisfies the identity xyxyxy = xyyxxy, that is, uvuvuv ¨∼ uvvuuv in C¨∗ 2, for any u, v ∈ C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first assume that uvuvuv is of highest weight, that is, ¨ε1, (uvuvuv), ¨ε2(uvuvuv) ∈ {0, +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 2 occurs in uvuvuv, then uvuvuv has a 2-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence 2 also occurs in uvuvuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this case, 2 and 2 occur in uv and vu, implying that both uvuvuv and uvvuuv have a decomposition of the form w12w22w32w4, for some w1, w2, w3, w4 ∈ C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, uvuvuv and uvvuuv are isolated words as they have 1- and 2-inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since wt(uvuvuv) = 3 wt(u) + 3 wt(v) = wt(uvvuuv), we get by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20 that uvuvuv ¨∼ uvvuuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we now assume that 2 does not occur in uvuvuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 1 occurs in uvuvuv, then uvuvuv has a 1-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 2 does not occur in uvuvuv, we get that 1 and 1 occur in uv and vu, implying that both uvuvuv and uvvuuv have a decomposition of the form w11w21w3, for some w1, w2, w3 ∈ � 1, 2, 1 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As 11 is an isolated word, we have by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that uvuvuv ¨∼ 13|u|1+3|v|123|u|2+3|v|21 3|u|1+3|v|1 ¨∼ uvvuuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, we further assume that 1 does not occur in uvuvuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 2 occurs in uvuvuv, then uvuvuv has a 1-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 2 and 1 do not occur in uvuvuv, we get that uv, vu ∈ {1, 2}∗, and 1 and 2 occur in uv, implying that there exist w1, w2, w3, w4 ∈ {1, 2}∗ such that uv = QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 55 w11w2 and uv = w32w4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As |w2uvw3|1 = |w2vuw3|1 and |w2uvw3|2 = |w2vuw3|2, we have by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='24 that uvuvuv = w11w2uvw32w4 ¨∼ w11|w2uvw3|1+12|w2uvw3|2+1w4 ¨∼ w11w2vuw32w4 = uvvuuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, if we also assume that 2 does not occur in uvuvuv, then u = 1|u| and v = 1|v|, which implies that uvuvuv = uvvuuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we obtain that uvuvuv ¨∼ uvvuuv, for any u, v, ∈ C∗ 2 such that uvuvuv is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that this also holds when uvuvuv is not of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose there exist u, v ∈ C∗ 2 such that uvuvuv ̸¨∼ uvvuuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The set W = � u′v′u′v′u′v′ �� u′, v′ ∈ C∗ 2, u′v′u′v′u′v′ ̸¨∼ u′v′v′u′u′v′, |u′| = |u|, |v′| = |v| � is nonempty and finite, so we can take words u′, v′ ∈ C∗ 2 such that u′v′u′v′u′v′ ̸¨∼ u′v′v′u′u′v′, |u′| = |u|, |v′| = |v|, and u′v′u′v′u′v′ has maximal weight among weights of words in W, that is, if w ∈ W and wt(w) ≥ wt(u′v′u′v′u′v′), then wt(w) = wt(u′v′u′v′u′v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As shown above, we have that u′v′u′v′u′v′ is not of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take i ∈ {1, 2} such that ¨ei is defined on u′v′u′v′u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u′v′u′v′u′v′ does not have an i-inversion, ¨εi(u′v′u′v′u′v′) = 3¨εi(u′) + 3¨εi(v′) ∈ Z>0, and ¨e¨εi(u′v′u′v′u′v′) i (u′v′u′v′u′v′) = ¨e¨εi(u′) i (u′)¨e¨εi(v′) i (v′)¨e¨εi(u′) i (u′)¨e¨εi(v′) i (v′)¨e¨εi(u′) i (u′)¨e¨εi(v′) i (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set u′′ = ¨e¨εi(u′) i (u′) and v′′ = ¨e¨εi(v′) i (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, wt(u′′v′′u′′v′′u′′v′′) > wt(u′v′u′v′u′v′), and by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10, |u′′| = |u′| = |u| and |v′′| = |v′| = |v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As the weight of u′v′u′v′u′v′ is maximal among weights of words in W, we get that u′′v′′u′′v′′u′′v′′ /∈ W, which implies that u′′v′′u′′v′′u′′v′′ ¨∼ u′′v′′v′′u′′u′′v′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4) that ¨fi is defined on u′′v′′u′′v′′u′′v′′, and so, u′′v′′u′′v′′u′′v′′ does not have an i-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u′′v′′u′′v′′u′′v′′ ¨∼ u′′v′′v′′u′′u′′v′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' we get that ¨ϕi(u′′v′′v′′u′′u′′v′′) = ¨ϕi(u′′v′′u′′v′′u′′v′′) = 3 ¨ϕi(u′′) + 3 ¨ϕi(v′′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' and as ¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) i (u′′v′′u′′v′′u′′v′′) = ¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) i � ¨e¨εi(u′v′u′v′u′v′) i (u′v′u′v′u′v′) � = u′v′u′v′u′v′ and ¨f ¨ϕi(u′′v′′u′′v′′u′′v′′) i (u′′v′′v′′u′′u′′v′′) = ¨f ¨ϕi(u′′) i (u′′) ¨f ¨ϕi(v′′) i (v′′) ¨f ¨ϕi(v′′) i (v′′) ¨f ¨ϕi(u′′) i (u′′) ¨f ¨ϕi(u′′) i (u′′) ¨f ¨ϕi(v′′) i (v′′) = u′v′v′u′u′v′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' we obtain that u′v′u′v′u′v′ ¨∼ u′v′v′u′u′v′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, for any u, v ∈ C∗ 2, we have that uvuvuv ¨∼ uvvuuv, that is, hypo(C2) satisfies the identity xyxyxy = xyyxxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We now turn our attention for whether hypo(Cn) satisfies identities when n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following results, we prove that hypo(Cn) does not satisfy any nontrivial identity, for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This is achieved by showing that it contains free submonoids with more than one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ � 1, 2 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ¨∼ v in C¨∗ n if and only if u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 56 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that u ¨∼ v and u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since wt(u) = wt(v), we have that |u|1 = |v|1 and |u|2 = |v|2, and as u ̸= v, both 1 and 2 occur in u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take u′, v′, w ∈ � 1, 2 �∗ such that u = w1u′ and v = w2v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since |u|1 = |v|1 and |u|2 = |v|2, we have that |v′|1 = |u′|1 + 1 and |u′|2 = |v′|2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The words u and v do not have 2-inversions, because neither 2 nor 3 occurs in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set u1 = ¨e |v′|2 2 (u) = w1¨e |v′|2 2 (u′) and v1 = ¨e |v′|2 2 (v) = w2¨e |v′|2 2 (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that ¨e |v′|2 2 (v′) is obtained from v′ by replacing each 2 by 3, and as |v′|2 = |u′|2 − 1, ¨e |v′|2 2 (u′) is obtained from u′ by replacing each 2 by 3 except for the left- most 2 that remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, 2 occurs in ¨e |v′|2 2 (u′) and does not occur in ¨e |v′|2 2 (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As u ¨∼ v, we also have that u1 ¨∼ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The words u1 and v1 do not have 1-inversions, because neither 2 nor 1 occurs in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set u2 = ¨f |w|+1 1 (u1) = ¨f |w| 1 (w)2¨e |v′|2 2 (u′) and v2 = ¨f |w|+1 1 (v1) = ¨f |w| 1 (w)1¨e |v′|2 2 (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that ¨f |w| 1 (w) is obtained from w by replacing each 1 by 2 and each 2 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 2 occurs in ¨e |v′|2 2 (u′), we have that u2 has a 2-inversion, and since neither 3 nor 2 occurs in v2, we have that v2 does not have a 2-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5, this is a contradiction, because u1 ¨∼ v1 implies that u2 ¨∼ v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let a, b ∈ Cn with a ̸= b, and let u, v ∈ {a, b}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ¨∼ v in C¨∗ n if and only if u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n} such that a ∈ � i, i � and b ∈ � j, j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since a ̸= b and the case a = b is trivial, without loss of generality we assume i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='10, we have that u ¨∼ v if and only if u ¨∼ v, and so, without loss of generality, we further assume that a = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose that u ¨∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As wt(u) = wt(v) and i ̸= j, we get that |u|a = |v|a and |u|b = |v|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If i = 1, set u′ = u and v′ = v, otherwise, u and v do not have (i − 1)-inversions, because neither i − 1 nor i occurs in them as i < j, and so, set u′ = ¨e|u|a 1 ¨e|u|a 2 · · ¨e|u|a i−1(u) and v′ = ¨e|v|a 1 ¨e|v|a 2 · · ¨e|v|a i−1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As a = i, note that u′ and v′ are respectively obtained from u and v by replacing each a by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, u′, v′ ∈ {1, b}∗ and |u′|b = |v′|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ v and |u|a = |v|a, we get that u′ ¨∼ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In the following cases, we obtain words u′′, v′′ ∈ � 1, 2 � by applying the same and in the same order quasi-Kashiwara operators to u′ and v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: b = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The words u′ and v′ do not have j-inversions, because neither j + 1 nor j occurs in them as 1 ≤ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set u′′ = ¨f |u′|b j ¨f |u′|b j+1 · · · ¨f |u′|b n ¨f |u′|b n−1 · · · ¨f |u′|b 2 (u′) and v′′ = ¨f |v′|b j ¨f |v′|b j+1 · · · ¨f |v′|b n ¨f |v′|b n−1 · · · ¨f |v′|b 2 (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: b = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If j = 2, set u′′ = u′ and v′′ = v′, otherwise, we have that j ≥ 3, as j > i ≥ 1, and the words u′ and v′ do not have (j − 1)-inversions, because neither j nor j − 1 occurs in them, and so, set u′′ = ¨f |u′|b j−1 ¨f |u′|b j−2 · · · ¨f |u′|b 2 (u′) and v′′ = ¨f |v′|b j−1 ¨f |v′|b j−2 · · · ¨f |v′|b 2 (v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 57 In either case, note that u′′ and v′′ are respectively obtained from u′ and v′ by replacing each b by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, u′′, v′′ ∈ � 1, 2 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u′ ¨∼ v′ and |u′|b = |v′|b, we get that u′′ ¨∼ v′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='33, we obtain that u′′ = v′′, and since the quasi- Kashiwara operators are injective when defined (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1(4)), we deduce that u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ From the previous result we have when n ≥ 3 that for a, b ∈ Cn with a ̸= b, the set {a, b} is free on hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This marks a difference when compared to hypo(C2), where {1, 2} is not free, as shown in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This also implies the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, hypo(Cn) does not satisfy nontrivial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first show that the hypoplactic monoid hypo(C2) does not admit a finite presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic congruence ¨∼ on the free monoid C∗ 2 is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, hypo(C2) has no finite presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let R be a finite subset of ¨∼, and denote by ∼R the monoid congruence on C∗ 2 generated by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, ∼R ⊆ ¨∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As R is finite, set m = 1 + max (u,v)∈R � |u|, |v| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first show that for any subword u of 121m2 with length at most m (that is, any word u consisting of at most m consecutive letters of 121m2) and v ∈ C∗ 2, if u ¨∼ v, then u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v ∈ C∗ 2 be such that u ¨∼ v and u is a subword of 121m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, u ∈ {1, 2}∗, and by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26, v ∈ {1, 2}∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As u is a subword of 121m2 with length at most m, we get that u lies in one of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: u = 1p where p ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ v, we have that wt(u) = wt(v), and since v ∈ {1, 2}∗, we get that |v|1 = |u|1 = p and |v|2 = |u|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, v = 1p = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: u = 1p121p2 where p1, p2 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As in the previous case, we have that |v|1 = p1 + p2 and |v|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, v = 1q121q2, for some q1, q2 ∈ Z≥0 such that q1 + q2 = p1 + p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='23, 1p121p2 ¨∼ 1q121q2 implies that p1 = q1 and p2 = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, v = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since R ⊆ ¨∼ and every pair (u, v) ∈ R satisfies |u| < m and |v| < m, we get that if (u, v) ∈ R and u or v are subwords of 121m2, then u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As R generates ∼R, we obtain for w ∈ C∗ 2 that 121m2 ∼R w if and only if w = 121m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, we have that 121m2 ̸= 1m+122, and by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='24, 121m2 ¨∼ 1m+122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, ∼R ̸= ¨∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And therefore, ¨∼ is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Although, hypo(C2) does not have a finite presentation, we are able to describe connected components of ΓC¨∗ 2 , and thus, find representatives for the hypoplactic congruence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let u, v, w ∈ � 1, 2, 1 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, 2u1v1w2 ¨∼ 1m+12p+21 q+1 in C¨∗ 2, for some m, p, q ∈ Z≥0 where m = 0 or q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set m = max � 0, |u|1 + |v|1 + |w|1 − |u|1 − |v|1 − |w|1 � , p = |u|2 + |v|2 + |w|2, and q = max � 0, |u|1 + |v|1 + |w|1 − |u|1 − |v|1 − |w|1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first show that each connected component ΓC¨∗ 2 � 2u1v1w2 � and ΓC¨∗ 2 � 1m+12p+21 q+1� is a path with p + 3 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The paths ΓC¨∗ 2 � 2u1v1w2 � and ΓC¨∗ 2 � 1m+12p+21 q+1� start respectively in 2u1v1w2 and 1m+12p+21 q+1, which are highest-weight words without 2-inversions, as 2 does not occur in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From these starting-points, there is a sequence of p+2 edges labelled by 2, each of which transforms a symbol 2 to a symbol 2, in order 58 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO from left to right through the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The end-points of the paths ΓC¨∗ 2 � 2u1v1w2 � and ΓC¨∗ 2 � 1m+12p+21 q+1� are 2 ¨f |u|2 2 (u)1 ¨f |v|2 2 (v)1 ¨f |w|2 2 (w)2 and 1m+12 p+21 q+1, re- spectively, which are lowest-weight words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, each vertex of the paths has a loop labelled by 1, because it admits a decomposition of the form w11w22w3 or w12w21w3, for some w1, w2, w3 ∈ C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a bijection ψ : C∗ 2 � 2u1v1w2 � → C∗ 2 � 1m+12p+21 q+1� that maps each word w ∈ C∗ 2 � 2u1v1w2 � to the word ψ(w) ∈ C∗ 2 � 1m+12p+21 q+1� such that the position of w in ΓC¨∗ 2 � 2u1v1w2 � is the same as ψ(w) in ΓC¨∗ 2 � 1m+12p+21 q+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As shown above, for u, v ∈ C∗ 2 � 2u1v1w2 � and i ∈ {1, 2}, we have that u i −−−→ v is an edge in ΓC¨∗ 2 � 2u1v1w2 � if and only if ψ(u) i −−−→ ψ(v) is an edge of ΓC¨∗ 2 � 1m+12p+21 q+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And since ψ � 2u1v1w2 � = 1m+12p+21 q+1, where wt � 2u1v1w2 � = (m − q, p + 2) = wt � 1m+12p+21 q+1� , we get that ψ preserves weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, ψ is a quasi-crystal isomorphism, which implies that 2u1v1w2 ¨∼ 1m+12p+21 q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Any connected component of C¨∗ 2 is quasi-crystal isomorphic to one and only one of the following: (1) C¨∗ 2 � 1m� , m ∈ Z≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (2) C¨∗ 2 � 2m11m2+12m3+11m4� , m1, m2, m3, m4 ∈ Z≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (3) C¨∗ 2 � 1m1+12m21 m3+1� , m1, m2, m3 ∈ Z≥0 with m1 = 0 or m3 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (4) C¨∗ 2 � 1m1+12m2+12 m3+11 m4+1� , m1, m2, m3, m4 ∈ Z≥0 with m1 = 0 or m4 = 0, and m2 = 0 or m3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, the elements in these connected components form a minimal set of rep- resentatives for the hypoplactic congruence classes on C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='11(2) any connected component of C¨∗ 2 has at least a highest- weight word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ 2 be a highest-weight word of C¨∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 2 occurs in w, then w has a 2-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that 2 occurs in w, and as w is of highest weight, w has a 1-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, w is an isolated word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each i ∈ {1, 2}, if |w|i ≥ |w|i, set mi = |w|i − |w|i and m5−i = 0, otherwise, set mi = 0 and m5−i = |w|i − |w|i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, wt(w) = (m1 − m4, m2 − m3) = wt � 1m1+12m2+12 m3+11 m4+1� , which implies by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20 that w ¨∼ 1m1+12m2+12 m3+11 m4+1, and so, we have a quasi-crystal isomorphism between C¨∗ 2(w) and C¨∗ 2 � 1m1+12m2+12 m3+11 m4+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, in the following, we assume that 2 does not occur in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 1 occurs in w, then w has a 1-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 2 does not occur in w, then we have in w that a 1 appears to the right of a 1, or otherwise, 2 appears to the right of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In this second case, as 1 occurs in w, we may have that a 1 appears to the right of a 2, or otherwise, every 1 occurs to the left of any 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Based on these decompositions, we get that w lies in one of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 1: w = w11w21w3, for some w1, w2, w3 ∈ � 1, 2, 1 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 11 is an isolated word, we get by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that w ¨∼ 1w1w2w31, and by iterating this process, w ¨∼ 1|w|12|w|21 |w|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set m2 = |w|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If |w|1 ≥ |w|1, set m1 = |w|1 − |w|1 and m3 = 0, otherwise, set m1 = 0 and m3 = |w|1 − |w|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 59 Since wt � 11 � = 0, we get by Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6 that w ¨∼ 1|w|12m21 |w|1 ¨∼ 1|w|11 |w|12m2 ¨∼ 1m1+11 m3+12m2 ¨∼ 1m1+12m21 m3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, there exists a quasi-crystal isomorphism between C¨∗ 2(w) and C¨∗ 2 � 1m1+12m21 m3+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 2: w = w12w21w31w42w5, for some w1, w2, w3, w4, w5 ∈ � 1, 2, 1 �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='37, we have that 2w21w31w42 ¨∼ 1p1+12p2+21 p3+1, for some p1, p2, p3 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As in the previous case, we get that w = w11p1+12p2+21 p3+1w5 ¨∼ 1m1+12m21 m3+1, for some m1, m2, m3 ∈ Z≥0 with m1 = 0 or m3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, there exists a quasi-crystal isomorphism between C¨∗ 2(w) and C¨∗ 2 � 1m1+12m21 m3+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Case 3: w = 1 pu, for some u ∈ {1, 2}∗ and p ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='25, we have that u ¨∼ 2q11q22q31q4, for some q1, q2, q3, q4 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' In particular, |w|2 = |u|2 = q1 + q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since w has a 1-inversion, then u has a 1-inversion, which implies by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that q2 > 0 and q3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that ¨ep+q1+q3 2 ¨ep 1 ¨f q1+q3 2 � 1 p2q11q22q31q4� = ¨ep+q1+q3 2 ¨ep 1 � 1 p2 q11q22 q31q4� = ¨ep+q1+q3 2 � 2 p2 q11q22 q31q4� = 2p+q11q22q31q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since u ¨∼ 2q11q22q31q4, we get that w ¨∼ 1 p2q11q22q31q4, and as |w|2 = q1 + q3, we obtain that ¨e|w|2+p 2 ¨ep 1 ¨f |w|2 2 (w) ¨∼ 2p+q11q22q31q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set m1 = p+q1, m2 = q2 −1, m3 = q3 −1 and m4 = q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, there exists a quasi-crystal isomorphism between C¨∗ 2(w) and C¨∗ 2 � 2m11m2+12m3+11m4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we further assume that 1 does not occur in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If 2 occurs in w, then w has a 1-inversion, because it is of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As neither 1 nor 2 occurs in w, we have that w ∈ {1, 2}∗, and by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='25, w ¨∼ 2p11p22p31p4, for some p1, p2, p3, p4 ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since w has a 1-inversion, we get by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 that p2 > 0 and p3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set m1 = p1, m2 = p2 − 1, m3 = p3 − 1 and m4 = p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, there exists a quasi-crystal isomorphism between C¨∗ 2(w) and C¨∗ 2 � 2m11m2+12m3+11m4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, if 2 does not occur in w, then w = 1m, where m = |w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And so, C¨∗ 2(w) coincides with C¨∗ 2(1m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have thus proved that any connected component of C¨∗ 2 is quasi-crystal iso- morphic to some connected component lying in (1) to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' It remains to show that it is quasi-crystal isomorphic to only one of such connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we now show that there are no quasi-crystal isomorphic connected components among the ones in (1) to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Each connected component C¨∗ 2 � 1m1+12m2+12 m3+11 m4+1� in (4) consists of an isolated word with weight (m1 − m4, m2 − m3), a 1-inversion and a 2-inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By the condition that m1 = 0 or m4 = 0, and m2 = 0 or m3 = 0, we have that all words in (4) have different weights, which implies by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='20 that they are not hypoplactic congruent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, all connected components in (1) to (3) contain a word without a 2-inversion, and so, there is no quasi-crystal isomorphism between any of them and some connected component in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Each connected component C¨∗ 2 � 1m1+12m21 m3+1� in (3) is formed by m2 + 1 words of the form 1m1+12 p12p21 m3+1, for p1, p2 ∈ Z≥0 such that p1 + p2 = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 60 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO In particular, each connected component has exactly one highest-weight word, namely: 1m1+12m21 m3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, if C¨∗ 2 � 1m1+12m21 m3+1� and C¨∗ 2 � 1m′ 1+12m′ 21 m′ 3+1� are quasi-crystal isomorphic connected components lying in (3), then we have that 1m1+12m21 m3+1 ¨∼ 1m′ 1+12m′ 21 m′ 3+1, which implies that m1 − m3 = m′ 1 − m′ 3 and m2 = m′ 2, and by the condition that m1 = 0 or m3 = 0 and the condition that m′ 1 = 0 or m′ 3 = 0, we obtain that m1 = m′ 1 and m3 = m′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, there are no quasi-crystal isomorphic connected components among the ones in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, all words lying in the connected components in (3) have 1-inversions, each connected component in (1) or (2) have at least one word without a 1-inversion (respectively, 1m or 2 m11m2+12 m3+11m4), and so, there is no quasi-crystal isomorphism between some connected component in (3) and some connected component in (1) or (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The only word lying in a connected component C¨∗ 2 � 2m11m2+12m3+11m4� in (2) and the set {1, 2}∗ is 2m11m2+12m3+11m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, if C¨∗ 2 � 2m11m2+12m3+11m4� and C¨∗ 2 � 2m′ 11m′ 2+12m′ 3+11m′ 4� are quasi-crystal isomorphic connected components in (2), then we have by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26 that 2m11m2+12m3+11m4 ¨∼ 2m′ 11m′ 2+12m′ 3+11m′ 4, which implies by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='28 that m1 = m′ 1, m2 = m′ 2, m3 = m′ 3 and m4 = m′ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, there are no quasi-crystal isomorphic connected components among the ones in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, each connected component in (2) contains at least one word with a 2-inversion (for instance, 1 m12m2+12 m3+11m4), while no word lying in some connected component in (1) has a 2-inversion, and so, there is no quasi- crystal isomorphism between some connected component in (2) and some connected component in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Finally, note that the only highest-weight word lying in a connected component C¨∗ 2 � 1m� in (1) and the set {1, 2}∗ is 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If C¨∗ 2 � 1m� and C¨∗ 2 � 1m′� are quasi-crystal isomorphic connected components in (1), then we have by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='26 that 1m ¨∼ 1m′, which implies that m = m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, there is no quasi-crystal isomorphism between distinct connected components among the ones in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Finally, we show that hypo(Cn) does not admit a finite presentation for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The hypoplactic congruence ¨∼ on C∗ n is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, hypo(Cn) has no finite presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let R be a finite subset of ¨∼, and denote by ∼R the monoid congruence on C∗ n generated by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Thus, ∼R ⊆ ¨∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As R is finite, take m = 1 + max (u,v)∈R � |u|, |v| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Set w = 1m2m1m2 m1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If u is a subword of w with length at most m (that is, a word u consisting of at most m consecutive letters of w), then u lies in {a, b}∗, for some a, b ∈ � 1, 2, 2, 1 � with a ̸= b, and if v ∈ C∗ n is such that u ¨∼ v, we get by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='27 that v ∈ {a, b}∗, and then, we obtain by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='34 that u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since R ⊆ ¨∼ and every pair (u, v) ∈ R satisfies |u| < m and |v| < m, we get that if (u, v) ∈ R and u or v are subwords of w, then u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As R generates ∼R, we have for w′ ∈ C∗ n that w ∼R w′ if and only if w′ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' On the other hand, we have that w ̸= 12m2m2 m1 m, and as 1m2m2 m1 m is an isolated word, w ¨∼ 12m2m2 m1 m, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that ∼R ̸= ¨∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' And therefore, ¨∼ is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From hypo(An−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We first show that an embedding of hypo(An) into hypo(Cn) cannot map each letter of An to a letter of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For m ≥ 2 and n ≥ 2, there exists no injective monoid ho- momorphism ψ : hypo(Am) → hypo(Cn) such that ψ(x), ψ(y) ∈ Cn, for some x, y ∈ Am with x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 61 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose ψ : hypo(Am) → hypo(Cn) is an injective monoid homomorphism such that ψ(x), ψ(y) ∈ Cn, for some x, y ∈ Am with x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Without loss of gener- ality assume x < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since xyx = xxy in hypo(Am), we get that ψ(x)ψ(y)ψ(x) = ψ(x)ψ(x)ψ(y) in hypo(Cn), which implies by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='30 that ψ(x) = ψ(y), or ψ(x) = 1 and ψ(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ψ is injective, we must have that ψ(x) = 1 and ψ(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 11 is an isolated word of C¨∗ n and wt � 11 � = 0, we get by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6 that ψ(xyxy) = 1111 = 11 = ψ(xy) in hypo(Cn), which is a contradiction as ψ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ We now show that an injective map between the relevant alphabets cannot be ex- tended to a (not necessarily injective) homomorphism from the hypoplactic monoid of type An to that of type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For m ≥ 3 and n ≥ 2, no injective map from Am to Cn can be extended to a monoid homomorphism from hypo(Am) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Suppose ψ : hypo(Am) → hypo(Cn) is a monoid homomorphism where its restriction ψ|Am is an injective map from Am to Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, we can take y ∈ {2, 3} such that ψ(y) ̸= 1 and ψ(y) ̸= ψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 1y1 = 11y in hypo(Am), we obtain that ψ(1)ψ(y)ψ(1) = ψ(1)ψ(1)ψ(y) in hypo(Cn), contradicting Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ Now, we show that hypo(An−1) can be embedded in hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Define a map ψ : A∗ n−1 → C∗ n by ψ(w) = � ǫ if w = ǫ wnnnn otherwise, for each w ∈ A∗ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then ψ factors to give an injective monoid homomorphism from hypo(An−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Denote the quasi-crystal structure of A¨∗ n−1 by wtA, ¨eA i , ¨f A i , ¨εA i and ¨ϕA i (i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 2}), and denote the hypoplactic congruence on A¨∗ n−1 by ¨∼A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Similarly, denote the quasi-crystal structure of C¨∗ n by wtC, ¨eC i , ¨f C i , ¨εC i and ¨ϕC i (i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n}), and denote the hypoplactic congruence on C¨∗ n by ¨∼C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='12 and Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, it is immediate that gA(w) = gC(w), for any w ∈ A∗ n−1 and g ∈ � ¨ei, ¨fi, ¨εi, ¨ϕi �� i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 2} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We now show that for u, v ∈ A∗ n−1, u ¨∼A v if and only if ψ(u) ¨∼C ψ(v), which implies that ψ induces a well-defined injective map from hypo(An−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' First, note that if u ¨∼A ǫ, for some u ∈ A∗ n−1, then wtA(u) = 0, which implies that u = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' If u′ ¨∼C ǫ, for some u′ ∈ C∗ n, then |u′|i + |u′|i+1 ≤ ¨ϕC i (u′) = ¨ϕC i (ǫ) = 0, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, implying that u′ = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ A∗ n−1 with w ̸= ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that wtC(wnnnn) = � |w|1, |w|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , |w|n−1, 0 � , which implies for w′ ∈ A∗ n−1 that wtC(wnnnn) = wtC(w′nnnn) if and only if wtA(w) = wtA(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, ¨εC n−1(wnnnn) = ¨εC n(wnnnn) = +∞, which implies that ¨eC n−1, ¨f C n−1, ¨eC n and ¨f C n are undefined on wnnnn, and for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 2}, ¨εC i (wnnnn) = ¨εC i (w) = ¨εA i (w) and ¨ϕC i (wnnnn) = ¨ϕC i (w) = ¨ϕA i (w), 62 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO because ¨εC i (nnnn) = ¨ϕC i (nnnn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since A¨∗ n−1 and C¨∗ n are seminormal, we get that ¨f C i is defined on wnnnn if and only if ¨f A i is defined on w, and if so, ¨f C i (wnnnn) = ¨f C i (w)nnnn = ¨f A i (w)nnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, for u, v ∈ A∗ n−1 and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 2}, we have an edge u i −−−→ v in ΓA¨∗ n−1 if and only if we have an edge unnnn i −−−→ vnnnn in ΓC¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that ΓC¨∗ n(wnnnn) is obtained from ΓA¨∗ n−1(w) by concatenating nnnn to each vertex, and adding (n−1)-labelled and n-labelled loops to each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Equivalently, ΓA¨∗ n−1(w) is obtained from ΓC¨∗ n(wnnnn) by removing the last four letters of each vertex, and removing all (n − 1)-labelled and n-labelled loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, for any u, v ∈ A∗ n−1, there exists a graph isomorphism between ΓA¨∗ n−1(u) and ΓA¨∗ n−1(v) mapping u to v if and only if there exists a graph isomorphism between ΓC¨∗ n(unnnn) and ΓC¨∗ n(vnnnn) mapping unnnn to vnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, we have that u ¨∼A v if and only if unnnn ¨∼C vnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' To obtain that ψ induces an injective monoid homomorphism from hypo(An−1) to hypo(Cn), it remains to prove that ψ(uv) ¨∼C ψ(u)ψ(v), for any u, v ∈ A∗ n−1, since ψ(ǫ) = ǫ follows from the definition of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As shown above, if uv ¨∼A ǫ, then uv = ǫ, which implies that u = v = ǫ, and so, ψ(uv) ¨∼C ψ(u)ψ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='18, we have that nnnn is a commutative and idempotent element of hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' So, for any u′, v′ ∈ C∗ n, we get that u′nnnnv′nnnn ¨∼C u′v′(nnnn)2 ¨∼C u′v′nnnn, in particular, for u, v ∈ A∗ n−1 with u ̸= ǫ or v ̸= ǫ, we obtain that ψ(uv) ¨∼C ψ(u)ψ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, we get that ψ induces an injective monoid homomorphism from hypo(An−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From hypo(Cn−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' The following result shows that we have a monoid embedding from hypo(Cn−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider ψ to be the monoid homomorphism from C∗ n−1 to C∗ n such that ψ(x) = (x + 1)11 and ψ(x) = � x + 1 � 11, for each x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ induces an injective monoid homomorphism from hypo(Cn−1) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let τ be the monoid homomorphism from C∗ n−1 to C∗ n such that τ(x) = x + 1 and τ(x) = x + 1, for each x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Equivalently, for w ∈ C∗ n−1, τ(w) is obtained from w by replacing each x by x + 1 and each x by x + 1, for x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' For each m ∈ {n−1, n}, denote the quasi-crystal structure of C¨∗ m by wt(m), ¨e(m) i , ¨f (m) i , ¨ε(m) i and ¨ϕ(m) i (i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m}), and denote the hypoplactic congruence on C¨∗ m by ¨∼(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' From Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='2, for w ∈ C∗ n−1 and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}, note that w has an i-inversion if and only if τ(w) has an (i + 1)-inversion, and so, we get that ¨ε(n) i+1 � τ(w) � = ¨ε(n−1) i (w) and ¨ϕ(n) i+1 � τ(w) � = ¨ϕ(n−1) i (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Moreover, ¨e(n) i+1 is defined on τ(w) if and only if ¨e(n−1) i is defined on w, and if so, ¨e(n) i+1 � τ(w) � = τ � ¨e(n−1) i (w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Analogously, ¨f (n) i+1 is defined on τ(w) if and only if ¨f (n−1) i is defined on w, and if so, ¨f (n) i+1 � τ(w) � = τ � ¨f (n−1) i (w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since ψ is a monoid homomorphism from C∗ n−1 to C∗ n, to prove that ψ induces an injective monoid homomorphism from hypo(Cn−1) to hypo(Cn), it suffices to show QUASI-CRYSTALS FOR ARBITRARY ROOT SYSTEMS 63 that for any u, v ∈ C∗ n−1, u ¨∼(n−1) v if and only if ψ(u) ¨∼(n) ψ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Note that for m ∈ {n − 1, n} and u ∈ C∗ m, if u ¨∼(m) ǫ, then |u|i + |u|i+1 ≤ ¨ϕ(m) i (u) = ¨ϕ(m) i (ǫ) = 0, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m}, implying that u = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let w ∈ C∗ n−1 with w ̸= ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Take x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , xk ∈ Cn−1 such that w = x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since 11 is an isolated word of C¨∗ n and wt � 11 � = 0, we have by Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='6 that ψ(w) = τ(x1)11τ(x2)11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' τ(xk)11 = τ(x1)τ(x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' τ(xk) � 11 �k = τ(w)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' We have that wt(n)� ψ(w) � = � 0, |w|1 − |w|1, |w|2 − |w|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , |w|n−1 − |w|n−1 � , which implies that for w′ ∈ C∗ n−1, wt(n)� ψ(w) � = wt(n)� ψ(w′) � if and only if wt(n−1)(w) = wt(n−1)(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Also, ¨ε(n) 1 � ψ(w) � = +∞, which implies that ¨e(n) 1 and ¨f (n) 1 are undefined on ψ(w), and for i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , n}, ¨ε(n) i � ψ(w) � = ¨ε(n) i � τ(w) � = ¨ε(n−1) i−1 (w) and ¨ϕ(n) i � ψ(w) � = ¨ϕ(n) i � τ(w) � = ¨ϕ(n−1) i−1 (w), because ¨ε(n) i � 11 � = ¨ϕ(n) i � 11 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Since C¨∗ n−1 and C¨∗ n are seminormal, we have that ¨f (n) i is defined on ψ(w) if and only if ¨f (n−1) i−1 is defined on w, and if so, ¨f (n) i � ψ(w) � = ¨f (n) i � τ(w) � 11 = τ � ¨f (n−1) i−1 (w) � 11 = ψ � ¨f (n−1) i−1 (w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Hence, for u, v ∈ C∗ n−1 and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', n − 1}, we have an edge u i −−−→ v in ΓC¨∗ n−1 if and only if we have an edge ψ(u) i −−−→ ψ(v) in ΓC¨∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' This implies that ΓC¨∗ n � ψ(w) � is obtained from ΓC¨∗ n−1(w) by applying ψ to each vertex, and adding 1-labelled loops to each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' As ψ is injective, ΓC¨∗ n−1(w) can also be obtained from ΓC¨∗ n � ψ(w) � by reversing the described process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Therefore, for any u, v ∈ C∗ n−1, there exists a graph isomorphism between ΓC¨∗ n−1(u) and ΓC¨∗ n−1(v) mapping u to v if and only if there exists a graph isomorphism between ΓC¨∗ n � ψ(u) � and ΓC¨∗ n � ψ(v) � mapping ψ(u) to ψ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='13, we have that u ¨∼(n−1) v if and only if ψ(u) ¨∼(n) ψ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' □ By composing the homomorphisms from the previous result, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Let n > m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Consider ψ to be the monoid homomorphism from C∗ m to C∗ n such that ψ(x) = (x + n − m)12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (n − m)(n − m) � n − m − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1 and ψ(x) = (x + n − m)12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' (n − m)(n − m) � n − m − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 1, for each x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Then, ψ induces an injective monoid homomorphism from hypo(Cm) to hypo(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' 64 ALAN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' CAIN, RICARDO P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' GUILHERME, AND ANTÓNIO MALHEIRO References [Bol98] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Bollobás.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Modern graph theory, volume 184 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Springer-Verlag, New York, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1007/978-1-4612-0619-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' [Bou02] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Bourbaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Lie groups and Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Chapters 4–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Elements of Mathematics (Berlin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Translated from the 1968 French original by Andrew Pressley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='1007/978-3-540-89394-3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Schensted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Longest increasing and decreasing subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Canadian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=', 13:179–191, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='4153/CJM-1961-015-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' [Sch97] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Schützenberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Pour le monoïde plaxique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Humaines, (140):5–10, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content=' Center for Mathematics and Applications (NovaMath), FCT NOVA, 2829–516 Ca- parica, Portugal Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='cain@fct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='pt Center for Mathematics and Applications (NovaMath), FCT NOVA, and Department of Mathematics, FCT NOVA, 2829–516 Caparica, Portugal Email address: rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='guilherme@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='fct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='pt Center for Mathematics and Applications (NovaMath), FCT NOVA, and Department of Mathematics, FCT NOVA, 2829–516 Caparica, Portugal Email address: ajm@fct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfbvf7/content/2301.00271v1.pdf'} +page_content='unl.' metadata={'source': 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Astrophys. Astr. (0000) 000: +DOI +Probing Cosmology beyond ΛCDM using the SKA +Shamik Ghosh1,2, Pankaj Jain3, Rahul Kothari4,*, Mohit Panwar3, Gurmeet Singh3, Prabhakar +Tiwari5 +1CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of +Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China +2School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, China +3Department of Physics, Indian Institute of Technology, Kanpur-208016, India. +4Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa. +5National Astronomical Observatories, Chinese Academy of Science, Beijing 100101, P.R.China. +*Corresponding author. E-mail: quantummechanicskothari@gmail.com +MS received 1 January 2022; accepted 1 January 2022 +Abstract. The cosmological principle states that the Universe is statistically homogeneous and isotropic at large +distance scales. There currently exist many observations which indicate a departure from this principle. It has been +shown that many of these observations can be explained by invoking superhorizon cosmological perturbations and +may be consistent with the Big Bang paradigm. Remarkably, these modes simultaneously explain the observed +Hubble tension, i.e., the discrepancy between the direct and indirect measurements of the Hubble parameter. We +propose several tests of the cosmological principle using SKA. In particular, we can reliably extract the signal of +dipole anisotropy in the distribution of radio galaxies. The superhorizon perturbations also predict a significant +redshift dependence of the dipole signal which can be nicely tested by the study of signals of reionization and +the dark ages using SKA. We also propose to study the alignment of radio galaxy axes as well as their integrated +polarization vectors over distance scales ranging from a few Mpc to Gpc. We discuss data analysis techniques that +can reliably extract these signals from data. +Keywords. +cosmological principle—superhorizon perturbations—square kilometre array. +1. Introduction +Current observations support an expanding universe. If +we extrapolate this back in time, we can infer that the +Universe started from a very hot and dense state. This +event, known as Big Bang, marked the origin of the +Universe in a very high temperature state. +In order to make the problem of expansion dynam- +ics tractable, we assume that the Universe is spatially +isotropic and homogeneous. This assumption is also +known as Cosmological Principle (hereafter CP) (Kolb +& Turner, 1994; Einstein, 1917; Aluri et al., 2022). It +turns out that Hubble’s law is a direct consequence of +CP (Coles & Lucchin, 2003). Furthermore, it can be +shown that the most general spacetime metric that de- +scribes a universe following CP is the FLRW metric +(Weinberg, 1972; Coles & Lucchin, 2003). It is also im- +portant to mention that CP is an independent assump- +tion and does not follow from symmetries of the Ein- +stein’s Equations. +The FLRW metric describes a Universe with a +smooth background having an exact isotropic and ho- +mogeneous matter distribution. +But observationally, +the Universe also possesses structure in the form of +stars, galaxies, etc. These structures arise due to cur- +vature perturbations which are seeded during the epoch +of exponential expansion called inflation. The resulting +cosmological model, including dark matter and dark +energy is called ΛCDM. +Although, these perturbations aren’t isotropic and +homogeneous per se, they satisfy these properties in a +statistical sense. For example, in the cosmic frame of +rest, the matter density is expected to be the same at +all points provided we average over a sufficiently large +distance scale. The precise value of this distance scale +is still not clear but is expected to be of order 100 Mpc +(see, for example Kim et al. (2021)). +It has been speculated that during an epoch, be- +fore inflation ensued, the Universe may be described +by a complicated metric whose nature is currently +poorly understood. However, it quickly evolves to the +isotropic and homogeneous FLRW metric during infla- +© Indian Academy of Sciences +1 +arXiv:2301.03065v1 [astro-ph.CO] 8 Jan 2023 + +Page 2 of +J. Astrophys. Astr. (0000) 000: +tion, perhaps within the first e-fold. Wald (1983) for the +first time, gave an explicit demonstration for Bianchi +Universes (except type IX). Some other results also +exist for inhomogeneous metric (Stein-Schabes, 1987; +Jensen & Stein-Schabes, 1986). We may speculate that +the idea generalizes to a larger class of metrics1. The +Big Bang paradigm is therefore consistent with an early +anisotropic and/or inhomogeneous phase of the Uni- +verse. Given the existence of such a phase, it is clearly +important to ask whether it has any observational con- +sequences. +Observationally, the Universe is found to be consis- +tent with CP to a good approximation. But currently +there exist many observations in CMB and large scale +structures (LSS henceforth) which appear to violate CP +(Ghosh et al., 2016). We review these anomalies later +in §2. For an expansive review, see Aluri et al. (2022). +There exist many theoretical attempts to explain these +observations. It has been suggested that superhorizon +modes, i.e., perturbations of wavelengths larger than +the horizon size (Grishchuk & Zeldovich, 1978a,b), +may explain some of these observations (Gordon et al., +2005; Erickcek et al., 2008a,b; Ghosh, 2014; Das et al., +2021; Tiwari et al., 2022). Additionally, these can ac- +count for low-ℓ alignments (Gao, 2011), though these +can’t extenuate the present accelerated expansion of +the Universe (Hirata & Seljak, 2005; Flanagan, 2005). +It is assumed that such large wavelength modes are +aligned with one another and hence do not obey CP. +An intriguing possibility is that such modes might orig- +inate during an anisotropic and/or inhomogeneous pre- +inflationary phase of the Universe (Aluri & Jain, 2012; +Rath et al., 2013). Hence, despite being in violation +with CP, they would be consistent with the Big Bang +paradigm. +1.1 Mathematical Formulation and Ramifications +In order to relate theory with observations, we seek en- +semble averages of the fields under consideration. Er- +godicity hypothesis (Ellis et al., 2012) allows us to re- +late this ensemble averaging to the space averaging. It +is known that for the gaussian random fields, all the +statistical information is contained in the 2 point cor- +relation functions (2PCF). However, in the presence of +non-gaussianities, we need higher order correlators like +bispectrum (3PCF) or trispectrum (4PCF), etc., in or- +der to extract optimal cosmological information. CP +dictates that the nPCF only be a function of distances +between the points xi ≡ (zi, ni). Thus +�ρ(x1)ρ(x2) . . . ρ(xn)� = f(x12, x13, . . . , xi j, . . .), +(1) +1There are exceptions to these results as well (Sato, 1988). +A +O +C +B +Figure 1. Illustration of statistical isotropy. In this Figure, +A, B and C are given points on the spherical surface such +that ∠AOB = ∠AOC. +where xij = |xi − xj| = xji and i � j. Clearly, this +makes this nPCF invariant under arbitrary translations +and rotations. The condition (1) for 2PCF in case of a +2D field, e.g., CMB temperature, takes the usual form +�T(x1)T(x2)� ≡ �T( ˆm)T(ˆn)� = f( ˆm · ˆn), +(2) +with x1 ≡ (z∗, ˆn) and x2 ≡ (z∗, ˆm), z∗ being the red- +shift to decoupling. Eq. (2) is the familiar result for +the 2PCF, which dictates that the temperature correla- +tion depends only upon the angle between the locations. +This is illustrated in Figure 1, where three points A, B +and C are chosen in a manner such that ∠AOC = ∠AOB. +Thus we must have �T(A)T(C)� = �T(A)T(B)�, since +A · C = A · B. +2. Observations at tension with ΛCDM +Our observations in the past two decades have firmly +planted the inflationary ΛCDM cosmology as the stan- +dard paradigm. A vast set of observables from CMB +to LSS broadly agree with ΛCDM predictions. Despite +the successes of ΛCDM, we have a growing set of ob- +servations that are at tension with our expectations from +ΛCDM. We will summarise some of the observed ten- +sions, in the context of the model discussed in this pa- +per. See Perivolaropoulos & Skara (2022) for a review. +2.1 Observed violations of Statistical Isotropy +As we discussed before, CP implies statistical isotropy +and homogeneity. Due to our fixed vantage point, it +is not possible to directly test statistical homogene- +ity. However, we can test statistical isotropy. Various + +J. Astrophys. Astr. (0000)000: +Page 3 of +observational tests, performed on different cosmologi- +cal datasets, amply attest statistical isotropy violations. +Some of these are reviewed in Ghosh et al. (2016). +2.1.1 The kinematic dipole: +As explained in the §1., +CP is valid only in the cosmic frame of rest. We as ob- +servers are not stationary with respect to this frame on +account of the motion of Earth, the Sun, and the Milky +Way. This gives rise to an effective peculiar velocity to +our observation frame. This peculiar velocity results in +a Doppler boost of the CMB temperature fluctuation, +further culminating in a kinematic dipole in the CMB +temperature fluctuations. Interpreting the CMB dipole +to be of kinematic origin (Planck Collaboration et al., +2014) leads to the peculiar velocity of our local frame +to be 384 ± 78 km s−1. +The peculiar velocity v of our observation is ex- +pected to give rise to a dipole in the observed num- +ber count of sources. The local motion would cause +a Doppler and aberration effects, both of which con- +tribute to a dipole in the observed number counts (Ellis +& Baldwin, 1984). For sources with flux following a +power law relation in frequency: S ∝ ν−α, and with dif- +ferential number count N(S, ˆn) = S −1−x, the expected +dipole is given by: +D = [2 + x(1 + α)] v/c, +(3) +where c is the speed of light, α is the frequency scal- +ing spectral index, and (1 + x) is the slope of ln N v/s +− ln S plot. We can use the estimates of our peculiar +velocity from the CMB and use it to predict the esti- +mated dipole in the large-scale structure data. Assum- +ing α ≈ 0.75 and x ≈ 1, we find the expected dipole +Dth ∼ 0.005. Measurements of the dipole in LSS sur- +veys at z ∼ 1 have all yielded results that are consis- +tent with CMB direction but the magnitude is found +to be double or more of the predicted value. In Table +1, we list the measured value of dipole in the NVSS, +NVSS+WENSS, NVSS+SUMSS and CatWISE cata- +logs. The dipole measured in the LSS has a much larger +magnitude than expected from CMB measurements but +is consistent with the CMB dipole direction. The devia- +tion is found to be at 4.9σ in the CatWISE data (Secrest +et al., 2021). +We point out that the assumed power law depen- +dence of number counts on S is not strictly valid (Ti- +wari et al., 2015). +This leads to a difference in the +dipole in number counts and in sky brightness. It also +introduces a dipole in the mean flux per source. Hence +this provides a nontrivial test of whether the dipole is +indeed of kinematic origin. This idea has been gener- +alised in Nadolny et al. (2021) who develop a method +to extract kinematic dipole independently from an in- +trinsic dipole. +Authors +|D| (×10−2) +(l, b) +Singal (2011) +1.8 ± 0.3 +(239◦, 44◦) +Rubart & Schwarz (2013) +1.6 ± 0.6 +(241◦, 39◦) +Tiwari et al. (2015) +1.25 ± 0.40 +(261◦, 37◦) +Tiwari & Nusser (2016) +0.9 ± 0.4 +(246◦, 38◦) +Colin et al. (2017) +1.6 ± 0.2 +(241◦, 28◦) +Secrest et al. (2021) +1.5 +(238◦, 29◦) +Table 1. Results for the dipole in LSS exceed the expected +value of 5 × 10−3. +2.1.2 Alignment of quadrupole (ℓ = 2) and octupole +(ℓ = 3): +Both ℓ = 2, 3 CMB multipoles are aligned +with preferred direction pointing roughly along the +CMB dipole (de Oliveira-Costa et al., 2004). Physi- +cally, both of these multipoles form a planar structure, +such that the perpendicular to this plane is aligned with +the CMB dipole. +2.1.3 Alignment of galaxy axes and polarizations: +There have been many observations, both in optical +(Hutsem´ekers, 1998) and radio (Tiwari & Jain, 2013; +Taylor & Jagannathan, 2016) data sets that suggest +alignment of galaxy axes and integrated linear polar- +izations. These observations can be nicely explained in +terms of the correlated magnetic field which may be of +primordial origin (Tiwari & Jain, 2016). Intriguingly, +the optical alignment is seen to be very prominent in the +direction of the CMB dipole (Ralston & Jain, 2004). +2.1.4 Dipole in radio polarization offset angles: +The +integrated polarizations of radio galaxies are known to +be aligned approximately perpendicular to the galaxy +position axes. Remarkably, the angle between these +two axes shows a dipole pattern in the sky with pre- +ferred axis again pointing roughly along the CMB +dipole (Jain & Ralston, 1999). Hence, we see that sev- +eral diverse observations appear to indicate the same +preferred direction. Taken together, they are strongly +suggestive of a violation of the CP (Ralston & Jain, +2004). +2.1.5 Dipole modulation and the Hemispherical Asym- +metry: +We find that the CMB temperature fluctua- +tions have slightly higher power in the southern ecliptic +hemisphere than the northern one. This is called the +hemispherical power asymmetry and was first observed +in the WMAP data (Hoftuft et al., 2009) and continues +to persist in the Planck measurements (Planck Collab- +oration et al., 2020). It is also observed that the CMB +temperature fluctuations appear to be modulated by a + +Page 4 of +J. Astrophys. Astr. (0000) 000: +dipole that points close to the south ecliptic pole. This +implies that the CMB temperature fluctuation along +line-of-sight direction ˆn is given by: +∆T(ˆn) = ∆Tiso [1 + Aλ · ˆn] , +(4) +where ∆Tiso satisfies CP, A is the amplitude of the +dipole and λ is the preferred direction. Current Planck +measurements (Planck Collaboration et al., 2020) give +A = 0.070+0.032 +−0.015 and λ = (221◦, −21◦) ± 31◦. Such a +dipole modulation would lead to difference in powers +in the two hemispheres along ˆλ. +2.1.6 Other CMB observations: +Other observations +of SI violations in the CMB are low in significance, +albeit they are present in both WMAP and Planck data. +For low-ℓ values, the even multipoles are anomalously +smaller than the odd multipole modes in power. This +is called the parity asymmetry. The largest asymme- +try are evidenced in the lowest multipoles, viz., ℓ ∈ +[2, 7]. +These low multipoles show an anomalously +small power, which is called the low power on large +scales in the CMB temperature fluctuations. +2.2 Hubble Tension +The Hubble tension is the disagreement in measured +value of the Hubble parameter H0 from different meth- +ods. +The local universe measurements of H0 using +the ‘distance ladder’ method with Cepheids and super- +novae type Ia (SNIa) or strong lensing systems dif- +fer from the measurements from the CMB assuming +ΛCDM. Other methods like tip of the red giant branch +(TRGB) (Reid et al., 2019) or gravitational wave events +(Gayathri et al., 2020; Mukherjee et al., 2020) have +measured value somewhere between the two. Broadly +speaking, H0 measurements from the local universe +is larger than the measurements from the CMB at +nearly 5σ significance (Anchordoqui & Perez Bergli- +affa, 2019). +The Cepheid-SNIa measurements use Cepheid +variables in host galaxies of SNIa, to calibrate the dis- +tance. +These calibrated type Ia supernovae are then +used to calibrate magnitude and redshift of a large +sample of SNIa. The full sample of SNIa probes the +Hubble flow and is used to directly infer the Hub- +ble parameter. Riess et al. (2019) estimate the value +H0 = 74.03 ± 1.42 kms−1Mpc−1. This agrees with the +Freedman et al. (2012) estimate of H0 = 74.3 ± 2.1 +kms−1Mpc−1. +The H0LiCOW team’s (Wong et al., +2020) recent measurement, using the time delay for +a system of six gravitationally lensed quasars, yields +H0 = 73.3+1.7 +−1.8 kms−1Mpc−1 that agrees very well with +Cepheid (Freedman et al., 2001) measurements. +In addition to the aforementioned ‘direct’ measure- +ments, CMB can also be used to infer the value of +the Hubble parameter ‘indirectly’. The CMB T and E +mode measurements are used to fit the ΛCDM model. +In its basic form, ΛCDM has only six parameters. The +Hubble parameter can be estimated indirectly from the +best fit. This indirect estimation of the H0 gives a value +lower than the direct measurements. Planck Collabora- +tion et al. (2018) gives H0 = 67.27 ± 0.60 kms−1Mpc−1 +using only T and E mode data. Estimates of H0 us- +ing other CMB experiments like ACT (Dunkley et al., +2011) and SPTpol (for ℓ < 1000) (Henning et al., 2018) +give consistent results with Planck. +3. Superhorizon perturbation model +It has been suggested that the superhorizon perturba- +tions can explain the observed violations of statisti- +cal isotropy. These are perturbations with wavelengths +larger than the particle horizon (Erickcek et al., 2008a). +Such modes necessarily exist in a cosmological model. +However, in order to explain the the observed viola- +tions of isotropy (Gordon et al., 2005) we also need +them to be aligned with one another. In Gordon et al. +(2005), such an alignment is attributed to a stochas- +tic phenomenon known as spontaneous breakdown of +isotropy. Alternatively, the alignment may be attributed +to an intrinsic violation of the cosmological principle. +A very interesting possibility is presented in Aluri & +Jain (2012) and Rath et al. (2013). It is argued that +during its very early phase, the Universe may not be +isotropic and homogeneous. As explained in §1., it ac- +quires this property during inflation (Wald, 1983). The +modes which originate during the early phase of in- +flation when the Universe deviates from isotropy and +homogeneity may not obey the cosmological principle +(Rath et al., 2013). +We postulate that these are the +aligned superhorizon modes. +3.1 Resolution of various anomalies +Cosmological implications of this phenomenon have +been obtained by assuming the existence of a single +adiabatic mode (Erickcek et al., 2008a,b; Ghosh, 2014; +Das et al., 2021; Tiwari et al., 2022). Working in the +conformal Newtonian Gauge, such a mode can be ex- +pressed as, +Ψp = ϱ sin(κx3 + ω) +(5) +Thus a superhorizon mode is characterised by its am- +plitude ϱ, wavenumber κ and phase factor ω � 0. In +Eq. (5), we have taken the mode to be aligned along +the x3 (or z) axis which we also assume to be the direc- +tion of CMB dipole. For a superhorizon mode, we have + +J. Astrophys. Astr. (0000)000: +Page 5 of +κ/H0 ≪ 1. +It has been shown that such a superhorizon mode is +consistent with all existing cosmological observations +(like CMB, NVSS constraints etc.) for a range of pa- +rameters (Ghosh, 2014; Das et al., 2021; Tiwari et al., +2022). Some parameter values are given in Table 2. +It can affect the large scale distribution of matter and +can potentially explain the enigmatic excess dipole sig- +nal observed in the radio galaxy distribution (Singal, +2011; Gibelyou & Huterer, 2012; Rubart & Schwarz, +2013; Tiwari et al., 2015; Tiwari & Jain, 2015; Tiwari +& Nusser, 2016; Colin et al., 2017). +The observed matter dipole, Dobs is expressed as: +Dobs = �Dkin + Dgrav + Dint +�ˆx3, +(6) +where Dkin, Dgrav and Dint respectively denote the am- +plitudes of the kinematic, gravitational and intrinsic +dipoles. +These components are redshift dependent. +Thus we can write the magnitude of the observed dipole +between the redshifts z1 and z2, due to the superhorizon +mode (5) as +Dobs(z1, z2) = +� +A1(z1, z2) + A2(z1, z2) ++ C(z1, z2) +�ϱκ cos ω +H0 ++ B +(7) +where the term +B = [2 + x(1 + α)]v +c +(8) +is the redshift independent kinematic dipole compo- +nent. The explicit expressions for other redshift depen- +dent factors A1(z1, z2), A2(z1, z2), C(z1, z2) are given in +(Das et al., 2021). Notice that in the absence of a su- +perhorizon mode, i.e., ϱ → 0, the dipole magnitude in +Eq. (7), as expected, becomes redshift independent and +equal to (3). +3.1.1 The Matter Dipole: +Due to the presence of an +aligned superhorizon mode, an additional contribution +to our velocity arises with respect to LSS in the CMB +dipole direction. This is given in Eq. (2.12) of (Das +et al., 2021). Hence it leads to a change in Dkin in com- +parison to its prediction based on CMB dipole (Das +et al., 2021). +Furthermore, the superhorizon mode +contributes through the Sachs-Wolfe (SW) and the in- +tegrated Sachs-Wolfe (ISW) effects (Erickcek et al., +2008a), thereby leading to Dgrav in Eq. (6). Finally, +the superhorizon mode leads to an intrinsic anisotropy +in the matter distribution and hence contributes to Dint. +Eq. (7) is the explicit expression considering all these +effects. From the equation, it is clear that for a given +value of ϱ > 0, the dipole contribution is maximum if +ω = π. All the contributions due to the mode depend on +redshift since the mode has a systematic dependence on +distance and hence the predicted dipole is redshift de- +pendent. +It is interesting to note that the contributions of the +superhorizon mode to the CMB dipole cancel out at +the leading order (Erickcek et al., 2008a). Such a can- +cellation does not happen in the case of matter dipole +(Das et al., 2021). We may understand this as follows. +As per Eq. (6), there are three different contributions +to the matter dipole – (a) the kinematic dipole which +arises due to our velocity relative to the source, (b) the +gravitational dipole (SW and ISW) and (c) the intrin- +sic dipole. In the case of CMB, these three add up to +zero. In the case of matter dipole, the kinematic dipole +explicitly depends on the parameter α which arises in +the spectral dependence of the flux from a source, as +well as the parameter x (see Eq. (8)) which arises in +the number count distribution. Furthermore, the gravi- +tational effect also depends on α. The intrinsic dipole, +however, does not depend on either of these parame- +ters. We point out that both of these parameters arise +at non-linear order in the theory of structure formation +and furthermore the assumed power law distribution is +only an approximation (Tiwari et al., 2015). These pa- +rameters are best extracted from observations and can- +not be reliably deduced theoretically. Hence, the sit- +uation is very different in the case of matter dipole in +comparison to CMB dipole and we do not expect that +the two would behave in the same manner. We clar- +ify that in the case of matter dipole, the superhorizon +perturbation is treated at first order in perturbation the- +ory. However, the small wavelength modes which are +responsible for structure formation have to be treated at +nonlinear order. In Das et al. (2021), the existence of +structures is assumed as given with their properties de- +duced observationally and the calculation focuses only +on the additional contribution due to the superhorizon +mode. However, a complete first principles calculation +would have to treat small wavelength modes at nonlin- +ear order. +There are some further issues, associated with +gauge invariance (Challinor & Lewis, 2011; Bonvin & +Durrer, 2011), which are not addressed in Das et al. +(2021). These issues are very important, but to the best +of our understanding, they are expected to lead to small +corrections to the calculational framework used in Das +et al. (2021) and not expected to qualitatively change +their results. It will be very interesting to repeat these +calculations using the gauge invariant framework, but +this is beyond the scope of the present paper. Such a +calculation must also take into account the fact that the +aligned superhorizon modes we are considering do not +arise within the ΛCDM model but perhaps due to an + +Page 6 of +J. Astrophys. Astr. (0000) 000: +anisotropic/inhomogeneous early phase of cosmic ex- +pansion (Aluri & Jain, 2012; Rath et al., 2013). +3.1.2 Alignment of Quadrupole and Octupole: +Fur- +ther, the superhorizon mode can also explain the align- +ment of CMB quadrupole and octupole (Gordon et al., +2005). With x3 axis along the CMB dipole, it leads +to non-zero spherical harmonic coefficients T10, T20, +T30 in the temperature anisotropy field (Erickcek et al., +2008a). +We obtain constraints on the mode param- +eters in Eq. +(5) by requiring that T20 and T30 are +less than three times the measured rms values of the +quadrupole and octupole powers respectively (Erickcek +et al., 2008a). It turns out that the dipole contribution +does not lead to a significant constraint. These con- +tributions can explain the alignment of quadrupole and +octupole if we assume the presence of an intrinsic con- +tribution to T10 and T20 which is partially cancelled by +the contribution due to the superhorizon mode. Note +that this intrinsic contribution is statistical in nature and +hence its exact value cannot be predicted. +3.1.3 Hubble Tension: +It has been shown by Das +et al. (2021) that a superhorizon mode leads to a per- +turbation in the gravitational potential between distant +galaxies and us. This culminates in a correction in ob- +served redshift of galaxies. +1 + zobs = (1 + z)(1 + zDoppler)(1 + zgrav) +(9) +Thus we see that in the presence of superhorizon +modes, the galaxy at redshift z is observed instead at a +redshift zobs. In the above equation, the redshifts zDoppler +and zgrav are respectively due to our velocity relative to +LSS and perturbation in potential introduced by the su- +perhorizon mode. We can express zobs as (Das et al., +2021; Tiwari et al., 2022), +zobs = ¯z + γ cos θ + . . . +(10) +where the first and the second terms on the RHS are the +monopole and dipole terms. Here θ is the polar angle +of the source with x3 axis along the CMB dipole and γ +the dipole amplitude. Interestingly, the monopole term +in Eq. (10) resolves the Hubble tension (Tiwari et al., +2022). For that we need to choose the phase ω � π. +The range of parameters which explain both the matter +dipole and the Hubble tension is given in (Tiwari et al., +2022). In Table 2, we quote some of those values. +The superhorizon modes are also likely to leave +their signatures in other cosmological observables like +Baryon Acoustic Oscillations, epoch of reionziation +etc. +ω +ϱ +κ/H0 +1 +0.81π +0.97 +2.58 × 10−3 +2 +0.81π +0.48 +6.4 × 10−3 +Table 2. Some parameter values for the superhorizon mode +(Eq. 5) explaining NVSS excess dipole and also resolving +the Hubble tension. +These values also satisfying CMB +constraints are taken from Tiwari et al. (2022). +4. Constraints using SKA +4.1 Superhorizon perturbation observation +The superhorizon model predicts several observations +which can be tested with SKA and other future surveys. +An interesting feature is the significant dependence of +dipole on the redshift in the presence of superhorizon +modes. +Hence, the dipole measurements in redshift +bins with SKA1 and SKA2 continuum survey can work +as a potential test of the model. For a radio contin- +uum survey, it is unlikely that we would have spec- +troscopic redshift information. +In the past, redshifts +of radio galaxies have been estimated by taking cross +correlation with well known redshift surveys (Blake & +Wall, 2002; Tiwari et al., 2015). Such strategies are +still viable by using data from the GAMA survey fields +(Baldry et al., 2018). However, new techniques like +template fitting (Duncan et al., 2018) or machine learn- +ing based photometric redshift computations (Brescia +et al., 2021) make it possible for the SKA radio con- +tinuum survey galaxies to contain redshift information. +This added information provides a unique possibility to +test superhorizon mode physics with the SKA. +4.2 Predictions +Here, we demonstrate how precisely evident the dipole +predictions with a superhorizon model would be. Fur- +ther, we demonstrate how much they are constrained +using SKA observations. +Assuming superhorizon +modes that (a) satisfy the present NVSS and Hubble pa- +rameter measurements and (b) are consistent with CMB +and other cosmological measurements (see Table 2); +we obtain the dipole magnitude Dobs in redshift bins +using the formalism described in (Tiwari et al., 2022). +The dependence of dipole signal on the redshift z is +shown in Figure 2 where we have shown the redshift +dependence of Dobs for two cases +1. Cumulative Redshift Bins: For this case, we fix +z1 = 0 in Eq. (7) and vary z2 = z. In other words, +we calculate Dobs(0, z). +2. Non-overlapping Redshift Bins: In this case, we +obtain the non overlapping z dependence by eval- + +J. Astrophys. Astr. (0000)000: +Page 7 of +1 +2 +3 +4 +5 +6 +z +0.010 +0.011 +0.012 +0.013 +0.014 +0.015 +0.016 +Dobs +cumulative redshift bins +non-overlapping redshift bins +Figure 2. The dipole signal observation in the presence of +superhorizon modes with SKA1 (z ≤ 5) and SKA2 (z ≤ 6) +continuum surveys. +The inner and outer shaded regions +respectively represent the optimistic and realistic uncer- +tainties for both SKA1 & SKA2. Here we have assumed a +superhorizon mode (Tiwari et al., 2022) satisfying present +NVSS dipole observation (Tiwari et al., 2015). +Flux Density +SKA1 +SKA2 +Optimistic +> 10 +> 1 +Realistic +> 20 +> 5 +Table 3. Optimistic and realistic flux densities (in µJy) for +SKA1 and SKA2 surveys. +uating Dobs(z − ∆z, z + ∆z) with ∆z = 0.25. For a +fix ∆z, this thus gives Dobs at z. +4.3 Estimating Uncertainties +We employ Alonso et al. (2015) ‘Ultra-large scales’ +codes2 (for continuum surveys) to determine the num- +ber densities for SKA surveys. We further assume that +SKA1 and SKA2 will observe the sky up to respective +declinations of 15◦ and 30◦. The optimistic and realis- +tic flux densities’ limits for SKA1 and SKA2 (Square +Kilometre Array Cosmology Science Working Group +et al., 2020; Bengaly et al., 2018) are given in Table 3. +Additionally, we note that SKA1 is expected to +probe up to 0 ≤ z ≤ 5, whereas the SKA2 will reach +up to redshift 6. We mock SKA1 and SKA2 contin- +uum sky to determine the observational implications +of the superhorizon model. +We produce 1000 num- +ber density simulation of SKA1 and SKA2 continuum +survey for each (optimistic and realistic) flux thresh- +old using HEALPix software (Go´rski et al., 2005), +with Nside = 64. +The mean number of galaxies in +2http://intensitymapping.physics.ox.ac.uk/codes.html +a pixel is determined using number density obtained +from Alonso et al. (2015) code and by modelling a +dipole with magnitude and direction expected in pres- +ence of a superhorizon mode. Given the mean number +density in a pixel, we call random Poisson distribution +to emulate the galaxy count in the pixel. The galaxy +mock thus neglects the cosmological galaxy clustering. +This is justified since the clustering dipole in LSS is +≈ 2.7×10−3 (Nusser & Tiwari, 2015; Tiwari & Nusser, +2016), which is roughly five times less than the appar- +ent dipole in LSS3 and inconsequential for our simula- +tions. This is roughly equal to the uncertainties in the +measured dipole is LSS using NVSS galaxies (see Ta- +ble 1). We consider SKA1, SKA2 sky coverage, i.e., +mask declination above 15◦ and 30◦, respectively. Ad- +ditionally, we mask the galactic latitudes (|b| < 10◦) in +order to remove Milky Way contamination. The galac- +tic plane cut is often chosen to be anywhere between +|b| < 5◦ and |b| < 15◦, and in most studies one tests +the robustness of the results with varying redshift cuts. +For tests on mock data, here we choose a typical cut +of |b| < 10◦ that should balance the exclusion of galac- +tic plane contamination and loss of sky fraction. Next, +we use Python Healpy4 (Go´rski et al., 2005; Zonca +et al., 2019) fit dipole function and obtain dipole +for each 1000 mock maps. From these 1000 dipole val- +ues, we calculate the standard deviation to determine +the uncertainty in measurements. The shaded regions +in Figure 2 show the results obtained for SKA1 and +SKA2 optimistic and realistic number densities in non- +overlapping and cumulative redshift bins. +4.4 Other Anisotropy tests with SKA +If the universe does not follow CP at large distance +scales then every observable should have directional +dependence characteristics. Out of all, three observ- +ables are of particular interest as these are independent +of the number density over the sky. So these are more +robust under unequal coverage and systematics of the +sky. These observables are +• Mean spectral index (¯α) – As we said in §2.1.1, +the spectral index for a radio source is defined +between flux density and frequency through S ∝ +ν−α. In the Healpy pixelation scheme, the sky is +divided into equal area pixels. For a given pixel p +with Np sources having spectral indices αi,p, we +3These estimates correspond to NVSS galaxies. +Assuming the +NVSS measured dipole in LSS is true, we expect similar numbers +from SKA continuum surveys. +4https://healpy.readthedocs.io/en/latest/index.html + +Page 8 of +J. Astrophys. Astr. (0000) 000: +define mean spectral index +¯αp = 1 +Np +� +i +αi,p +(11) +here i runs over all the sources in pixel p +• Exponent (x) of differential number count – De- +fined using N(S, ˆn) ∝ S −1−x +• Average Flux Density ( ¯S ) – We define this quan- +tity for a pixel p +¯S p = 1 +Np +� +i +S i,p +(12) +where again i runs over all the sources in p and +Np is the number of sources in the pixel +The spectral index characterises the morphology of +an astronomical source. Angular dependence of ¯α has +not been much looked at in the literature. Analysing +the dipole anisotropy in ¯α has been a challenge since it +requires reliable multi-frequency continuum radio sky +survey. Such an SKA survey can be used to estimate +this anisotropy if the flux density of the sources at dif- +ferent frequencies is measured with sufficient accuracy. +For x, the angular dependency was analysed by (Ghosh +& Jain, 2017) in NVSS data using likelihood maximisa- +tion and the results were found to be consistent with CP. +However, with a larger expected source count of SKA +that is almost twice in comparison to NVSS, angular +dependence analysis of x may provide a more stringent +test of CP. +SKA will also be able to test the phenomenon of +alignment of radio galaxy axes and integrated polariza- +tions, as claimed in earlier radio observations (Tiwari +& Jain, 2013, 2016; Taylor & Jagannathan, 2016). +5. Conclusion and Outlook +In this paper, we have reviewed several cosmological +signals which appear to show a violation of CP. We +have also reviewed a model, based on aligned super- +horizon modes which can explain some of these obser- +vations along with the Hubble tension (Tiwari et al., +2022). The model can be theoretically justified by pos- +tulating a pre-inflationary phase during which the Uni- +verse may not be homogeneous and isotropic (Rath +et al., 2013). The model leads to several cosmolog- +ical predictions which can be tested at SKA. By us- +ing the best fit parameters with current observations, +we have determined the redshift dependence of the pre- +dicted dipole in radio galaxy number counts and associ- +ated uncertainties. As can be seen from Figure 2, SKA +can test this very reliably. If this prediction is confirmed +by SKA, it may provide us with a first glimpse into the +Physics of the pre-inflationary phase of the Universe. +We have also suggested other isotropy tests with +SKA using other variables which are independent of +number density and thus are more robust under unequal +coverage and systematics of the sky. These variables +are (a) mean spectral index ¯α, (b) exponent of the dif- +ferential number count x & (c) average flux density ¯S . +Acknowledgements +RK is supported by the South African Radio Astron- +omy Observatory and the National Research Founda- +tion (Grant No. 75415). PT acknowledges the support +of the RFIS grant (No. 12150410322) by the National +Natural Science Foundation of China (NSFC) and the +support by the National Key Basic Research and De- +velopment Program of China (No. 2018YFA0404503) +and NSFC Grants 11925303 and 11720101004. +SG +is supported in part by the National Key R & D Pro- +gram of China (2021YFC2203100), by the Fundamen- +tal Research Funds for the Central Universities under +grant no: WK2030000036, and the NSFC grant no: +11903030. We are also very thankful to the anonymous +referee whose comments were really helpful in improv- +ing the presentation of the paper. +References +Alonso, D., Bull, P., Ferreira, P. G., Maartens, R., & +Santos, M. 2015, Astrophys. J., 814, 145 +Aluri, P. K., & Jain, P. 2012, Modern Physics Letters +A, 27, 50014 +Aluri, P. K., et al. 2022, arXiv:2207.05765 +Anchordoqui, L. A., & Perez Bergliaffa, S. E. 2019, +Phys. Rev. D, 100, 123525 +Baldry, I. K., Liske, J., Brown, M. J. 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F., et al. 2020, +MNRAS, 498, 1420 +Zonca, A., Singer, L., Lenz, D., et al. 2019, Journal of +Open Source Software, 4, 1298 + diff --git a/3tE1T4oBgHgl3EQfSQPo/content/tmp_files/load_file.txt b/3tE1T4oBgHgl3EQfSQPo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..21c918552a26d7f0c6d42c6e3919677b7b6c53ff --- /dev/null +++ b/3tE1T4oBgHgl3EQfSQPo/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf,len=889 +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000) 000: DOI Probing Cosmology beyond ΛCDM using the SKA Shamik Ghosh1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Pankaj Jain3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Rahul Kothari4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Mohit Panwar3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Gurmeet Singh3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Prabhakar Tiwari5 1CAS Key Laboratory for Researches in Galaxies and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' China 2School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Indian Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Kanpur-208016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 5National Astronomical Observatories, Chinese Academy of Science, Beijing 100101, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' E-mail: quantummechanicskothari@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='com MS received 1 January 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' accepted 1 January 2022 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The cosmological principle states that the Universe is statistically homogeneous and isotropic at large distance scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' There currently exist many observations which indicate a departure from this principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It has been shown that many of these observations can be explained by invoking superhorizon cosmological perturbations and may be consistent with the Big Bang paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Remarkably, these modes simultaneously explain the observed Hubble tension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', the discrepancy between the direct and indirect measurements of the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We propose several tests of the cosmological principle using SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In particular, we can reliably extract the signal of dipole anisotropy in the distribution of radio galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The superhorizon perturbations also predict a significant redshift dependence of the dipole signal which can be nicely tested by the study of signals of reionization and the dark ages using SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We also propose to study the alignment of radio galaxy axes as well as their integrated polarization vectors over distance scales ranging from a few Mpc to Gpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We discuss data analysis techniques that can reliably extract these signals from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' cosmological principle—superhorizon perturbations—square kilometre array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Introduction Current observations support an expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' If we extrapolate this back in time, we can infer that the Universe started from a very hot and dense state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This event, known as Big Bang, marked the origin of the Universe in a very high temperature state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In order to make the problem of expansion dynam- ics tractable, we assume that the Universe is spatially isotropic and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This assumption is also known as Cosmological Principle (hereafter CP) (Kolb & Turner, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Einstein, 1917;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Aluri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It turns out that Hubble’s law is a direct consequence of CP (Coles & Lucchin, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Furthermore, it can be shown that the most general spacetime metric that de- scribes a universe following CP is the FLRW metric (Weinberg, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Coles & Lucchin, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is also im- portant to mention that CP is an independent assump- tion and does not follow from symmetries of the Ein- stein’s Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The FLRW metric describes a Universe with a smooth background having an exact isotropic and ho- mogeneous matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' But observationally, the Universe also possesses structure in the form of stars, galaxies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These structures arise due to cur- vature perturbations which are seeded during the epoch of exponential expansion called inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The resulting cosmological model, including dark matter and dark energy is called ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Although, these perturbations aren’t isotropic and homogeneous per se, they satisfy these properties in a statistical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For example, in the cosmic frame of rest, the matter density is expected to be the same at all points provided we average over a sufficiently large distance scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The precise value of this distance scale is still not clear but is expected to be of order 100 Mpc (see, for example Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It has been speculated that during an epoch, be- fore inflation ensued, the Universe may be described by a complicated metric whose nature is currently poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, it quickly evolves to the isotropic and homogeneous FLRW metric during infla- © Indian Academy of Sciences 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='03065v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='CO] 8 Jan 2023 Page 2 of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000) 000: tion, perhaps within the first e-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Wald (1983) for the first time, gave an explicit demonstration for Bianchi Universes (except type IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Some other results also exist for inhomogeneous metric (Stein-Schabes, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Jensen & Stein-Schabes, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We may speculate that the idea generalizes to a larger class of metrics1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The Big Bang paradigm is therefore consistent with an early anisotropic and/or inhomogeneous phase of the Uni- verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Given the existence of such a phase, it is clearly important to ask whether it has any observational con- sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Observationally, the Universe is found to be consis- tent with CP to a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' But currently there exist many observations in CMB and large scale structures (LSS henceforth) which appear to violate CP (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We review these anomalies later in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For an expansive review, see Aluri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' There exist many theoretical attempts to explain these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It has been suggested that superhorizon modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', perturbations of wavelengths larger than the horizon size (Grishchuk & Zeldovich, 1978a,b), may explain some of these observations (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Ghosh, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Additionally, these can ac- count for low-ℓ alignments (Gao, 2011), though these can’t extenuate the present accelerated expansion of the Universe (Hirata & Seljak, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Flanagan, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is assumed that such large wavelength modes are aligned with one another and hence do not obey CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' An intriguing possibility is that such modes might orig- inate during an anisotropic and/or inhomogeneous pre- inflationary phase of the Universe (Aluri & Jain, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Rath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence, despite being in violation with CP, they would be consistent with the Big Bang paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 Mathematical Formulation and Ramifications In order to relate theory with observations, we seek en- semble averages of the fields under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Er- godicity hypothesis (Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2012) allows us to re- late this ensemble averaging to the space averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is known that for the gaussian random fields, all the statistical information is contained in the 2 point cor- relation functions (2PCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, in the presence of non-gaussianities, we need higher order correlators like bispectrum (3PCF) or trispectrum (4PCF), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', in or- der to extract optimal cosmological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' CP dictates that the nPCF only be a function of distances between the points xi ≡ (zi, ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Thus �ρ(x1)ρ(x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' ρ(xn)� = f(x12, x13, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' , xi j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' ), (1) 1There are exceptions to these results as well (Sato, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' A O C B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Illustration of statistical isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In this Figure, A, B and C are given points on the spherical surface such that ∠AOB = ∠AOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' where xij = |xi − xj| = xji and i � j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Clearly, this makes this nPCF invariant under arbitrary translations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The condition (1) for 2PCF in case of a 2D field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', CMB temperature, takes the usual form �T(x1)T(x2)� ≡ �T( ˆm)T(ˆn)� = f( ˆm · ˆn), (2) with x1 ≡ (z∗, ˆn) and x2 ≡ (z∗, ˆm), z∗ being the red- shift to decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2) is the familiar result for the 2PCF, which dictates that the temperature correla- tion depends only upon the angle between the locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is illustrated in Figure 1, where three points A, B and C are chosen in a manner such that ∠AOC = ∠AOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Thus we must have �T(A)T(C)� = �T(A)T(B)�, since A · C = A · B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Observations at tension with ΛCDM Our observations in the past two decades have firmly planted the inflationary ΛCDM cosmology as the stan- dard paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' A vast set of observables from CMB to LSS broadly agree with ΛCDM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Despite the successes of ΛCDM, we have a growing set of ob- servations that are at tension with our expectations from ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We will summarise some of the observed ten- sions, in the context of the model discussed in this pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' See Perivolaropoulos & Skara (2022) for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 Observed violations of Statistical Isotropy As we discussed before, CP implies statistical isotropy and homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Due to our fixed vantage point, it is not possible to directly test statistical homogene- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, we can test statistical isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Various J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000)000: Page 3 of observational tests, performed on different cosmologi- cal datasets, amply attest statistical isotropy violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Some of these are reviewed in Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 The kinematic dipole: As explained in the §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', CP is valid only in the cosmic frame of rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We as ob- servers are not stationary with respect to this frame on account of the motion of Earth, the Sun, and the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This gives rise to an effective peculiar velocity to our observation frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This peculiar velocity results in a Doppler boost of the CMB temperature fluctuation, further culminating in a kinematic dipole in the CMB temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Interpreting the CMB dipole to be of kinematic origin (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2014) leads to the peculiar velocity of our local frame to be 384 ± 78 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The peculiar velocity v of our observation is ex- pected to give rise to a dipole in the observed num- ber count of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The local motion would cause a Doppler and aberration effects, both of which con- tribute to a dipole in the observed number counts (Ellis & Baldwin, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For sources with flux following a power law relation in frequency: S ∝ ν−α, and with dif- ferential number count N(S, ˆn) = S −1−x, the expected dipole is given by: D = [2 + x(1 + α)] v/c, (3) where c is the speed of light, α is the frequency scal- ing spectral index, and (1 + x) is the slope of ln N v/s − ln S plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We can use the estimates of our peculiar velocity from the CMB and use it to predict the esti- mated dipole in the large-scale structure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Assum- ing α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='75 and x ≈ 1, we find the expected dipole Dth ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Measurements of the dipole in LSS sur- veys at z ∼ 1 have all yielded results that are consis- tent with CMB direction but the magnitude is found to be double or more of the predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In Table 1, we list the measured value of dipole in the NVSS, NVSS+WENSS, NVSS+SUMSS and CatWISE cata- logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The dipole measured in the LSS has a much larger magnitude than expected from CMB measurements but is consistent with the CMB dipole direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The devia- tion is found to be at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='9σ in the CatWISE data (Secrest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We point out that the assumed power law depen- dence of number counts on S is not strictly valid (Ti- wari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This leads to a difference in the dipole in number counts and in sky brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It also introduces a dipole in the mean flux per source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence this provides a nontrivial test of whether the dipole is indeed of kinematic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This idea has been gener- alised in Nadolny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021) who develop a method to extract kinematic dipole independently from an in- trinsic dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Authors |D| (×10−2) (l, b) Singal (2011) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3 (239◦, 44◦) Rubart & Schwarz (2013) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='6 (241◦, 39◦) Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2015) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='40 (261◦, 37◦) Tiwari & Nusser (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='4 (246◦, 38◦) Colin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2017) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2 (241◦, 28◦) Secrest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='5 (238◦, 29◦) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Results for the dipole in LSS exceed the expected value of 5 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2 Alignment of quadrupole (ℓ = 2) and octupole (ℓ = 3): Both ℓ = 2, 3 CMB multipoles are aligned with preferred direction pointing roughly along the CMB dipole (de Oliveira-Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Physi- cally, both of these multipoles form a planar structure, such that the perpendicular to this plane is aligned with the CMB dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3 Alignment of galaxy axes and polarizations: There have been many observations, both in optical (Hutsem´ekers, 1998) and radio (Tiwari & Jain, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Taylor & Jagannathan, 2016) data sets that suggest alignment of galaxy axes and integrated linear polar- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These observations can be nicely explained in terms of the correlated magnetic field which may be of primordial origin (Tiwari & Jain, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Intriguingly, the optical alignment is seen to be very prominent in the direction of the CMB dipole (Ralston & Jain, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='4 Dipole in radio polarization offset angles: The integrated polarizations of radio galaxies are known to be aligned approximately perpendicular to the galaxy position axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Remarkably, the angle between these two axes shows a dipole pattern in the sky with pre- ferred axis again pointing roughly along the CMB dipole (Jain & Ralston, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence, we see that sev- eral diverse observations appear to indicate the same preferred direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Taken together, they are strongly suggestive of a violation of the CP (Ralston & Jain, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='5 Dipole modulation and the Hemispherical Asym- metry: We find that the CMB temperature fluctua- tions have slightly higher power in the southern ecliptic hemisphere than the northern one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is called the hemispherical power asymmetry and was first observed in the WMAP data (Hoftuft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2009) and continues to persist in the Planck measurements (Planck Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is also observed that the CMB temperature fluctuations appear to be modulated by a Page 4 of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000) 000: dipole that points close to the south ecliptic pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This implies that the CMB temperature fluctuation along line-of-sight direction ˆn is given by: ∆T(ˆn) = ∆Tiso [1 + Aλ · ˆn] , (4) where ∆Tiso satisfies CP, A is the amplitude of the dipole and λ is the preferred direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Current Planck measurements (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020) give A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='070+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='015 and λ = (221◦, −21◦) ± 31◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such a dipole modulation would lead to difference in powers in the two hemispheres along ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='6 Other CMB observations: Other observations of SI violations in the CMB are low in significance, albeit they are present in both WMAP and Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For low-ℓ values, the even multipoles are anomalously smaller than the odd multipole modes in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is called the parity asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The largest asymme- try are evidenced in the lowest multipoles, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', ℓ ∈ [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These low multipoles show an anomalously small power, which is called the low power on large scales in the CMB temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2 Hubble Tension The Hubble tension is the disagreement in measured value of the Hubble parameter H0 from different meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The local universe measurements of H0 using the ‘distance ladder’ method with Cepheids and super- novae type Ia (SNIa) or strong lensing systems dif- fer from the measurements from the CMB assuming ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Other methods like tip of the red giant branch (TRGB) (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2019) or gravitational wave events (Gayathri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020) have measured value somewhere between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Broadly speaking, H0 measurements from the local universe is larger than the measurements from the CMB at nearly 5σ significance (Anchordoqui & Perez Bergli- affa, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The Cepheid-SNIa measurements use Cepheid variables in host galaxies of SNIa, to calibrate the dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These calibrated type Ia supernovae are then used to calibrate magnitude and redshift of a large sample of SNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The full sample of SNIa probes the Hubble flow and is used to directly infer the Hub- ble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2019) estimate the value H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='42 kms−1Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This agrees with the Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2012) estimate of H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 kms−1Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The H0LiCOW team’s (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020) recent measurement, using the time delay for a system of six gravitationally lensed quasars, yields H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='8 kms−1Mpc−1 that agrees very well with Cepheid (Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2001) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In addition to the aforementioned ‘direct’ measure- ments, CMB can also be used to infer the value of the Hubble parameter ‘indirectly’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The CMB T and E mode measurements are used to fit the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In its basic form, ΛCDM has only six parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The Hubble parameter can be estimated indirectly from the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This indirect estimation of the H0 gives a value lower than the direct measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2018) gives H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='60 kms−1Mpc−1 using only T and E mode data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Estimates of H0 us- ing other CMB experiments like ACT (Dunkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2011) and SPTpol (for ℓ < 1000) (Henning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2018) give consistent results with Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Superhorizon perturbation model It has been suggested that the superhorizon perturba- tions can explain the observed violations of statisti- cal isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These are perturbations with wavelengths larger than the particle horizon (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such modes necessarily exist in a cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, in order to explain the the observed viola- tions of isotropy (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2005) we also need them to be aligned with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2005), such an alignment is attributed to a stochas- tic phenomenon known as spontaneous breakdown of isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Alternatively, the alignment may be attributed to an intrinsic violation of the cosmological principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' A very interesting possibility is presented in Aluri & Jain (2012) and Rath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is argued that during its very early phase, the Universe may not be isotropic and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' As explained in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', it ac- quires this property during inflation (Wald, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The modes which originate during the early phase of in- flation when the Universe deviates from isotropy and homogeneity may not obey the cosmological principle (Rath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We postulate that these are the aligned superhorizon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 Resolution of various anomalies Cosmological implications of this phenomenon have been obtained by assuming the existence of a single adiabatic mode (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Ghosh, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Working in the conformal Newtonian Gauge, such a mode can be ex- pressed as, Ψp = ϱ sin(κx3 + ω) (5) Thus a superhorizon mode is characterised by its am- plitude ϱ, wavenumber κ and phase factor ω � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (5), we have taken the mode to be aligned along the x3 (or z) axis which we also assume to be the direc- tion of CMB dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For a superhorizon mode, we have J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000)000: Page 5 of κ/H0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It has been shown that such a superhorizon mode is consistent with all existing cosmological observations (like CMB, NVSS constraints etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=') for a range of pa- rameters (Ghosh, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Some parameter values are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It can affect the large scale distribution of matter and can potentially explain the enigmatic excess dipole sig- nal observed in the radio galaxy distribution (Singal, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Gibelyou & Huterer, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Rubart & Schwarz, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari & Jain, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari & Nusser, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Colin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The observed matter dipole, Dobs is expressed as: Dobs = �Dkin + Dgrav + Dint �ˆx3, (6) where Dkin, Dgrav and Dint respectively denote the am- plitudes of the kinematic, gravitational and intrinsic dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These components are redshift dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Thus we can write the magnitude of the observed dipole between the redshifts z1 and z2, due to the superhorizon mode (5) as Dobs(z1, z2) = � A1(z1, z2) + A2(z1, z2) + C(z1, z2) �ϱκ cos ω H0 + B (7) where the term B = [2 + x(1 + α)]v c (8) is the redshift independent kinematic dipole compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The explicit expressions for other redshift depen- dent factors A1(z1, z2), A2(z1, z2), C(z1, z2) are given in (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Notice that in the absence of a su- perhorizon mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', ϱ → 0, the dipole magnitude in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (7), as expected, becomes redshift independent and equal to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 The Matter Dipole: Due to the presence of an aligned superhorizon mode, an additional contribution to our velocity arises with respect to LSS in the CMB dipole direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='12) of (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence it leads to a change in Dkin in com- parison to its prediction based on CMB dipole (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Furthermore, the superhorizon mode contributes through the Sachs-Wolfe (SW) and the in- tegrated Sachs-Wolfe (ISW) effects (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a), thereby leading to Dgrav in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Finally, the superhorizon mode leads to an intrinsic anisotropy in the matter distribution and hence contributes to Dint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (7) is the explicit expression considering all these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' From the equation, it is clear that for a given value of ϱ > 0, the dipole contribution is maximum if ω = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' All the contributions due to the mode depend on redshift since the mode has a systematic dependence on distance and hence the predicted dipole is redshift de- pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It is interesting to note that the contributions of the superhorizon mode to the CMB dipole cancel out at the leading order (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such a can- cellation does not happen in the case of matter dipole (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We may understand this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' As per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (6), there are three different contributions to the matter dipole – (a) the kinematic dipole which arises due to our velocity relative to the source, (b) the gravitational dipole (SW and ISW) and (c) the intrin- sic dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In the case of CMB, these three add up to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In the case of matter dipole, the kinematic dipole explicitly depends on the parameter α which arises in the spectral dependence of the flux from a source, as well as the parameter x (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (8)) which arises in the number count distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Furthermore, the gravi- tational effect also depends on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The intrinsic dipole, however, does not depend on either of these parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We point out that both of these parameters arise at non-linear order in the theory of structure formation and furthermore the assumed power law distribution is only an approximation (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These pa- rameters are best extracted from observations and can- not be reliably deduced theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence, the sit- uation is very different in the case of matter dipole in comparison to CMB dipole and we do not expect that the two would behave in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We clar- ify that in the case of matter dipole, the superhorizon perturbation is treated at first order in perturbation the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, the small wavelength modes which are responsible for structure formation have to be treated at nonlinear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021), the existence of structures is assumed as given with their properties de- duced observationally and the calculation focuses only on the additional contribution due to the superhorizon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, a complete first principles calculation would have to treat small wavelength modes at nonlin- ear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' There are some further issues, associated with gauge invariance (Challinor & Lewis, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Bonvin & Durrer, 2011), which are not addressed in Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These issues are very important, but to the best of our understanding, they are expected to lead to small corrections to the calculational framework used in Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021) and not expected to qualitatively change their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It will be very interesting to repeat these calculations using the gauge invariant framework, but this is beyond the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such a calculation must also take into account the fact that the aligned superhorizon modes we are considering do not arise within the ΛCDM model but perhaps due to an Page 6 of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000) 000: anisotropic/inhomogeneous early phase of cosmic ex- pansion (Aluri & Jain, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Rath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2 Alignment of Quadrupole and Octupole: Fur- ther, the superhorizon mode can also explain the align- ment of CMB quadrupole and octupole (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' With x3 axis along the CMB dipole, it leads to non-zero spherical harmonic coefficients T10, T20, T30 in the temperature anisotropy field (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We obtain constraints on the mode param- eters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (5) by requiring that T20 and T30 are less than three times the measured rms values of the quadrupole and octupole powers respectively (Erickcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' It turns out that the dipole contribution does not lead to a significant constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These con- tributions can explain the alignment of quadrupole and octupole if we assume the presence of an intrinsic con- tribution to T10 and T20 which is partially cancelled by the contribution due to the superhorizon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Note that this intrinsic contribution is statistical in nature and hence its exact value cannot be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3 Hubble Tension: It has been shown by Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2021) that a superhorizon mode leads to a per- turbation in the gravitational potential between distant galaxies and us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This culminates in a correction in ob- served redshift of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 1 + zobs = (1 + z)(1 + zDoppler)(1 + zgrav) (9) Thus we see that in the presence of superhorizon modes, the galaxy at redshift z is observed instead at a redshift zobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In the above equation, the redshifts zDoppler and zgrav are respectively due to our velocity relative to LSS and perturbation in potential introduced by the su- perhorizon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We can express zobs as (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022), zobs = ¯z + γ cos θ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (10) where the first and the second terms on the RHS are the monopole and dipole terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Here θ is the polar angle of the source with x3 axis along the CMB dipole and γ the dipole amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Interestingly, the monopole term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (10) resolves the Hubble tension (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For that we need to choose the phase ω � π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The range of parameters which explain both the matter dipole and the Hubble tension is given in (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In Table 2, we quote some of those values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The superhorizon modes are also likely to leave their signatures in other cosmological observables like Baryon Acoustic Oscillations, epoch of reionziation etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' ω ϱ κ/H0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='81π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='58 × 10−3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='81π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='4 × 10−3 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Some parameter values for the superhorizon mode (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 5) explaining NVSS excess dipole and also resolving the Hubble tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These values also satisfying CMB constraints are taken from Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Constraints using SKA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1 Superhorizon perturbation observation The superhorizon model predicts several observations which can be tested with SKA and other future surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' An interesting feature is the significant dependence of dipole on the redshift in the presence of superhorizon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Hence, the dipole measurements in redshift bins with SKA1 and SKA2 continuum survey can work as a potential test of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For a radio contin- uum survey, it is unlikely that we would have spec- troscopic redshift information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In the past, redshifts of radio galaxies have been estimated by taking cross correlation with well known redshift surveys (Blake & Wall, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such strategies are still viable by using data from the GAMA survey fields (Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, new techniques like template fitting (Duncan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2018) or machine learn- ing based photometric redshift computations (Brescia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2021) make it possible for the SKA radio con- tinuum survey galaxies to contain redshift information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This added information provides a unique possibility to test superhorizon mode physics with the SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='2 Predictions Here, we demonstrate how precisely evident the dipole predictions with a superhorizon model would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Fur- ther, we demonstrate how much they are constrained using SKA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Assuming superhorizon modes that (a) satisfy the present NVSS and Hubble pa- rameter measurements and (b) are consistent with CMB and other cosmological measurements (see Table 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' we obtain the dipole magnitude Dobs in redshift bins using the formalism described in (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The dependence of dipole signal on the redshift z is shown in Figure 2 where we have shown the redshift dependence of Dobs for two cases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Cumulative Redshift Bins: For this case, we fix z1 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (7) and vary z2 = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In other words, we calculate Dobs(0, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Non-overlapping Redshift Bins: In this case, we obtain the non overlapping z dependence by eval- J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000)000: Page 7 of 1 2 3 4 5 6 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='016 Dobs cumulative redshift bins non-overlapping redshift bins Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The dipole signal observation in the presence of superhorizon modes with SKA1 (z ≤ 5) and SKA2 (z ≤ 6) continuum surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The inner and outer shaded regions respectively represent the optimistic and realistic uncer- tainties for both SKA1 & SKA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Here we have assumed a superhorizon mode (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022) satisfying present NVSS dipole observation (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Flux Density SKA1 SKA2 Optimistic > 10 > 1 Realistic > 20 > 5 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Optimistic and realistic flux densities (in µJy) for SKA1 and SKA2 surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' uating Dobs(z − ∆z, z + ∆z) with ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For a fix ∆z, this thus gives Dobs at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='3 Estimating Uncertainties We employ Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2015) ‘Ultra-large scales’ codes2 (for continuum surveys) to determine the num- ber densities for SKA surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We further assume that SKA1 and SKA2 will observe the sky up to respective declinations of 15◦ and 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The optimistic and realis- tic flux densities’ limits for SKA1 and SKA2 (Square Kilometre Array Cosmology Science Working Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Bengaly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2018) are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Additionally, we note that SKA1 is expected to probe up to 0 ≤ z ≤ 5, whereas the SKA2 will reach up to redshift 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We mock SKA1 and SKA2 contin- uum sky to determine the observational implications of the superhorizon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We produce 1000 num- ber density simulation of SKA1 and SKA2 continuum survey for each (optimistic and realistic) flux thresh- old using HEALPix software (Go´rski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2005), with Nside = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The mean number of galaxies in 2http://intensitymapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='uk/codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='html a pixel is determined using number density obtained from Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (2015) code and by modelling a dipole with magnitude and direction expected in pres- ence of a superhorizon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Given the mean number density in a pixel, we call random Poisson distribution to emulate the galaxy count in the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The galaxy mock thus neglects the cosmological galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is justified since the clustering dipole in LSS is ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='7×10−3 (Nusser & Tiwari, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Tiwari & Nusser, 2016), which is roughly five times less than the appar- ent dipole in LSS3 and inconsequential for our simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' This is roughly equal to the uncertainties in the measured dipole is LSS using NVSS galaxies (see Ta- ble 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We consider SKA1, SKA2 sky coverage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', mask declination above 15◦ and 30◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Ad- ditionally, we mask the galactic latitudes (|b| < 10◦) in order to remove Milky Way contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The galac- tic plane cut is often chosen to be anywhere between |b| < 5◦ and |b| < 15◦, and in most studies one tests the robustness of the results with varying redshift cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For tests on mock data, here we choose a typical cut of |b| < 10◦ that should balance the exclusion of galac- tic plane contamination and loss of sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Next, we use Python Healpy4 (Go´rski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2019) fit dipole function and obtain dipole for each 1000 mock maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' From these 1000 dipole val- ues, we calculate the standard deviation to determine the uncertainty in measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The shaded regions in Figure 2 show the results obtained for SKA1 and SKA2 optimistic and realistic number densities in non- overlapping and cumulative redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='4 Other Anisotropy tests with SKA If the universe does not follow CP at large distance scales then every observable should have directional dependence characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Out of all, three observ- ables are of particular interest as these are independent of the number density over the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' So these are more robust under unequal coverage and systematics of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These observables are Mean spectral index (¯α) – As we said in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='1, the spectral index for a radio source is defined between flux density and frequency through S ∝ ν−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' In the Healpy pixelation scheme, the sky is divided into equal area pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For a given pixel p with Np sources having spectral indices αi,p, we 3These estimates correspond to NVSS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Assuming the NVSS measured dipole in LSS is true, we expect similar numbers from SKA continuum surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 4https://healpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='io/en/latest/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content='html Page 8 of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' (0000) 000: define mean spectral index ¯αp = 1 Np � i αi,p (11) here i runs over all the sources in pixel p Exponent (x) of differential number count – De- fined using N(S, ˆn) ∝ S −1−x Average Flux Density ( ¯S ) – We define this quan- tity for a pixel p ¯S p = 1 Np � i S i,p (12) where again i runs over all the sources in p and Np is the number of sources in the pixel The spectral index characterises the morphology of an astronomical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Angular dependence of ¯α has not been much looked at in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Analysing the dipole anisotropy in ¯α has been a challenge since it requires reliable multi-frequency continuum radio sky survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Such an SKA survey can be used to estimate this anisotropy if the flux density of the sources at dif- ferent frequencies is measured with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' For x, the angular dependency was analysed by (Ghosh & Jain, 2017) in NVSS data using likelihood maximisa- tion and the results were found to be consistent with CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' However, with a larger expected source count of SKA that is almost twice in comparison to NVSS, angular dependence analysis of x may provide a more stringent test of CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' SKA will also be able to test the phenomenon of alignment of radio galaxy axes and integrated polariza- tions, as claimed in earlier radio observations (Tiwari & Jain, 2013, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Taylor & Jagannathan, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Conclusion and Outlook In this paper, we have reviewed several cosmological signals which appear to show a violation of CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We have also reviewed a model, based on aligned super- horizon modes which can explain some of these obser- vations along with the Hubble tension (Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The model can be theoretically justified by pos- tulating a pre-inflationary phase during which the Uni- verse may not be homogeneous and isotropic (Rath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' The model leads to several cosmolog- ical predictions which can be tested at SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' By us- ing the best fit parameters with current observations, we have determined the redshift dependence of the pre- dicted dipole in radio galaxy number counts and associ- ated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' As can be seen from Figure 2, SKA can test this very reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' If this prediction is confirmed by SKA, it may provide us with a first glimpse into the Physics of the pre-inflationary phase of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We have also suggested other isotropy tests with SKA using other variables which are independent of number density and thus are more robust under unequal coverage and systematics of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' These variables are (a) mean spectral index ¯α, (b) exponent of the dif- ferential number count x & (c) average flux density ¯S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' Acknowledgements RK is supported by the South African Radio Astron- omy Observatory and the National Research Founda- tion (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 75415).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' PT acknowledges the support of the RFIS grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 12150410322) by the National Natural Science Foundation of China (NSFC) and the support by the National Key Basic Research and De- velopment Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2018YFA0404503) and NSFC Grants 11925303 and 11720101004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' SG is supported in part by the National Key R & D Pro- gram of China (2021YFC2203100), by the Fundamen- tal Research Funds for the Central Universities under grant no: WK2030000036, and the NSFC grant no: 11903030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' We are also very thankful to the anonymous referee whose comments were really helpful in improv- ing the presentation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' References Alonso, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', Bull, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=', Ferreira, P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} +page_content=' 2019, Journal of Open Source Software, 4, 1298' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfSQPo/content/2301.03065v1.pdf'} diff --git a/49AzT4oBgHgl3EQffvx7/content/tmp_files/2301.01457v1.pdf.txt b/49AzT4oBgHgl3EQffvx7/content/tmp_files/2301.01457v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3eea20f619671b1cbf548d8ce8f08dcf8e490124 --- /dev/null +++ b/49AzT4oBgHgl3EQffvx7/content/tmp_files/2301.01457v1.pdf.txt @@ -0,0 +1,1971 @@ +Bootstrap Embedding on a Quantum Computer +Yuan Liu,∗,† Oinam R. Meitei,‡ Zachary E. Chin,† Arkopal Dutt,¶ Max Tao,† +Troy Van Voorhis,∗,‡ and Isaac L. Chuang†,§ +†Department of Physics, Co-Design Center for Quantum Advantage, Massachusetts +Institute of Technology, Cambridge, Massachusetts 02139, USA +‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge, +Massachusetts 02139, USA +¶Department of Mechanical Engineering, Massachusetts Institute of Technology, +Cambridge, Massachusetts 02139, USA +§Department of Electrical Engineering and Computer Science, Massachusetts Institute of +Technology, Cambridge, Massachusetts 02139, USA +E-mail: yuanliu@mit.edu; tvan@mit.edu +Abstract +We extend molecular bootstrap embedding to make it appropriate for implementa- +tion on a quantum computer. This enables solution of the electronic structure problem +of a large molecule as an optimization problem for a composite Lagrangian governing +fragments of the total system, in such a way that fragment solutions can harness the +capabilities of quantum computers. By employing state-of-art quantum subroutines +including the quantum SWAP test and quantum amplitude amplification, we show how +a quadratic speedup can be obtained over the classical algorithm, in principle. Utiliza- +tion of quantum computation also allows the algorithm to match – at little additional +computational cost – full density matrices at fragment boundaries, instead of being +limited to 1-RDMs. Current quantum computers are small, but quantum bootstrap +1 +arXiv:2301.01457v1 [quant-ph] 4 Jan 2023 + +embedding provides a potentially generalizable strategy for harnessing such small ma- +chines through quantum fragment matching. +1 +Introduction +Determining the ground state of large-scale interacting fermionic systems is an important +challenge in quantum chemistry, materials science, and condensed matter physics. Just as +electronic properties of molecules underpin their chemical reactivity,1–3 phase diagrams of +solid state materials are also determined to a large degree by their ground state electronic +structure.4–6 However, close to exact solution to the time-independent Schrodinger equation +of a practical many-electron system remains a daunting task because the dimension of the +underlying Hilbert space grows exponentially with the number of orbitals, and the computa- +tional resources required to perform calculations over such a large space can quickly exceed +the capacity of current classical or quantum hardware. +One promising approach to fit a large electronic structure problem into a limited amount +of computational resources is to break the original system into smaller fragments, where +each fragment can be solved individually from which a solution to the whole is then ob- +tained.7–9 Efforts along this direction have successfully led to various embedding schemes +that significantly expand the complexity of the systems solvable using classical computa- +tional resources, such as density-based embedding theories,10,11 density-matrix embedding +theories (DMET),12–16 various Green’s function embedding theories6,17–21 and the bootstrap +embedding theory.22–24 The essence of such embedding-based methods is to add an additional +external potential to each fragment Hamiltonian and then iteratively update the potential +until some conditions on certain observables of the system are matched. Nevertheless, due to +the significant cost in solving the fragment Hamiltonian itself as the fragment size increases, +the applicability of such methods are limited to relatively small fragments, which may lead to +incorrect predictions in systems with long-range correlations.25 While approximate fragment +2 + +solvers such as the coupled-cluster theory or many-body perturbation theory have greatly +enhanced the applicability of such embedding methods at a reduced cost,26–28 these approx- +imations tend to fail for strongly correlated systems due to limited treatment of electron +correlation. In addition, because of limitations on computing k-electron reduced density +matrices (k-RDMs for k > 2), embedding and observable calculations beyond 2-RDM are +difficult in general. +Quantum computers are believed to be promising in tackling electronic structure prob- +lems more efficiently,29 despite the possibility of an exponential speedup still being unclear.30 +One natural idea to circumvent the problems of classical eigensolvers is to use a quantum +computer to treat the fragments. By mapping each orbital to a constant (small) number of +qubits, the exponentially large (in the number of orbitals) Hilbert space of an interacting +fermionic system can be encoded in only a polynomial number of qubits and terms. Indeed, +quantum eigensolvers such as the quantum phase estimation (QPE)31 algorithm has been +proposed to achieve an exponential advantage given a properly prepared input state32 with +non-exponentially small overlap with the exact ground state. More recently, various variants +of the variational quantum eigensolver (VQE)33–37 have been demonstrated experimentally +on NISQ devices to achieve significant speedup without sacrificing accuracy as compared +to classical methods. Moreover, k-RDMs (for any k) can be measured through quantum +eigensolvers38,39 that may circumvent the difficulty encountered on classical computers. +To take the full advantage of these quantum eigensolvers within the embedding frame- +work,18,40–44 two open questions immediately arise as a result of the intrinsic nature of +quantum systems. Firstly, the wave function of a quantum system collapses when measured. +This means any measurement of the fragment wave function is but a statistical sample (akin +to Monte Carlo methods), and many measurements are needed to obtain statistical averages +with sufficiently low uncertainty in order to achieve a good matching condition for the em- +bedding. Secondly, the best way to perform matching between fragments using results from +quantum eigensolvers is not clear, and most likely a new approach needs to be formulated +3 + +to match fragments. Admittedly, it would be straightforward to first estimate the density +matrices by collecting a number of quantum samples and then use the estimated density ma- +trices to minimize the cost function as in classical embedding theories.12,22 But this approach +would be very costly especially given the increasing number of elements in qubit reduced +density matrices (RDMs) that need to be estimated.45 Could there be a quantum method +for matching, as opposed to a statistical sampling-based classical approach? +We address the two challenges by providing a quantum coherent matching algorithm and +an adaptive sampling schedule, leading to a quantum bootstrap embedding (QBE) method +based on classical bootstrap embedding.22 Instead of matching the RDM element-by-element, +the quantum matching algorithm employs a SWAP test46,47 to match the full RDM between +overlapping regions of the fragments in parallel. Moreover, the quantum amplitude estima- +tion algorithm48,49 allows an extra quadratic speedup to reach a target accuracy on estimating +the fragment overlap. In addition, the adaptive sampling changes the number of samples +as the optimization proceeds in order to achieve an increasingly better matching conditions. +The present work invites a viewpoint of treating quantum computers as coherent sampling +machines which have three major advantages, as compared to their classical counterparts. +First, the exponentially large Hilbert space provided by a quantum computer allows more +efficient exact ground state solver (QPE) than their classical counterpart (exact diagonal- +ization). Second, in the case of truncation for seeking approximate solutions, the abundant +Hilbert space of quantum computers enable more flexible and expressive variational ansatz +than classical computers, leading to more accurate solutions. Third, the coherent nature of +quantum computers allows sampling to be performed at a later stage, e.g. after quantum +amplitude amplification of matching conditions to extract just the feedback desired, instead +of having to read out full state of a system. +The rest of the paper is organized as follows. Sec. 2 overviews bootstrap embedding +method at a high level and analyzes its scaling on classical computers, in order to motivate +the need for bootstrap embedding on quantum computers. This section serves to set the +4 + +notation and baseline of comparison for the rest of the paper. Sec. 3 presents the theoretical +framework of quantum bootstrap embedding in detail as constraint optimization problems. +In Sec. +4, we give details of the QBE algorithm to solve the optimization problem. +In +Sec. 5, we apply our methods to hydrogen chains under minimal basis where both classical +and quantum simulation results are shown to demonstrate the convergence and sampling +advantage of our QBE method. We conclude the paper in Sec. 6 with prospects and future +directions. +2 +Ideas of Bootstrap Embedding +The idea of Bootstrap Embedding (BE) for quantum chemistry has recently led to a promis- +ing path to tackle large-scale electronic structure problems.22,23,50 In this section, we establish +the terminology and framework that will be used in the rest of the paper. We first briefly +review BE and outline the main framework of BE for computation on a classical computer in +Sec. 2.1 and 2.2 for non-chemistry readers, to set up the notation. We then begin presenting +new material by discussing typical behavior and computational resource requirements for BE +on classical computers in Sec. 2.3, which leads to the quest for performing BE on a quantum +computer in Sec. 2.4. +2.1 +Fragmentation and Embedding Hamiltonians +To provide a foundation for a more concrete exposition of the bootstrap embedding method, +we first establish some rigorous notation for discussing molecular Hamiltonians and their +associated Hilbert spaces. We will work with the molecular Hamiltonian under the second +quantization formalism. +Specifically, given a particular molecule of interest, define O = +{φµ | µ = 1, . . . , N} to be an orthonormal set of single-particle local orbitals (LOs), where +N is the total number of orbitals; in this work, these LOs are generated through L¨owdin’s +symmetric orthogonalization method.51 The full Hilbert space H for the entire molecular +5 + +system is thus given by H = F(O), where F(O) denotes the Fock space determined by +the LOs in the set O. Further define the creation (annihilation) operator c† +µ (cµ) which +creates (annihilates) an electron in the LO φµ, the molecular Hamiltonian is written in the +second-quantized notation +ˆH = +N +� +µν=1 +hµνc† +µcν + 1 +2 +N +� +µνλσ=1 +Vµνλσc† +µc† +νcσcλ +(1) +where hµν and Vµνλσ are the standard one- and two-electron integrals. +Note that the number of terms in the full molecular Hamiltonian ˆH scales polynomially +with the total number of orbitals N, but the dimension of H scales exponentially with N. +Clearly, for large N, it will become prohibitively expensive to directly compute the exact +full ground state. To circumvent this issue, we divide the full molecule into multiple smaller +fragments, each equipped with its own “embedding Hamiltonian” which contains a number of +terms that only scales polynomially with the number of orbitals in the fragment. Given that +there are potentially far fewer orbitals in each fragment than in the whole molecular system, +computing the ground state of each fragment’s embedding Hamiltonian can be significantly +less expensive than computing the ground state of the full system. +Furthermore, using +the bootstrap embedding procedure to be described later, the ground states of individual +fragments can, to a high degree of accuracy, be algorithmically combined to recover the +desired electron densities prescribed by the exact ground state of the full system. Thus, this +combination of fragmentation and bootstrap embedding can be used to reconstruct the full +molecular ground state more efficiently than by direct computation alone. +We now briefly review the construction of embedding Hamiltonians for each fragment. +Consider a single fragment associated with a label A, without loss of generality, define +O(A) = {φµ | µ = 1, . . . , NA} with NA ≤ N to be the set of LOs contained in fragment A; +we will refer to O(A) as the set of fragment orbitals. Note that O(A) ⊆ O, the set of LOs +for the entire molecular system. The construction of the embedding Hamiltonian ˆH(A) for +6 + +fragment A begins with any solution of the ground state of the full system ˆH. For simplicity, +the Hartree-Fock (HF) solution |ΦHF⟩ is often used because it is easy to obtain on a classical +computer. By invoking a Schmidt decomposition, we can write |ΦHF⟩ with the following +tensor product structure for ∀ A +|ΦHF⟩ = +� NA +� +i=1 +λ(A) +i +|f (A) +i +⟩ ⊗ |b(A) +i +⟩ +� +⊗ |Ψ(A) +env⟩ . +(2) +In the above decomposition, the |f (A) +i +⟩ represent single-particle fragment states contained +in the Fock space F(O(A)) of fragment orbitals. On the other hand, the |b(A) +i +⟩ and |Ψ(A) +env⟩ +represent Slater determinants contained in the “environment” Fock space F(O \ O(A)) of +the N − NA orbitals not included in the fragment. The key difference between the single +environment state |Ψ(A) +env⟩ and the various “bath” states |b(A) +i +⟩ is that the bath states |b(A) +i +⟩ are +entangled with the fragment states |f (A) +i +⟩ while |Ψ(A) +env⟩ is not; this entanglement is quantified +by the Schmidt coefficients λ(A) +i +. Crucially, since the HF solution is used, the sum in Eq. +(2) only has NA terms (as opposed to 2NA for a general many-body wave function). Denote +the collection of the NA entangled bath orbitals as O(A) +bath = {βµ |µ = 1, . . . , NA}, where each +of the LOs βµ are linear combinations of the original LOs not included in the fragment, +βµ ∈ Span{O \ O(A)}. Furthermore, we denote the Fock space that corresponds to this set +of entangled bath orbitals as F(O(A) +bath). +This tensor product structure of |ΦHF⟩ allows us to naturally decompose the Hilbert space +H for the full molecular system into the direct product of two smaller Hilbert spaces, namely +H = H(A) ⊗ H(A) +env, +(3) +where +H(A) = F(O(A)) ⊗ F(O(A) +bath) +(4) +7 + +is the active fragment embedding space and H(A) +env contains the remaining states, including +|Ψ(A) +env⟩. Note that since both sets O(A) and O(A) +bath have size NA, the fragment Hilbert space +H(A) is a Fock space spanned of just 2NA single-particle orbitals. The core intuition mo- +tivating this decomposition is that, in the exact ground state of the full system, states in +H(A) +env are unlikely to be strongly entangled with the many-body fragment states (consider the +approximate HF ground state in Eq. (2), where they are perfectly disentangled); therefore, +in a mean-field approximation, it is reasonable to entirely disregard the states in H(A) +env when +calculating the ground state electron densities on fragment A. Following this logic, we can +define an embedding Hamiltonian ˆH(A) for fragment A only on the 2NA LOs in H(A), which +will have the form +ˆH(A) = +2NA +� +pq +h(A) +pq a(A)† +p +a(A) +q ++ 1 +2 +2NA +� +pqrs +V (A) +pqrsa(A)† +p +a(A)† +q +a(A) +s a(A) +r +, +(5) +given some creation and annihilation operators a(A)† +p +and a(A) +p , which respectively create and +annihilate electrons in orbitals from the combined set O(A) ∪ O(A) +bath for H(A). The new one- +and two- electron integrals h(A) +pq and V (A) +pqrs can be computed by projecting ˆH into the smaller +Hilbert space H(A) (consult the Supporting Information (SI) Sec. S1 for details). Note that +since we can choose 2NA ≪ N, the ground state of this embedding Hamiltonian can be +solved at a significantly reduced cost when compared to that of the full system Hamiltonian. +We are hence prepared to generate an embedding Hamiltonian for any arbitrary frag- +ment of the original molecular system. However, the ground state electron densities of the +fragment embedding Hamiltonian are unlikely to exactly match those of the full system +Hamiltonian because, as mentioned above, the embedding process may neglect some small +(but nonzero) entanglement of the fragment orbitals with the environment. Because we can +expect interactions in the molecular Hamiltonian to be reasonably local, we anticipate that +the electron densities on orbitals near the edge of the fragment (those closest to the “envi- +ronment”) will deviate most significantly from their true values, while electron densities on +8 + +orbitals toward the center of the fragment will be most accurate. +To improve the accuracy of the fragment ground state wave function near the fragment +edge, we employ the technique of bootstrap embedding. Broadly speaking, we first divide the +full molecule into overlapping fragments such that the edge of each fragment overlaps with +the center of another. Fig. 1i illustrates this fragmentation strategy: for example, we see +that the edge of fragment A (labeled as orbital 3) coincides with the center of fragment B. +We then apply additional local potentials to the edge sites of each fragment to match their +electron densities to those on overlapping center sites of adjacent fragments. Because we +expect the electron densities computed on the center sites to be closer to their true values, +these added local potentials should improve the accuracy of each fragment wave function +near the edges. In the next section, we will formalize this edge-to-center matching process +rigorously and discuss its implementation on a classical computer. +2.2 +Matching Electron Densities: an Optimization Problem +As mentioned in the previous section, we intend to correct the electron density error near +a fragment’s edge by applying a local potential to the edge; this local potential serves to +match the edge electron density of the fragment to the center electron density of an adjacent +overlapping fragment, which we expect to be more accurate. In principle, to achieve an +exact density matching, all k-electron reduced density matrices (k-RDM, for any k) on the +overlapping region have to be matched. However, in practice, such matching beyond the 2- +RDM is difficult on a classical computer due to the mathematical challenge that the number +of terms in k-RDM in general increases exponentially as k. In addition, almost all electronic +structure codes available on classical computers are programmed to deal with only 1- and +2-RDMs, despite the importance of k-RDMs (k > 2) for computing observables such as +entropy and other multi-point correlation functions.52 Due to this reason, the discussion of +density matching process in classical BE in this section will be based on 1-RDMs. We note +that the matching process applies similarly if k-RDMs are matched. +9 + +Figure 1: Schematic of bootstrap embedding on classical (left, blue arrows) and quantum +(right, red arrows) computers. The arrows indicate BE iterative loops that are used to +optimize the corresponding objective functions. Starting from panel (i) (upper center), the +original system is first broken into overlapping fragments (Fragmentation), where each +fragment is solved using a classical (iic) (upper left) or quantum eigensolver (iiq) (upper +right). In classical matching, the 1-electron reduced density matrices (1-RDM) on the +overlapping sites of adjacent fragments are used to obtain the matching condition (iiic) +(lower left), while in the quantum case a coherent matching protocol based on SWAP tests of +overlapping sites combined with a single qubit measurement (iiiq) (lower right). The +matching results are then used by classical computers to generate the bootstrap embedding +potential VBE (iv) (lower center) and the updated fragment embedding Hamiltonian +Hemb + VBE (back to panel (i) in order to minimize a target objective function L in both +classical and quantum case. +10 + +(iic) +(i) +Fragmentation +(ig) Quantum Eigensolver +Classical +TTOTOTTOOTOTO +(QPE, VQE, ...) +Eigensolver +000000 +Frag A +Hemb+VBE +0-010! +FragA +FCI +CCSD +000000 +FragB @10i0 +Frag B +VMC +FragA +Frag C +000000 +010010 +Frag B +4 +FragD +000000 +Classical BE +Quantum BE +1-RDMs +VBE +RDMs +L=(Hemb)+Qx +L = +Frag A +QESA +[29] +[亚) +1-RDM +Subsystem +P(2) +P(c) +[[] +Frag B +difference +overlap +Frag B +QESB +[P() +ad +P(P)We begin by introducing some rigorous notation. Recall that a fragment A is defined by +a set of local orbitals O(A) which constitute the fragment. We partition this set of LOs into +a subset of edge sites (or orbitals), denoted E(A), and a subset of center sites, denoted C(A), +such that E(A) ∪ C(A) = O(A) and E(A) ∩ C(A) = ∅. Given the ground state wave function +|Ψ(A)⟩ of the embedding Hamiltonian, we further define the 1-electron reduced density matrix +(1-RDM) P(A) according to +P (A) +pq += ⟨Ψ(A)| a(A)† +p +a(A) +q +|Ψ(A)⟩ +(6) +where p, q = 1, . . . , 2NA and the operators a(A)† +p +and a(A) +q +are defined in the previous section. +Suppose, for example, that the edge of fragment A overlaps with the center of another +fragment B so that E(A) ∩ C(B) ̸= ∅. On a high level, the goal of bootstrap embedding is to +find a ground state wave function |Ψ(A)⟩, perturbed by local potentials on the edge sites of +A, such that |P (A) +pq +− P (B) +pq | → 0 for indices p and q that correspond to orbitals in the set of +overlapping sites E(A) ∩ C(B). More generally, and more rigorously, the goal is to find a wave +function which minimizes the fragment Hamiltonian energy +|Ψ(A)⟩ = arg min +Ψ(A)⟨ ˆH(A)⟩A +(7) +subject to the constraints +⟨a(A)† +p +a(A) +q +⟩A − P (B) +pq += 0 +(8) +for all other fragments B with E(A) ∩ C(B) ̸= ∅ and for all p, q corresponding to orbitals in +E(A) ∩C(B). Here, we explicitly write the expectation ⟨·⟩A = ⟨Ψ(A)|·|Ψ(A)⟩ in terms of |Ψ(A)⟩ +to indicate that the optimization is over the wave function of A. +We can formulate this constrained optimization problem as finding the stationary solution +to a Lagrangian by associating a scalar Lagrange multiplier (λ(A) +B )pq to Eq. (8). Since Eq. (8) +11 + +has to be satisfied for any p, q and B that overlaps with A, these constraint can be rewritten +in a more compact vector form λ(A) +B +· Q1-RDM(Ψ(A); P(B)) where the dot product conceals +the implicit sum over p, q, and each component of the vector Q1-RDM(Ψ(A); P(B))pq represents +the constraint associated with Lagrange multiplier (λ(A) +B )pq, given by the left hand side of Eq. +(8). With this notation, we arrive at the following Lagrangian with the constraint added as +an additional term +L(A) =⟨ ˆH(A)⟩A + E(A) � +⟨Ψ(A)| Ψ(A)⟩ − 1 +� ++ +� +B +λ(A) +B +· Q1-RDM(Ψ(A); P(B)), +(9) +where once again the B are fragments adjacent to A with E(A) ∩C(B) ̸= ∅ and p, q are indices +of orbitals contained in the overlapping set E(A) ∩ C(B). Here, the additional constraint with +Lagrange multiplier E(A) is also included to ensure normalization of the ground state wave +function |Ψ(A)⟩. Solving for the stationary solution of the Lagrangian in Eq. (9) will only +result in a ground state wave function for fragment A whose 1-RDM elements at the edge +sites match those at the center sites of adjacent overlapping fragments. However, we would +instead like to solve for such a ground state for all fragments in the molecule simultaneously. +Toward this regard, we can combine all individual fragment Lagrangians (of the form of Eq. +(9)) into a single composite Lagrangian for the whole molecule, given by +L = +Nfrag +� +A=1 +L(A) + µP +(10) +where Nfrag is the number of fragments in the molecule. Observe that we have added one +additional constraint +P = +� +� +Nfrag +� +A=1 +� +p′∈C(A) +⟨a(A)† +p′ +a(A) +p′ ⟩A +� +� − Ne +(11) +with Lagrange multiplier µ to restore the desired total number of electrons in the molecule, +12 + +Ne. Note in Eq. (11) that p′ is summed over indices corresponding to orbitals only in C(A); +this is to ensure that there is no double-counting of electrons in the whole molecule. By +self-consistently finding ground states |Ψ(A)⟩ for A = 1, . . . , Nfrag which make the composite +Lagrangian in Eq. (10) stationary, we will have completed the density matching procedure +for all fragments, and the process of bootstrap embedding will be complete. +We can gain insight into which wave functions |Ψ(A)⟩ will make the composite Lagrangian +L stationary by differentiating L with respect to |Ψ(A)⟩ for some fixed fragment A and setting +the resulting expression equal to zero. Upon some algebraic manipulation, we can recover +the eigenvalue equation +( ˆH(A) + VBE) |Ψ(A)⟩ = −E(A) |Ψ(A)⟩ , +(12) +where VBE, the local bootstrap embedding potential, is given by +VBE = +� +B +� +p,q +(λ(A) +B )pqa(A)† +p +a(A) +q ++ µ +� +p′ +a(A)† +p′ +a(A) +p′ +(13) +where the p, q are indices of orbitals in the overlapping set E(A) ∩C(B), and the p′ are indices +of orbitals in the fragment center C(A). We see that, when the composite Lagrangian is +made stationary with respect to the fragment wave functions, the bare fragment embedding +Hamiltonians become dressed with a potential VBE that contains a component local to the +edge sites of each fragment (see the left term of Eq. (13)). This observation confirms our +intuition that adding a local potential to the edge of one fragment will allow the edge site +electron density to be matched to that of a center site on an overlapping neighbor. Note +that VBE also contains an additional potential on the center sites of each fragment (see the +right term of Eq. (13)); this is simply to conserve the total electron number in the molecule. +Moreover, VBE as in Eq. (13) only contains one-body terms because only 1-RDM is used for +density matching. In general, VBE will contain up to k-body terms if k-RDMs are used for +matching. +13 + +On a classical computer, the composite Lagrangian in Eq. (10) is made stationary through +an iterative optimization algorithm22 until the edge-to-center matching condition for all +fragments is satisfied by some criterion. One possible criterion is to terminate the algorithm +when the root-mean-squared 1-RDM mismatch, given by +ϵ = +� +� +1 +Nsites +Nfrag +� +A +� +B +� +p,q +(P (A) +pq +− P (B) +pq )2 +� +� +1 +2 +, +(14) +drops below some predetermined threshold. Note again that p, q are indices corresponding to +orbitals in the overlapping set E(A)∩C(B); also, Nsites denotes the total number of overlapping +sites in the whole molecule, equal to Nsites = �Nfrag +A +� +B +� +p,q 1. The final set of density- +matched fragment wave functions {|Ψ(A)⟩} for A = 1, . . . , Nfrag which solve the composite +Lagrangian can then be used to reconstruct the electron densities and other observables for +the full molecular system, as desired. +2.3 +Resource Requirement and Typical Behavior of BE on Clas- +sical Computers +Given the notation established for classical BE, we now begin presenting new material. We +discuss the computational resource requirement and typical behaviors of performing BE on +classical computers to set the stage for a quantum BE theory. The details of the classical BE +algorithms are omitted for succinctness, and we refer the reader to Ref.22–24,50 for details. +The space and time resource requirement to perform the classical BE can be broken +down into two parts: a) the number of iteration steps to reach a fixed accuracy for ϵ (Eq. +(14)); b) the runtime of the fragment eigensolver. For a), numerical evidence suggests an +exponentially fast convergence on total system energy as the number of bootstrap iteration +increases (black trace in Fig. 2 for FCI), while a proof of the convergence rate has yet to be +established. +14 + +We focus on resource requirement in b) in the following. Admittedly, an exact classical +eigensolver such as full configuration interaction (FCI) can be used to solve the embedding +Hamiltonian in Eq. +(5). +However, both the storage space and time requirement scales +exponentially as the the number of orbitals (see blue symbols and dashed line in Fig. 3 for +the runtime scaling of FCI). Even with the state-of-the-art classical computational resources, +exact solutions using FCI are only tractable for systems up to 20 electrons in 20 orbitals.53 +As a result, classical computation of BE resorts to approximate eigensolvers with only +polynomial cost in practice, by properly truncating or sampling from the fragment Hilbert +space. One example for truncation is the coupled-cluster singles and doubles (CCSD),54 +which scales with N 6 with N being the number of orbitals. Alternately, different flavors of +stochastic electronic structure solvers can be employed as fragment solvers in BE. Depending +on implementation, these stochastic solvers can be biased or unbiased (if unbiased, with a +cost of introducing the phase problem in general).55–58 Collecting each sample on a classical +computer usually has similar cost as a mean field theory (roughly O(N 3)), while the overall +target accuracy ϵ on observable estimation can be achieved with a sampling overhead of +roughly O( 1 +ϵ2) with a constant prefactor depending on the severity of the sign problem. +Importantly, the sampling feature of these stochastic electronic structure methods on +classical computers are strikingly similar to the nature of quantum computers where mea- +surement necessarily collapses the wave function. As a result, only a classical sample (in +terms of measurement results) can be obtained from a quantum computer. This similarity +suggests a general strategy that many sampling techniques in stochastic classical algorithms +can be deployed to design better quantum algorithms. For example, sophisticated impor- +tance sampling techniques59,60 can be employed to greatly improve the sampling efficiency +in both classical58 and quantum cases.61 +Due their shared feature on sampling between classical stochastic algorithm and quantum +eigensolvers, we shall use one approximate sign-problem-free flavor of stochastic electronic +structure method, the variational Monte Carlo (VMC), to serve as an additional baseline +15 + +scenario for comparison with quantum BE in later sections. In addition to BE convergence +behavior with a FCI solver, Fig. 2 also shows, for a VMC eigensolver, the density mismatch +converges exponentially fast initially as iteration number increases with varying number of +samples. However, due to the statistic noise on estimating the 1-RDM (thus the gradient for +the optimization), the final density mismatch plateaus to a finite biased value. Comparing +among the VMC solver with different number of samples, the accuracy improves as the +number of samples increases (dashed horizontal lines). +Figure 2: Typical convergence of density mismatch with respect to the number of +eigensolver calls in classical bootstrap embedding with a deterministic eigensolver (FCI, +black circle) and a stochastic eigensolver (VMC) with different number of samples (grey, +blue, and orange solid lines). The horizontal dashed lines shows the final plateaued value of +the density mismatch for VMC, while the FCI data converges to 10−6 after 700 eigensolver +calls (not shown on the figure). The discrete jumps around 200 and 300 eigensolver calls +are due to switching to the next BE iteration. The data is obtained for an H8 linear chain +under STO-3G basis. See SI Sec. S9 B for computational details. +The increasing accuracy of density mismatch with respect to BE iteration also suggests +an increasing number of samples are needed. Thus, an optimal number of samples at each +BE iteration must be determined to achieve the desired accuracy in the matching conditions. +A careful design of such a sampling schedule can potentially save a large amount of compu- +tational resources. We defer a thorough discussion of this point to later sections on quantum +BE. +16 + +4 × 10-3 +3 × 10-3 +Density Mismatch +M +2X1 +FCI +Q +VMC (40k samples) +6 × 10-4 +VMC (160k samples) +VMC (640k samples) +4 × 10-4 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Eigensolver Calls2.4 +The Quest for BE on Quantum Computers +By employing the coherent superposition and entanglement of quantum states, the limita- +tion of an exact classical solver can be overcome by substituting it with an exact quantum +eigensolver such as the quantum phase estimation (QPE) algorithm.31 Fig. 3 compares the +runtime (gate depth) of FCI and QPE for finding the ground state of linear hydrogen chain +Hn for different system size n. Clearly, the QPE runtime scales only polynomially as the +system size increases as expected,30,32 while its classical counterpart (FCI) has an exponen- +tially increasing runtime. Note the runtime is normalized to the case of n = 1 for each +solver separately (see SI Sec. S9 for details). The dramatic advantage in the runtime scaling +of quantum over classical eigensolvers demonstrated above suggests formulating BE on a +quantum computer can bring significant benefits. +Figure 3: Runtime (normalized) as a function of system size n for finding the ground state +of a linear hydrogen chain Hn at STO-3G basis, comparing an exact classical solver (FCI, +blue square) and an exact quantum solver (QPE, red circle) on real classical and quantum +devices. Red (blue) dashed line shows a polynomial (exponential) fit to the QPE (FCI) +runtime. Note the crossover at large system size. +One might think that the eigensolver at the heart of the classical BE algorithm could +simply be replaced with a quantum one. +However, as mentioned before, there are two +outstanding challenges for such a quantum bootstrap embedding (QBE) method. First, just +17 + +108 +QPE +FCI +Time (normalized) +106 +104 +/ +102 +/ +Q +口 +口 +口 +口 +100 +口 +3 +1 +5 +7 +9 +1113 +15 +17 +19 +System Sizeas in classical stochastic methods, the results of a quantum eigensolver need to be measured +for later use, but quantum wave functions collapse after measurement. Therefore, sampling +from the quantum eigensolver is required, and the optimal sampling strategy is unclear. +Secondly, with quantum wave function from quantum eigensolvers, it is not wise to achieve +matching between fragments in the same way as classical BE, as many incoherent samples are +needed to obtain a good estimation of the 1-RDM elements. Clearly, performing matching +in a quantum way is desired. +In the next two sections (Secs. 3 and 4), we present how we address these two challenges +by an adaptive quantum sampling scheduling algorithm and a quantum coherent matching +algorithm in detail. +3 +Quantum Bootstrap Embedding Methods +In previous sections, we have seen potential advantages of performing bootstrap embedding +on quantum computers, and discussed two major challenges of doing so. In this section, +we present the theoretical formulation of our bootstrap embedding method on a quantum +computer that addresses these challenges. +Sec. 3.1 first set up notations and discuss a few aspects of locality and global symmetry on +performing embedding of fermions on quantum computers. Sec. 3.2 discuss a naive extension +of the classical BE algorithm on quantum computers by matching individual elements of the +RDMs directly, and highlight the disadvantage of doing so. Sec. 3.3 introduces the SWAP +test circuit and show that it achieves the matching between two RDMs coherently. In 3.4, +we discuss some subtleties on why it is impossible to incorporate this coherent matching +condition into the Lagrange multiplier optimization method, and present an alternative +quadratic penalty method to perform the optimization. +18 + +3.1 +Fermion-Qubit Mapping - Global Symmetry vs. Locality +When mapping electronic structure problem to qubits on quantum computers, it is well- +known that the global anti-symmetric property of fermionic wave functions necessarily leads +to an overhead in operator lengths or qubit counts.62 On the other hand, chemical informa- +tion is usually local if represented using localized single-particle orbitals.63,64 In the case of +performing bootstrap embedding, this tension between locality of chemical information and +global fermionic anti-symmetry is more subtle. Because bootstrap embedding intrinsically +uses the fermionic occupation number in the local orbitals (LOs) to perform matching, it is +therefore convenient to preserve such locality when constructing the mapping. Throughout +the discussion, without loss of generality, we assume a mapping that preserves fermionic local +occupation number, such as the Jordan-Wigner mapping where each spin-orbital is mapped +to one qubit. Our discussion equally applies to cases where a non-local mapping is used (such +as parity mapping). In that case, a unitary transformation from the non-local mapping to +a local mapping will be required before actually computing the matching conditions. It is +usually more convenient to work with qubit reduced density matrices (RDMs)65 on quantum +computers instead of k-electron RDMs.66 Due to this reason, we shall formulate our QBE +method based on these qubit RDMs. The full density matrix of fragment A is thus provided +by ρ(A) = |ΨA⟩ ⟨ΨA|. Given an orbital set R ⊂ O(A) for O(A) being set of orbitals in fragment +A. Let ρ(A) +R +signify the RDM obtained from ρ(A) by tracing out the set of qubits not in R. +Specially, if R only contains orbitals on the edge (center) of fragment A, then ρ(A) +R +represents +information about the density information (for example the occupation number) on the edge +(center) of A. +These RDMs can be expanded under an arbitrary set of orthonormal basis {Σα} as follows +ρ(A) +R += I + �4m−1 +α=1 ⟨Σα⟩A Σα +2m +(15) +where ⟨Σα⟩A = ⟨ΨA| Σα |ΨA⟩ = Tr +� +ρ(A) Σα +� +, ∀α ∈ [1, 4m − 1], and m = |R| is the number +19 + +of orbitals in the set R. One convenient orthonormal basis set is the generalized Gell-Mann +basis.67 In the special case of a 1-qubit RDM, {Σα} (α = x, y, z) is the familiar Pauli +matrices. +3.2 +Naive RDM Linear Matching and its Disadvantage +A naive implementation of BE on a quantum computer is to simply replace 1-RDM in +Eq. +(6) with the qubit RDM in Eq. +(15) on the fragment overlapping regions. +Such +an extension imposes matching constraints on each elements of the RDMs, resulting the +following constraint vector in analogous to Eq. (8) +Qlin(ρ(A) +R ; ρ(B) +R ) = +� +����� +⟨Σ1⟩A − ⟨Σ1⟩B +... +⟨Σ4m−1⟩A − ⟨Σ4m−1⟩B +� +����� += 0. +(16) +It is obvious that ρ(A) +R +− ρ(B) +R += 0, if and only if all the (4m − 1) components in the above +constraint are satisfied. +Similarly, we can associate a scalar Lagrange multiplier to each constraint in Eq. (16) +and use this linear RDM constraint in place of the 1-RDM constraint Q1-RDM(Ψ(A); P(B)) +in Eq. (9). Finding the stationary point of this new Lagrangian gives the same eigenvalue +equation as Eq. (12) with a new BE potential given by +VBE = +� +B̸=A,CB∩EA̸=∅ +λ(A) +B +· [I ⊗ Σr ⊗ I] +(17) +where Σr = +� +Σ1, · · · , Σα, · · · , Σ4m−1 +� +is a (4m − 1)-dimensional vector of the orthonormal +basis in Eq. (15), and λ(A) +B +is the Lagrange multipliers now modulating the local potentials +on each qubit basis, and n is the number of overlapping sites between A and B. +To perform the optimization, the eigenvalue equation Eq. (12) with the above new BE +20 + +potential in (17) can be solved on a quantum computer to obtain an updated wave function +for fragment A. By iteratively solving the eigenvalue equation and updating the Lagrange +multipliers {λ, µ} using either gradient-based or gradient-free methods,68 an algorithm can +be formulated to solve the optimization problem. For completeness, we document the algo- +rithm from the naive linear matching of RDMs in Sec. S8 of the SI. +The above is a convenient way to impose the constraint on quantum computers, but it +is computationally costly as the number of constraints in (16) increases exponentially as +the number of overlapping sites n on neighboring fragments. For each constraint equation, +the expectation values ⟨Σα⟩ has to be measured on the quantum computer, which therefore +introduces an exponential overhead on the sampling complexity. +In the next section, we introduce a simple alternative to evaluate the mismatch between +two RDMs on a quantum computer much faster based on a SWAP test. +3.3 +Coherent Quantum Matching from SWAP Test +The wave functions of two overlapping fragments are stored coherently as many amplitudes +that suppose with each other. The beauty of quantum computers and algorithms lies at the +ability to coherently manipulating such amplitudes simultaneously. We may naturally ask: +are there quantum algorithms or circuits that can coherently achieve matching between an +exponentially large number of amplitudes, without explicitly measuring each amplitude? +In quantum information, there is a class of quantum protocols to perform the task of +estimating the overlap between two wave functions or RDMs under various assumptions.69 +Among these protocols, the SWAP test is widely used.47,70 Such a SWAP test on a quantum +computer can also be naturally implemented by simple controlled-SWAP operations as in Fig. +4, showing a SWAP test between two qubits. The essence of a SWAP test is to entangle the +symmetric and anti-symmetric subspaces of the two quantum states (|φ⟩ and |ψ⟩) to a single +21 + +ancillary qubit, such that the quantum state of the system before the final measurement is +|Ψ⟩ = 1 +2 +� +|0⟩ +� +|φ⟩ |ψ⟩ + |ψ⟩ |φ⟩ +� ++ |1⟩ +� +|φ⟩ |ψ⟩ − |ψ⟩ |φ⟩ +�� +. +(18) +By measuring the top single ancillary qubit in the usual computational Z-basis (collapsing +it to either the |0⟩ or |1⟩ state), the overlap of the two qubit wave function, |⟨φ|ψ⟩|, can be +directly obtained from the measurement outcome probability: +Prob[M = 0] = 1 + |⟨φ|ψ⟩|2 +2 +, +(19) +without requiring explicit estimation of the density matrix elements of each individual qubit. +|0⟩ +H +• +H +M +|φ⟩ +× +|ψ⟩ +× +Figure 4: Quantum circuit of a SWAP test between two qubits (lower, with state |φ⟩ and +|ψ⟩). The circuit is composed of two Hadamard gate (H), a controlled-SWAP operation in +between, and a final Z-basis measurement M on an additional ancilla qubit (top), where +M = 0, 1. +Can we recast the linear matching conditions as linear combination of several SWAP tests? +Observe that an equivalent condition alternative to Eq. (16) is the following quadratic match- +ing condition (see Sec. S3 of SI for a proof of the equivalence between the two quantum +matching conditions) +Qquad(ρ(A) +R ; ρ(B) +R ) = Tr +�� +ρ(A) +R +− ρ(B) +R +�2� += 0. +(20) +Interestingly, the above quadratic constraint can be rewritten as a linear combination of +three different multi-qubit generalization of the SWAP tests (with each repeated multiple +times), regardless of the number of overlapping sites (Fig. 1iiiq). Two of the SWAP tests are +22 + +to estimate the purity of ρ(A) +R +and ρ(B) +R +each, while the third one is to estimate the overlap +between ρ(A) +R +and ρ(B) +R . See Sec. S4 how to generalize the SWAP test on two qubits to a +multi-qubit setting and how to relate the SWAP test results to the quadratic constraint. +The reformulation of the quadratic constraint allows us to estimate the mismatch be- +tween two fragments by measuring only a single ancilla qubit (estimating three different +amplitudes). As compared to the linear constraint case where an exponentially large num- +ber of constraints have to be estimated individually (4m − 1 where m = |R| is the number of +overlapping sites again), the quadratic matching based on SWAP tests achieves an exponential +saving in the types of measurements required. +Furthermore, the reduction of the mismatch to the estimation of only a few (three) +amplitudes in SWAP tests allows an additional quadratic speedup by amplifying the amplitude +of the ancilla qubit before measure it. We will discuss more details on how to achieve the +quadratic speedup in Sec. 4.3. Admittedly, such amplitude amplification algorithm may be +applied even to the naive linear RDM matching by boosting individual RDM amplitude, but +the resulting quantum circuit will be much more complicated. +3.4 +Optimization Using the Quadratic Penalty Method +With an efficient way to estimate the quadratic penalty constraint established in Eq. (20), it +now appears feasible to use this new constraint in Eq. (9) as in the case of linear constraint. +However, the nature of the quadratic matching in Eq. (20) makes the same Lagrange mul- +tiplier optimization method used in the linear case invalid. We first discuss in more detail +why this approach fails, in Sec. 3.4.1; we then describe an alternative way of treating the +quadratic constraint as a penalty term to optimize the resulting objective function in Sec. +3.4.2. +23 + +3.4.1 +Violation of the Constraint Qualification +A necessary condition to use the Lagrange multiplier method for constraint optimization +is that the gradient of the constraint itself with respect to system variables has to be +non-zero at the solution point (this guarantees a non-zero effective potential to be added +to the original Hamiltonian), a.k.a., constraint qualification.71,72 Specifically, we require +∇Qquad(ρ(A) +R ; ρ(B) +R ) ̸= 0 when ρ(A) +R += ρ(B) +R . +Unfortunately, in the quadratic case, we have +∇Qquad(ρ(A) +R ; ρ(B) +R ) ∝ ρ(A) +R +− ρ(B) +R += 0 +(21) +when ρ(A) +R +and ρ(B) +R +matches, which violates the above condition. Note that any high-order +constraint other than linear order will violate the constraint qualification. The existence +of such constraint qualification makes sense also from a physical point of view. Because +the gradient ∇Qquad(ρ(A) +R ; ρ(B) +R ) enters the eigenvalue equation (13) as the BE potential VBE +modulated by the Lagrange multipliers. The vanishing of this potential near the solution +point means there is no way to modulate VBE by adjusting the Lagrange multipliers, and +therefore will lead to failure of convergence of the Lagrange multiplier. +Alternatively, the quadratic constraint can be treated as a penalty by using λ(A) +B Qquad(ρ(A) +R ; ρ(B) +R ) +to substitute the constraint λ(A) +B +· Q1-RDM(Ψ(A); P(B)) in Eq. (9). We can then employ the +quadratic penalty method73 to minimize this cost function. To highlight the distinction +of quadratic penalty method from the Lagrange multiplier method, we use “cost function” +instead of “Lagrangian” to refer to the objective function in the quadratic penalty case. +3.4.2 +Details of the Quadratic Penalty Method +The idea of the penalty method is to use the constraint as a penalty where the magnitude +of λ(A) +B +serves as a weight to the penalty. Initially, λ(A) +B +is set to a small constant, and then +we treat the resulting cost function as an unconstrained minimization where its minimum +24 + +is found by varying the wave functions. The next step is to increase λ(A) +B +to a larger value +leading to a new Lagrangian, which is then minimize again by varying the wave function +parameters. This procedure is repeated until the penalty parameter λ(A) +B +is large enough to +guarantee a small mismatch Qquad(ρ(A) +r +; ρ(B) +r +). In our case, we choose all λ(A) +B += λ for all pairs +of adjacent fragments. +It is helpful to note that optimization of the wave function is done again using the +eigenvalue equation as in Eq. (12) by tuning the BE potential VBE. In other words, for a +fixed penalty parameter λ, the fragment Lagrangian LA({VBE}) is minimized with respect +to VBE. For a particular parametrization in terms of local potentials {vα} on the edge sites +of fragment A +VBE({vα}) = +M +� +α=0 +vα I ⊗ Σα ⊗ I, +(22) +where {Σα} is a set of Hermitian generator basis of size M on the edge sites of fragment A +(can be Pauli operators for a single edge site), and {vα} is the corresponding local potential +(real numbers). Note that M in Eq. (22) can be much smaller than the total number of +generators (4m) on the edge sites, because in each bootstrap embedding iteration, only a +small local potential is added to the Hamiltonian. This perturbative nature of the bootstrap +embedding iteration allows us to expand the BE potential VBE in each iteration under the +Hermitian generator basis from the previous iteration, such that the BE potential in each +iteration is diagonal dominant, i.e., M ≪ 4m where n is the number of edge sites on any +fragment A. +To update {vα}, we derive the following gradient +dL(A) +dvα += +� +n′̸=0 +� +C†(I ⊗ W(n′) +α +⊗ I)C(n′)� +× +� +C(n′)† � +H(A) + E(A) +0 ++ 2λ (I ⊗ (ρEA − ρCB) ⊗ I) +� +C +� +(23) +∀α ∈ [0, M], that can, in principle, be used to perform the updating of VBE to minimize +25 + +L(A). In the above, C(n) is the eigenvector of the n-th eigenstate (n ≥ 1) while C is the +eigenvector of the ground state, W(n′) +α +is a perturbation matrix between ground state and +the n′-th eigenstate for the α-th Pauli basis at the edge site of fragment A, whereas ρEA and +ρCB are the RDM at the edge and center sites of fragment A and B, respectively (see SI Sec. +S5 for detailed derivation). +The above gradient in Eq. (23) is only formally useful, but computing it exactly requires +all the eigenstates to be known (not only the ground state) which is clearly very costly if +possible. Nevertheless, it serves as a good starting point to develop approximated updating +scheme or to perform bootstrap embedding for excited states. +We leave such topics for +future investigation. +In the present work, instead of using Eq. +(23) to update VBE, we +employ gradient-free schemes to update {vα} and measure the required expectation values +using SWAP test to obtain the mismatch to evaluate the cost function L(A). +We note that one additional advantage of this quadratic penalty method is that it can +be easily integrated with variational eigensolvers34 by treating the quadratic penalty as +an additional term in the VQE cost function.74 The drawback is that the optimized wave +function only exactly equals to the true wave function when the penalty goes to infinity +λ → ∞. Practically, we find that choosing the penalty parameter large enough is sufficient +to obtain satisfactory results. +4 +Quantum Bootstrap Embedding Algorithms +Given the theoretical formulation of QBE method in Sec. 3, we present a general hybrid +quantum-classical algorithm in this section that can be practically used to solve the BE +problem on quantum computers to find the BE potentials VBE that satisfies the matching +condition. +In our quantum bootstrap embedding algorithm, the electronic structure problem of +the total system is formulated as a minimization of a composite objective function with a +26 + +penalty term constructed from the matching conditions on the full qubit RDMs on overlap- +ping regions of adjacent fragments. We then design an iterative hybrid quantum-classical +algorithm to solve the optimization problem, where a quantum subroutine as an eigensolver +is employed to prepare the ground state of fragment Hamiltonian. The quantum matching +algorithm employs a SWAP test46,47 between wave functions of two fragments to evaluate the +matching conditions, which is a dramatic improvement as compared to the straightforward +method of measuring an exponential number (with respect to the number of qubits on the +fragment edge) of RDM elements. Additionally, the quantum bootstrap embedding frame- +work is internally self-consistent without the need to match fragment density matrices to +external more accurate solutions. The adaptive sampling changes the number of samples as +the optimization proceeds in order to achieve an increasingly better matching conditions. +We note that the SWAP test adds only little computational cost to quantum eigensolvers +which can be readily performed on current NISQ devices. The amplitude amplified coherent +quantum matching requires iterative application of eigensolvers multiple times which are +more suitable for small fault-tolerant quantum computers. +The rest of this section is organized as follows. Sec. 4.1 gives an outline of the QBE +algorithm with the quadratic penalty method. Sec. 4.2 discusses possible choices of quantum +eigensolvers with an analysis on sampling complexities. We then present a way to achieve +an additional quadratic speedup by using coherent amplitude estimating algorithm in Sec. +4.3. +4.1 +The Algorithm +We present a high-level framework of the main algorithm in this section. As a comparison, +the QBE algorithm with naive linear matching can be found in SI Sec. S8. Code for the +algorithms and data for generating the plots are available as open source on github.75 +To quantify the mismatch across all fragments, we define ∆ρ to be the root mean square +density matrix mismatch averaged over all the overlapping sites of all the fragments according +27 + +to +∆ρ = +� +� +� +� +1 +Nsites +� +A,B +� +r∈E(A)∩C(B) +Tr +�� +ρ(B) +r +− ρ(A) +r +�2� +(24) +where Tr +�� +ρ(B) +r +− ρ(A) +r +�2� += Qquad(ρ(A) +r +; ρ(B) +r +) as in Eq. (20), which may also be recognized +as the Frobenius norm of (ρ(B) +r +− ρ(A) +r +). Nsites is the total number of terms in the double sum +in Eq. (24), Nsites = � +A̸=B |E(A) ∩ C(B)|, with |S| denoting the number of elements in set S. +The cost function L(A)(λ) being optimized is discussed in Sec. 3.4.1. For clarity, we write +it explicitly here +L(A)(λ) =⟨ ˆH(A)⟩A + +� +B +λQquad(ρ(A) +R ; ρ(B) +R ), +(25) +with Qquad given by Eq. +(20). +We have omitted the term E(A) for simplicity since the +normalization of the wave function is guaranteed for a fault-tolerant quantum computer. +However, this term can be important on a noisy quantum computer where the purity of +the wave function can be contaminated. Note the expectation value in Eq. (25) has to be +estimated by collecting samples on a quantum computer. +The quantum bootstrap embedding algorithm with quadratic penalty method is presented +below in Alg. 1. The algorithm takes as its input the total Hamiltonian of the original system, +and then perform the fragmentation and parameter initialization, followed by the main +optimization loop to achieve the matching. Finally, it returns the optimized BE potential +V (A) +BE for any fragment A and the final mismatch ∆ρ. Inside the main loop (line 9 of Alg. 1), +the cost function L(A)(λ) for each fragment A is minimized for a fixed penalty parameter λ +(line 10 and 11). The penalty λ is then increased geometrically (line 12) until the mismatch +28 + +criteria is met, i.e., ∆ρ ≤ ε. +Algorithm 1: +Quantum bootstrap embedding algorithm: +quadratic penalty +method +1 Input: Geometry of the total molecular system and the associated ab initio +Hamiltonian. +2 +/* Initialization +*/ +3 Fragmentation: Divide the full molecular system into Nfrag overlapping fragments; +4 for A = 1 to Nfrag do +5 +Generate H(A) using Eq. (S1) of SI Sec. S1; +6 +Set V (A) +BE = 0; +7 Parameter initialization: set initial penalty factor λ = 1; set initial mismatch +∆ρ > ϵ. +8 +/* Main loop: +*/ +9 while ∆ρ > ε do +10 +for A = 1 to Nfrag do +11 +Minimize L(A)(λ) as in Eq. (25) : Repeatedly generate V (A) +BE and estimate +the penalty loss function L(A)(λ) using SWAP test. +12 +Increase penalty parameter: λ ← γλ, for some fixed γ > 1. +13 +Update mismatch: for A = 1, Nfrag do +14 +Estimate Qquad(ρ(A) +r +; ρ(B) +r +) using N SWAP +samp (Eq. (27)) samples for each SWAP test. +15 +Classically compute the mismatch ∆ρ using Eq. (24). +16 Returns: +� +H(A) + V (A) +BE +� +for all A, ∆ρ. +A key step of the algorithm is the minimization of L(A)(λ) at line 11, which consists of +repeatedly generating the BE potential V (A) +BE and estimate the mismatch using SWAP test. +BE potentials V (A) +BE are generated differently for different optimization algorithms. In our +29 + +implementation, a quasi-Newton method, the L-BFGS-B76 algorithm, is used at line 11 for +minimizing L(A)(λ), where V (A) +BE is proposed by the optimizer in order to estimate the inverse +Hessian matrix to steer the optimization properly. Alternatively, if derivative-free methods +such as Nelder-Mead77 is used, V (A) +BE will be generated in a high-dimensional simplex defined +by the coefficients {vα} in Eq. (22), which is repeatedly refined. +Once V (A) +BE is generated, the first term in the cost function in Eq. (25) is estimated by +invoking the quantum eigensolver for the Hamiltonian +� +H(A) + V (A) +BE +� +. The second term, the +mismatch in Eq. (25) can be estimated by measurement outcomes of the ancilla qubit in +the SWAP test (SI Sec. S4). The mismatch estimation at line 13 is performed in the same +way as those in line 11. Note that the number of samples N SWAP +samp (Eq. (27)) for the SWAP test +estimation can be changed adaptively in different BE iterations for different accuracy, which +we discuss in detail in the next section. +4.2 +Eigensolver Subroutines and Sampling Complexity +Two major quantum eigensolvers, QPE78 and VQE34 can be used in line 11 and 14 of Alg. 1 +to estimate the cost function. QPE is an exact eigensolver, where the system wave function +collapses to the exact ground state regardless of the number of evaluation qubits used. In +contrast to QPE, VQE is an approximate eigensolver and the results depends on the choice +of ansatz and the optimization algorithm used. +A crucial feature of a quantum eigensolver is its probabilistic nature, in a sense that +any measurement collapses the entire quantum state. This perspective allows us to treat a +quantum eigensolver as a sign-problem-free sampling oracle for correlated electronic structure +problems where Ref.79 provides a concrete example. +The stochastic nature also means a more careful treatment on the number of samples is +required to fully quantify any potential quantum speedup. In general, for typical iterative +mixed quantum-classical algorithms, some parameters are usually passed from one iteration +to the next, where the parameters are estimated by repeatedly sampling from a quantum +30 + +eigensolver oracle through proper measurement. This means the uncertainty on these pa- +rameters estimated from one iteration has to be small enough to avoid a divergence of the +algorithm as iteration continues. +In particular in the bootstrap embedding case, the sampling accuracy on the fragment +overlap of each iteration has to be good enough such that the uncertainty of the mismatch +passed to the next iteration will not spoil the iteration and lead to diverging results as +iterations continue. When estimating the overlap S to an accuracy ϵ naively by density +matrix tomography of individual RDM elements, it is shown under mild assumptions that +the total number of samples required (see Sec. S6 in SI) +N TMG +samp (S, ϵ, n) = O(en) +�D +ϵ2 +� +, +(26) +where n is the number of qubits on the overlapping region, and D is a system-dependent +constant as a function of the two RDMs. In contrast, the quantum matching based on SWAP +test costs +N SWAP +samp(S, ϵ) = +�1 − S2 +8 +� 1 +ϵ2, +(27) +which is independent of the size n of the overlapping region of two fragments. This demon- +strates that our quadratic quantum matching achieves an exponential speedup compared to +naive tomography of density matrices. This dramatic speedup is perhaps not that surpris- +ing because we only care about one particular observable (the overlap) instead of the full +subsystem RDMs. Therefore, if the observable can be mapped to measurement outcome of +few qubits by some quantum operations (SWAP test in this case), advantages are expected in +general. +Moreover, the dependence of N SWAP +samp(S, ϵ) on the overlap S and estimation accuracy ϵ +allows an adaptive sampling schedule to be implemented for line 11 and 14 of Alg. 1. For +example, we may use the overlap S estimated from the previous BE iteration to compute +31 + +the required N SWAP +samp in the current BE iteration. The accuracy ϵ can also be dynamically +tuned according to the error of the first term in Eq. +(25), as well as the value of the +penalty parameter λ. For example, at the beginning BE iterations, the mismatch (∆ρ or +more precisely Qquad(ρ(A) +r +; ρ(B) +r +)) is large so that a moderate ϵ suffices. As the BE iteration +proceeds, the overlap converges exponentially, therefore an exponentially decreasing ϵ has to +be used as well. A numerical value of ϵ needs be determined from case to case. +In addition, Eq. (27) suggests an interesting behavior. As the QBE algorithm proceeds +and the overlap S increases, fewer samples are needed to achieve a target accuracy. If S +approaches 1 exponentially fast as S ∼ 1 − e−γ·niter for some constant γ, then the required +number of samples for SWAP will degrees exponentially as BE iteration niter goes N SWAP +samp ∼ +e−γ·niter/ϵ2. In practice, the overlap of two subsystem can never approach 1 but saturates +to a constant 0 < c < 1 when matching is achieved, and therefore N SWAP +samp ∼ (1 − c)/ϵ2 still +obeys the 1/ϵ2 scaling generally. This, on the other hand, suggests that a larger overlapping +region is advantageous to reduce N SWAP +samp because the RDM of a larger subsystem of a pure +state will have greater purity (hence larger c) in general. +4.3 +Additional Quadratic Speedup +The above perspective of treating quantum eigensolver as oracle where some amplitude is +estimated through proper measurements allows us to achieve an additional quadratic speedup +in our quantum bootstrap embedding algorithm. The intuition is that instead of directly +measure a small quantum amplitude to accumulate enough counts to reduce the error bar, +we may use quantum algorithms to first amplify the amplitude before the measurement. +There are well-established ways of performing such amplitude amplification task via coherent +quantum algorithms.48 +In particular, in each iteration of the algorithm, it can be shown (SI Sec. +S7) that +by combining oblivious amplitude amplification and a binary search protocol, estimating +the overlap up to precision ϵ between adjacent fragments takes N SWAP+AE +samp +samples (state +32 + +preparation and SWAP tests) +N SWAP+AE +samp += +√ +2 +2 ln(2)ϵ ln2(1 +ϵ), +(28) +regardless of the overlap S. +Comparing (28) with (27), the above analysis suggests that our coherent quantum match- +ing algorithm achieves a quadratic speed up (up to a factor of polylog(1 +ϵ)) as compared to the +SWAP test based quantum matching algorithm, which is consistent with typical behavior of a +Grover-type of search algorithm. Moreover, in contrast to (26), an exponential advantage is +present with respect to the size of the overlapping region, indicating the benefit of using our +quadratic QBE algorithm for fragment matching in the presence of large overlapping region. +5 +Results and Discussions +With the theoretical foundation and algorithms discussed in previous sections, we present +numerical results in this section, demonstrating the convergence of the QBE algorithm in +Sec. 5.1 with an exact solver (at infinite sampling limit). In Sec. 5.2, we present numerical +evidence for the sampling advantage of the QBE algorithm by considering its behavior with +a finite number of samples. +We use a typical benchmark system in quantum chemistry, +hydrogen chains under minimal basis, to perform the numerical calculations. More numerical +results using approximate variational quantum eigensolvers (VQE) on a random spin model +can be found in Sec. S9 E of SI. +5.1 +Convergence of QBE in Infinite Sampling Limit +We focus on demonstrating the convergence of QBE in the infinite sampling limit by using +exact deterministic solver with the quadratic constraint in Eq. (20) and linear constraint +in Eq. (16). As a standard benchmark system for electronic structure, we perform QBE +on a H8 chain under a minimal STO-3G basis, which is fragmented into six overlapping +33 + +Figure 5: Convergence of the quantum bootstrap embedding algorithms on (a) density +mismatch and (b) energy error for the linear constraint (pink) and quadratic penalty +method (red) in the infinite sample limit for an H8 molecule. The dashed trend lines in +both panels indicate an exponential fit. +fragments each with six embedding orbitals. Fig. 5a shows the exponential convergence of +the density mismatch for an H8 molecule in both linear and quadratic constraint cases. A +similar convergence is established for a toy spin model and a perturbed H4 molecule using a +VQE eigensolver with the linear RDM matching (more details can be found in Sec. S9 E of +the SI). +To quantify how much energy error the final converged result has, Fig. 5b shows the +absolute value of the error in energy using the energy in the last (11th) iteration as a reference. +We can see that the energy errors from both the linear and quadratic constraint algorithm +exhibit similar exponential convergence as the density mismatch. Moreover, the energy in +both cases converge to the same value within 10−6 in the last iteration (not shown in the +figure). We note that the linear constraint case shows a slightly oscillatory convergence, +while the quadratic case is free of such oscillatory behavior. The fact that quadratic appears +to converge slightly faster than linear may be coincidence for the system investigated, and +34 + +10- +QBE (Linear) +Density Mismatch +QBE (Quadratic) +10-5 +10-6 +10-7 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +10- +10- +10-5 +10-6. +10-7 +10-8 +2 +0 +3 +8 +9 +10 +1 +6 +7 +Iteration Numberthe convergence rate in general depends on the optimization algorithm chosen. See Sec. S9 D +of the SI for a detailed description on definition of the energy. +5.2 +Sampling Advantage of Coherent Quantum Matching +In the previous section, we have seen that our quantum bootstrap embedding algorithm +convergence as expected in the infinite sampling limit. It is also seen (in the SI) that the +approximate VQE leads to biased behavior on the density matching. In practice, only a +finite number of samples can be collected on a quantum computer, and we will focus on theis +scenario in this section. In particular, we present numerical data demonstrating the sampling +advantage of our coherent quantum matching algorithm. Sec. 5.2.1 discusses the sampling +advantage of the quantum matching algorithm for an overlapping region of increasing size. +In Sec. +5.2.2, the additional quadratic speedup in estimating the overlap via amplitude +amplification and binary search (AE) is presented. +5.2.1 +Advantage in Fragment Overlap Size +To perform bootstrap embedding, it is usually advantageous to partition the system into +fragments with large overlapping region to increase the convergence rate, because a large +overlapping region necessarily means more information is provided to update the local po- +tential for the following BE iteration. However, a larger overlapping size also lead to an +exponentially higher sampling complexity versus the number of qubits in the overlapping +region if estimating the overlap naively from density matrix tomography as in Eq. (26). +The quantum matching algorithm implemented by a SWAP test (Fig. 1iiiq) bypass the need +for density matrix tomography, and therefore leads to a sample complexity as in Eq. (27) +independent of the size of the overlapping region. +To validate our theoretical sample complexity, a simulation of the quantum matching +algorithm with QPE as an eigensolver for two identical H4 chain is performed using a noiseless +Qiskit AerSimulator (see SI Sec. S9 C for more details) for an increasing overlap region +35 + +ranging from 2 to 4, 6, and 8 qubits (schematic in Fig. 6). In the simulation, we first +use QPE to prepare the ground state for two non-interacting H4 molecules separately. A +SWAP test is then performed on relevant qubits in the overlapping region between the two +H4 molecules. The evaluation qubits for QPE and the ancilla qubit for SWAP test are all +measured afterwards. Post-selection on the QPE evaluation qubits are performed in order to +select the ground states of H4 molecules. The SWAP test results are processed and converted +to the estimation on the overlap S. +Figure 6: Sampling complexity ratio of naive density matrix tomography (TMG) and SWAP +test versus number of qubits in the overlapping region for a target precision ϵ = 0.001 on +overlap S. The inset shows a simulated convergence of overlap (S) estimation using +quantum matching (SWAP) for the case of two overlapping qubits. Data are obtained from a +non-interacting chain of H4 (see SI Sec. S9 C for details). +The inset of Fig. 6 shows the estimated overlap S as a function of sample size (number +of eigensolver calls) in the case of two overlap qubits. +The estimated overlap converges +to the exact value (black dashed horizontal line) for roughly four million samples within +5 × 10−4 (error bar invisible for the last data point). This demonstrates the correctness of +our quantum matching algorithm. +By repeating similar estimation as described above for increasingly larger overlapping +regions, the exponential sampling advantage of the quantum matching algorithm over naive +36 + +105 +Exact +S +0.550 +亚 +Simulated +rlap +Over +0.525 +104 +des +0.500 +104 +106 +EigensolverCalls +103 +102 +2 +:3 +4 +5 +6 +7 +8 +9 +Number of Overlap Qubitsdensity matrix tomography is evident in Fig. +6. +As we can see, to achieve a constant +target precision of ϵ = 0.001 on the overlap S, the ratio between the SWAP test estimation +and the naive tomography estimation for the required number of eigensolver calls increases +exponentially as the number of qubits. +We note that in general, overlaps between density matrices are not low-rank observables, +so the sampling complexity of estimating it is likely to be high. However, more efficient +sampling schemes may exist than the naive density matrix tomography as presented in Eq. +(26). For example, by sampling the differences in the RDMs between the current and the +previous BE iterations, the sampling complexity could be much better than exponential. We +leave this for future investigation. +5.2.2 +Additional Quadratic Speedup in Accuracy +We have seen in the previous section that the quantum matching implemented by a SWAP +test shows an exponential sampling advantage in terms of the size of the overlapping region +as compared to naive density matrix tomography. However, the sample complexity in the +estimation accuracy ϵ follows the same scaling of 1/ϵ2 as classical sampling based algorithms. +As is derived in Sec. 4.3, we see that the sample complexity can be reduced to roughly 1/ϵ +with a coherent quantum matching algorithm, by combining amplitude estimation and a +binary search protocol, thus achieving a quadratic speedup. +In this section, we present +concrete numerical data demonstrating this quadratic speedup. +Fig. 7 shows that for a single BE iteration, the required number of samples (eigensolver +calls) on estimating the RDM overlap S between two adjacent fragments as a function of the +required precision on the overlap, comparing the SWAP test based quantum matching (blue) +and the coherent overlap estimation combining the SWAP test and amplitude estimation +(SWAP+AE) (red). We can see that the required number of samples increases quadratically +as the accuracy ϵ increases for the SWAP test based estimation. In contrast, the slope of the +SWAP+AE sample complexity is reduced to roughly half of the SWAP test. It is worthwhile to +37 + +Figure 7: Number of eigensolver calls required as a function of target precision at overlap +S = 0.4, comparing incoherent (blue) and coherent (red) estimation. The blue scatter +points for the incoherent are obtained from classical variational Monte Carlo estimation +and the blue dashed line shows the incoherent sample as derived in Eq. (27). The red data +points are obtain from the linear constraint convergence in Fig. 5, while the red dashed +line shows the complexity as derived in the SI. The inset plots the eigensolver calls as a +function of the overlap S for a target precision ϵ = 0.001. Note the crossover in both plots. +The coherent estimation shows a square-root advantage at high target precision. +note that this quadratic speedup is only advantageous in the high precision (small ϵ) limit, +as is evident from the existence of a crossing point in Fig. 7 (between 10−4 and 10−2), which +defines a critical ϵ∗. For ϵ < ϵ∗, SWAP+AE is favored whereas the SWAP test wins when ϵ > ϵ∗. +Moreover, the dependence of the sampling complexity on the value of the overlap S is +very different. This difference is clear from the inset of Fig. 7, comparing the SWAP (blue) +and the SWAP+AE estimation (red). In more detail, the sample complexity for the SWAP +test decreases quadratically as the overlap S approaches 1 (Eq. (27)). As a comparison, +the SWAP+AE stays roughly a constant for the coherent quantum matching ((28)), because +the amplitude amplification process used in the present work is agnostic to the value of the +amplitude (overlap S), i.e., oblivious amplitude amplification.80,81 The slight drop in sample +complexity in the SWAP+AE approach (red line, inset of Fig. 7) is due to the discrete bit +representation of S (see Sec. S7 B of SI for details). The different scaling on S between +these two algorithms leads to a crossover of the sampling complexity at roughly S = 0.8 for +38 + +1015 +100000 +Number of Eigensolver Calls +1012 +50000 +0 +109 +0000010000 +0.0 +0.5 +1.0 +Overlap S +106 +103 +SWAP+AE +SWAP +100 +10-8 +10-6 +10-4 +10-2 +100 +Target Precisiona target precision of ϵ = 0.001. This crossover suggests again that the plain SWAP test is +advantageous for a large overlap, while amplitude estimation works better for small overlap +S. +In addition, as mentioned in the previous section, as the bootstrap embedding iteration +proceeds, the exponential convergence of the density mismatch (overlap S) suggests the need +for an exponentially increasing accuracy ϵ on the overlap estimation. This further means +the number of samples per iteration in the SWAP test should increases exponentially as the +the number of iterations. Similarly, SWAP+AE achieves a square-root speedup in the total +sample numbers (remains exponential). We note that there may exist ways of sampling the +overlap in the current BE iteration normalized by the previous BE iteration to accelerate +this requirement on a large number of samples, which we leave for future investigation. +6 +Conclusion and Outlook +In conclusion, we have developed a general quantum bootstrap embedding method to find +the ground state of large electronic structure problems on a quantum computer by taking +advantage of quantum algorithms. We formulated the original electronic structure problem as +a optimization problem using a quadratic penalty to impose matching condition of adjacent +fragments. A coherent quantum matching algorithm based on the SWAP test achieves efficient +matching with an exponential sampling advantage compared to naive RDM tomography. +By estimating the amplitude that encodes the overlap information combing an amplitude +amplification and binary search protocol, an additional quadratic speedup is achieved. In +addition, an adaptive sampling scheme is used based on previous overlap information and +the desired target accuracy to improve the sampling efficiency. +We demonstrate the performance of the QBE algorithm using a linear hydrogen molecule +under minimal basis. +Our QBE algorithm is shown to achieve exponential convergence +in density mismatch and energy error similar to classical bootstrap embedding. However, +39 + +instead of the exponential cost of an exact classical solver (full configuration interaction), +quantum eigensolvers such as quantum phase estimation can solve the fragment electronic +structure exactly without incurring the exponential cost. +While we have made progress toward solving electronic structure problems employing +quantum resources in bootstrap embedding, there are several open questions to explore in +the future. At the algorithmic level, it is important to reconstruct the total system density +matrices from subsystem ones82 in order to compute observables other than the energy. +Ideally, quantum algorithms that can perform the reconstruction process would be desired. +Moreover, we have established how the bootstrap embedding potential can affect the system +energy including the excited states in Eq. (23). Future works on developing a QBE algorithm +targeting excited states83 or finite temperature electronic structures58,84,85 would be of great +interest. Alternative constraint optimization methods such as the augmented Lagrangian +method can also be explored to achieve potentially better convergence.16 +In addition, the idea of quantum matching proposed in the present work could also be +exploited further in other embedding theories to harness quantum computers and resources, +including but not limited to embedding schemes based on wave functions, density matrices, +and Green’s functions.9 In these contexts, it is likely that more sophisticated quantum prim- +itives and algorithms could accomplish quantum matching more efficiently than the simple +SWAP test we employ. For example, it is possible that higher order matching, or matching +of derivatives, could be accomplished quantum-mechanically, thus side-stepping sampling +noise. +More broadly, these quantum embedding theories and algorithms enabled by quantum +computation resources open new possibilities in chemistry, physics, and quantum informa- +tion. For example, large molecular systems in catalysis86,87 and protein-ligand binding com- +plexes88,89 likely can be simulated at a much higher accuracy by combining state-of-the- +art quantum and classical computational resources in embedding properly. In condensed +matter and material science, quantum bootstrap embedding may be adapted to periodic +40 + +systems20,90,91 for quantum material design92 and probing phase diagrams of various lattice +models93 close to the thermodynamic limit. +Finally, from a viewpoint of quantum information, the concept of embedding is closely +related to entanglement. Understanding the connection between the performance of quan- +tum embedding algorithms and fragment-bath entanglement entropy may provide a general +way to describe and understand the complexity of chemical and physical problems from a +quantum information perspective.94–96 Current quantum computers are small – we believe +our quantum bootstrap embedding method provides a general strategy to use multiple small +quantum machines to solve large problems in chemistry and beyond. We look forward to +future development in these directions. +Acknowledgement +YL thanks Di Luo, Minh Tran, and Daniel Ranard for helpful discussions. The work on +analysis and numerical simulation was supported by the U.S. Department of Energy, Office +of Science, National Quantum Information Science Research Centers, Co-Design Center for +Quantum Advantage, under contract number DE-SC0012704. +The conceptual algorithm +development was supported in part by NTT Research. +Supporting Information Available +Additional theoretical and numerical details. +References +(1) Fukui, K.; Yonezawa, T.; Shingu, H. A molecular orbital theory of reactivity in aromatic +hydrocarbons. The Journal of Chemical Physics 1952, 20, 722–725. +41 + +(2) Parr, R. G.; Yang, W. Density functional approach to the frontier-electron theory of +chemical reactivity. Journal of the American Chemical Society 1984, 106, 4049–4050. +(3) Greeley, J.; Nørskov, J. K.; Mavrikakis, M. Electronic structure and catalysis on metal +surfaces. Annual Review of Physical Chemistry 2002, 53, 319–348. +(4) LeBlanc, J. P.; Antipov, A. E.; Becca, F.; Bulik, I. W.; Chan, G. 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Quantum Information Theory; Cambridge University Press, 2013. +52 + diff --git a/49AzT4oBgHgl3EQffvx7/content/tmp_files/load_file.txt b/49AzT4oBgHgl3EQffvx7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d31c805533effb3eb62cb841e3ae0fb7c0a7bc3 --- /dev/null +++ b/49AzT4oBgHgl3EQffvx7/content/tmp_files/load_file.txt @@ -0,0 +1,1700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf,len=1699 +page_content='Bootstrap Embedding on a Quantum Computer Yuan Liu,∗,† Oinam R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Meitei,‡ Zachary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Chin,† Arkopal Dutt,¶ Max Tao,† Troy Van Voorhis,∗,‡ and Isaac L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Chuang†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='§ †Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Co-Design Center for Quantum Advantage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' USA ‡Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' USA ¶Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' USA §Department of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' USA E-mail: yuanliu@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' tvan@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='edu Abstract We extend molecular bootstrap embedding to make it appropriate for implementa- tion on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' This enables solution of the electronic structure problem of a large molecule as an optimization problem for a composite Lagrangian governing fragments of the total system, in such a way that fragment solutions can harness the capabilities of quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' By employing state-of-art quantum subroutines including the quantum SWAP test and quantum amplitude amplification, we show how a quadratic speedup can be obtained over the classical algorithm, in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Utiliza- tion of quantum computation also allows the algorithm to match – at little additional computational cost – full density matrices at fragment boundaries, instead of being limited to 1-RDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Current quantum computers are small, but quantum bootstrap 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='01457v1 [quant-ph] 4 Jan 2023 embedding provides a potentially generalizable strategy for harnessing such small ma- chines through quantum fragment matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 1 Introduction Determining the ground state of large-scale interacting fermionic systems is an important challenge in quantum chemistry, materials science, and condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Just as electronic properties of molecules underpin their chemical reactivity,1–3 phase diagrams of solid state materials are also determined to a large degree by their ground state electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='4–6 However, close to exact solution to the time-independent Schrodinger equation of a practical many-electron system remains a daunting task because the dimension of the underlying Hilbert space grows exponentially with the number of orbitals, and the computa- tional resources required to perform calculations over such a large space can quickly exceed the capacity of current classical or quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' One promising approach to fit a large electronic structure problem into a limited amount of computational resources is to break the original system into smaller fragments, where each fragment can be solved individually from which a solution to the whole is then ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='7–9 Efforts along this direction have successfully led to various embedding schemes that significantly expand the complexity of the systems solvable using classical computa- tional resources, such as density-based embedding theories,10,11 density-matrix embedding theories (DMET),12–16 various Green’s function embedding theories6,17–21 and the bootstrap embedding theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='22–24 The essence of such embedding-based methods is to add an additional external potential to each fragment Hamiltonian and then iteratively update the potential until some conditions on certain observables of the system are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Nevertheless, due to the significant cost in solving the fragment Hamiltonian itself as the fragment size increases, the applicability of such methods are limited to relatively small fragments, which may lead to incorrect predictions in systems with long-range correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='25 While approximate fragment 2 solvers such as the coupled-cluster theory or many-body perturbation theory have greatly enhanced the applicability of such embedding methods at a reduced cost,26–28 these approx- imations tend to fail for strongly correlated systems due to limited treatment of electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In addition, because of limitations on computing k-electron reduced density matrices (k-RDMs for k > 2), embedding and observable calculations beyond 2-RDM are difficult in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Quantum computers are believed to be promising in tackling electronic structure prob- lems more efficiently,29 despite the possibility of an exponential speedup still being unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='30 One natural idea to circumvent the problems of classical eigensolvers is to use a quantum computer to treat the fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' By mapping each orbital to a constant (small) number of qubits, the exponentially large (in the number of orbitals) Hilbert space of an interacting fermionic system can be encoded in only a polynomial number of qubits and terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Indeed, quantum eigensolvers such as the quantum phase estimation (QPE)31 algorithm has been proposed to achieve an exponential advantage given a properly prepared input state32 with non-exponentially small overlap with the exact ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' More recently, various variants of the variational quantum eigensolver (VQE)33–37 have been demonstrated experimentally on NISQ devices to achieve significant speedup without sacrificing accuracy as compared to classical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Moreover, k-RDMs (for any k) can be measured through quantum eigensolvers38,39 that may circumvent the difficulty encountered on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' To take the full advantage of these quantum eigensolvers within the embedding frame- work,18,40–44 two open questions immediately arise as a result of the intrinsic nature of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Firstly, the wave function of a quantum system collapses when measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' This means any measurement of the fragment wave function is but a statistical sample (akin to Monte Carlo methods), and many measurements are needed to obtain statistical averages with sufficiently low uncertainty in order to achieve a good matching condition for the em- bedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Secondly, the best way to perform matching between fragments using results from quantum eigensolvers is not clear, and most likely a new approach needs to be formulated 3 to match fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Admittedly, it would be straightforward to first estimate the density matrices by collecting a number of quantum samples and then use the estimated density ma- trices to minimize the cost function as in classical embedding theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='12,22 But this approach would be very costly especially given the increasing number of elements in qubit reduced density matrices (RDMs) that need to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='45 Could there be a quantum method for matching, as opposed to a statistical sampling-based classical approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We address the two challenges by providing a quantum coherent matching algorithm and an adaptive sampling schedule, leading to a quantum bootstrap embedding (QBE) method based on classical bootstrap embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='22 Instead of matching the RDM element-by-element, the quantum matching algorithm employs a SWAP test46,47 to match the full RDM between overlapping regions of the fragments in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Moreover, the quantum amplitude estima- tion algorithm48,49 allows an extra quadratic speedup to reach a target accuracy on estimating the fragment overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In addition, the adaptive sampling changes the number of samples as the optimization proceeds in order to achieve an increasingly better matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The present work invites a viewpoint of treating quantum computers as coherent sampling machines which have three major advantages, as compared to their classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' First, the exponentially large Hilbert space provided by a quantum computer allows more efficient exact ground state solver (QPE) than their classical counterpart (exact diagonal- ization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Second, in the case of truncation for seeking approximate solutions, the abundant Hilbert space of quantum computers enable more flexible and expressive variational ansatz than classical computers, leading to more accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Third, the coherent nature of quantum computers allows sampling to be performed at a later stage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' after quantum amplitude amplification of matching conditions to extract just the feedback desired, instead of having to read out full state of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2 overviews bootstrap embedding method at a high level and analyzes its scaling on classical computers, in order to motivate the need for bootstrap embedding on quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' This section serves to set the 4 notation and baseline of comparison for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 3 presents the theoretical framework of quantum bootstrap embedding in detail as constraint optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 4, we give details of the QBE algorithm to solve the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 5, we apply our methods to hydrogen chains under minimal basis where both classical and quantum simulation results are shown to demonstrate the convergence and sampling advantage of our QBE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We conclude the paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 6 with prospects and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2 Ideas of Bootstrap Embedding The idea of Bootstrap Embedding (BE) for quantum chemistry has recently led to a promis- ing path to tackle large-scale electronic structure problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='22,23,50 In this section, we establish the terminology and framework that will be used in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We first briefly review BE and outline the main framework of BE for computation on a classical computer in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='2 for non-chemistry readers, to set up the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We then begin presenting new material by discussing typical behavior and computational resource requirements for BE on classical computers in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='3, which leads to the quest for performing BE on a quantum computer in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='1 Fragmentation and Embedding Hamiltonians To provide a foundation for a more concrete exposition of the bootstrap embedding method, we first establish some rigorous notation for discussing molecular Hamiltonians and their associated Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We will work with the molecular Hamiltonian under the second quantization formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Specifically, given a particular molecule of interest, define O = {φµ | µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' , N} to be an orthonormal set of single-particle local orbitals (LOs), where N is the total number of orbitals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' in this work, these LOs are generated through L¨owdin’s symmetric orthogonalization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='51 The full Hilbert space H for the entire molecular 5 system is thus given by H = F(O), where F(O) denotes the Fock space determined by the LOs in the set O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Further define the creation (annihilation) operator c† µ (cµ) which creates (annihilates) an electron in the LO φµ, the molecular Hamiltonian is written in the second-quantized notation ˆH = N � µν=1 hµνc† µcν + 1 2 N � µνλσ=1 Vµνλσc† µc† νcσcλ (1) where hµν and Vµνλσ are the standard one- and two-electron integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Note that the number of terms in the full molecular Hamiltonian ˆH scales polynomially with the total number of orbitals N, but the dimension of H scales exponentially with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Clearly, for large N, it will become prohibitively expensive to directly compute the exact full ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' To circumvent this issue, we divide the full molecule into multiple smaller fragments, each equipped with its own “embedding Hamiltonian” which contains a number of terms that only scales polynomially with the number of orbitals in the fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Given that there are potentially far fewer orbitals in each fragment than in the whole molecular system, computing the ground state of each fragment’s embedding Hamiltonian can be significantly less expensive than computing the ground state of the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Furthermore, using the bootstrap embedding procedure to be described later, the ground states of individual fragments can, to a high degree of accuracy, be algorithmically combined to recover the desired electron densities prescribed by the exact ground state of the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Thus, this combination of fragmentation and bootstrap embedding can be used to reconstruct the full molecular ground state more efficiently than by direct computation alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We now briefly review the construction of embedding Hamiltonians for each fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Consider a single fragment associated with a label A, without loss of generality, define O(A) = {φµ | µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' , NA} with NA ≤ N to be the set of LOs contained in fragment A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' we will refer to O(A) as the set of fragment orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Note that O(A) ⊆ O, the set of LOs for the entire molecular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The construction of the embedding Hamiltonian ˆH(A) for 6 fragment A begins with any solution of the ground state of the full system ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' For simplicity, the Hartree-Fock (HF) solution |ΦHF⟩ is often used because it is easy to obtain on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' By invoking a Schmidt decomposition, we can write |ΦHF⟩ with the following tensor product structure for ∀ A |ΦHF⟩ = � NA � i=1 λ(A) i |f (A) i ⟩ ⊗ |b(A) i ⟩ � ⊗ |Ψ(A) env⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' (2) In the above decomposition, the |f (A) i ⟩ represent single-particle fragment states contained in the Fock space F(O(A)) of fragment orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' On the other hand, the |b(A) i ⟩ and |Ψ(A) env⟩ represent Slater determinants contained in the “environment” Fock space F(O \\ O(A)) of the N − NA orbitals not included in the fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The key difference between the single environment state |Ψ(A) env⟩ and the various “bath” states |b(A) i ⟩ is that the bath states |b(A) i ⟩ are entangled with the fragment states |f (A) i ⟩ while |Ψ(A) env⟩ is not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' this entanglement is quantified by the Schmidt coefficients λ(A) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Crucially, since the HF solution is used, the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' (2) only has NA terms (as opposed to 2NA for a general many-body wave function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Denote the collection of the NA entangled bath orbitals as O(A) bath = {βµ |µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' , NA}, where each of the LOs βµ are linear combinations of the original LOs not included in the fragment, βµ ∈ Span{O \\ O(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Furthermore, we denote the Fock space that corresponds to this set of entangled bath orbitals as F(O(A) bath).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' This tensor product structure of |ΦHF⟩ allows us to naturally decompose the Hilbert space H for the full molecular system into the direct product of two smaller Hilbert spaces, namely H = H(A) ⊗ H(A) env, (3) where H(A) = F(O(A)) ⊗ F(O(A) bath) (4) 7 is the active fragment embedding space and H(A) env contains the remaining states, including |Ψ(A) env⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Note that since both sets O(A) and O(A) bath have size NA, the fragment Hilbert space H(A) is a Fock space spanned of just 2NA single-particle orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The core intuition mo- tivating this decomposition is that, in the exact ground state of the full system, states in H(A) env are unlikely to be strongly entangled with the many-body fragment states (consider the approximate HF ground state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' (2), where they are perfectly disentangled);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' therefore, in a mean-field approximation, it is reasonable to entirely disregard the states in H(A) env when calculating the ground state electron densities on fragment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Following this logic, we can define an embedding Hamiltonian ˆH(A) for fragment A only on the 2NA LOs in H(A), which will have the form ˆH(A) = 2NA � pq h(A) pq a(A)† p a(A) q + 1 2 2NA � pqrs V (A) pqrsa(A)† p a(A)† q a(A) s a(A) r , (5) given some creation and annihilation operators a(A)† p and a(A) p , which respectively create and annihilate electrons in orbitals from the combined set O(A) ∪ O(A) bath for H(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The new one- and two- electron integrals h(A) pq and V (A) pqrs can be computed by projecting ˆH into the smaller Hilbert space H(A) (consult the Supporting Information (SI) Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' S1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Note that since we can choose 2NA ≪ N, the ground state of this embedding Hamiltonian can be solved at a significantly reduced cost when compared to that of the full system Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We are hence prepared to generate an embedding Hamiltonian for any arbitrary frag- ment of the original molecular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' However, the ground state electron densities of the fragment embedding Hamiltonian are unlikely to exactly match those of the full system Hamiltonian because, as mentioned above, the embedding process may neglect some small (but nonzero) entanglement of the fragment orbitals with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Because we can expect interactions in the molecular Hamiltonian to be reasonably local, we anticipate that the electron densities on orbitals near the edge of the fragment (those closest to the “envi- ronment”) will deviate most significantly from their true values, while electron densities on 8 orbitals toward the center of the fragment will be most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' To improve the accuracy of the fragment ground state wave function near the fragment edge, we employ the technique of bootstrap embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Broadly speaking, we first divide the full molecule into overlapping fragments such that the edge of each fragment overlaps with the center of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 1i illustrates this fragmentation strategy: for example, we see that the edge of fragment A (labeled as orbital 3) coincides with the center of fragment B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We then apply additional local potentials to the edge sites of each fragment to match their electron densities to those on overlapping center sites of adjacent fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Because we expect the electron densities computed on the center sites to be closer to their true values, these added local potentials should improve the accuracy of each fragment wave function near the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In the next section, we will formalize this edge-to-center matching process rigorously and discuss its implementation on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='2 Matching Electron Densities: an Optimization Problem As mentioned in the previous section, we intend to correct the electron density error near a fragment’s edge by applying a local potential to the edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' this local potential serves to match the edge electron density of the fragment to the center electron density of an adjacent overlapping fragment, which we expect to be more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In principle, to achieve an exact density matching, all k-electron reduced density matrices (k-RDM, for any k) on the overlapping region have to be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' However, in practice, such matching beyond the 2- RDM is difficult on a classical computer due to the mathematical challenge that the number of terms in k-RDM in general increases exponentially as k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In addition, almost all electronic structure codes available on classical computers are programmed to deal with only 1- and 2-RDMs, despite the importance of k-RDMs (k > 2) for computing observables such as entropy and other multi-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='52 Due to this reason, the discussion of density matching process in classical BE in this section will be based on 1-RDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' We note that the matching process applies similarly if k-RDMs are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 9 Figure 1: Schematic of bootstrap embedding on classical (left, blue arrows) and quantum (right, red arrows) computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The arrows indicate BE iterative loops that are used to optimize the corresponding objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' Starting from panel (i) (upper center), the original system is first broken into overlapping fragments (Fragmentation), where each fragment is solved using a classical (iic) (upper left) or quantum eigensolver (iiq) (upper right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' In classical matching, the 1-electron reduced density matrices (1-RDM) on the overlapping sites of adjacent fragments are used to obtain the matching condition (iiic) (lower left), while in the quantum case a coherent matching protocol based on SWAP tests of overlapping sites combined with a single qubit measurement (iiiq) (lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' The matching results are then used by classical computers to generate the bootstrap embedding potential VBE (iv) (lower center) and the updated fragment embedding Hamiltonian Hemb + VBE (back to panel (i) in order to minimize a target objective function L in both classical and quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' 10 (iic) (i) Fragmentation (ig) Quantum Eigensolver Classical TTOTOTTOOTOTO (QPE, VQE, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=') Eigensolver 000000 Frag A Hemb+VBE 0-010!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQffvx7/content/2301.01457v1.pdf'} +page_content=' FragA FCI CCSD 000000 FragB @10i0 Frag B VMC FragA Frag C 000000 010010 Frag B 4 FragD 000000 Classical BE Quantum BE 1-RDMs VBE RDMs L=(Hemb)+Qx L = 0, (i.e. at the below-critical +level,) is already simple, see [KL93, Proposition 2.12] in the simply-laced case. +However, for k > h∨, the affine VOA V−h∨+k(g) living on the “positive” bound- +ary is not simple: It is very well understood [KK79], has the proper maximal +ideal, and the simple quotient denoted L−h∨+k(g). It was argued in [CDG20] +that the nonperturbative boundary monopoles implement this simple quotient, +at least on the “positive” boundary, turning V−h∨+k(g) into L−h∨+k(g). +Thus we expect the exact chiral algebra (for k > h∨) to contain V−h∨−k(g)⊗ +L−h∨+k(g), not V−h∨−k(g) ⊗ V−h∨+k(g), which already differs from Dk[GC]. +Furthermore, the 3D bulk is expected to admit only finitely many line operators +in the IR. Indeed, it is gapped (at least if the interval is long enough) and given +by the Gk−h∨ CS for levels3 k > h∨, which only admits finitely many Wilson +lines labeled by the integrable representations of L−h∨+k(g) [Wit89; Kac95]. +We, therefore, expect the exact VOA to be, roughly, V−h∨−k(g) ⊗ L−h∨+k(g) +extended by such a finite set of bimodules. This appears to be significantly +different from Dt[GC] (e.g., the latter is an infinite extension for generic t). +In this paper, we fully address the abelian case, G = U(1), and give par- +tial results on the nonabelian G, including the perturbative treatment outlined +earlier. We also speculate on the kind of nonperturbative corrections in the non- +abelian GC NLSM that are expected to modify Dk[GC]. We leave the detailed +study of such nonperturbative effects for future work. +In the abelian case, GC = C∗, so the IR regime is described by the NLSM +into C∗, which was analyzed previously in [DGP17]. This theory of course has a +C∗-valued βγ system for its perturbative chiral algebra Dt[C∗]. We think of it ei- +ther multiplicatively, i.e., γ is C∗-valued, or additively, i.e., �γ ∼ �γ +2πiR, where +R is the radius of the compact boson (to be determined by the CS level). Since +the theory is free, the distinction between perturbative and non-perturbative +3The IR behavior at arbitrary k is described in [Bas+18], see also [AHW82; Oht99; GKS18]. +5 + +is slightly formal. In the case of the C∗ model, the natural nonperturbative +completion amounts to including twisted sectors into the theory, which corre- +spond to windings around the nontrivial one-cycle in the target. The twist fields +are vortex-like disorder operators, whose 3D origin is, expectedly, the boundary +monopoles. The presence of such sectors extends the βγ system by the so-called +spectrally flowed modules [RW14; AW22]. We show that this results in the lat- +tice VOA for the simplest Narain lattice [Nar86; NSW87], i.e., Z2 ⊂ R2 with the +scalar product whose Gram matrix is (0 1 +1 0). In addition to showing this, we also +provide the 3D perspective, where each boundary supports a 1D lattice VOA +(abelian WZW [DGP18]), and Wilson lines extend them by the bi-modules. +In total, we find three presentation of the same VOA: (1) as a Narain lattice +VOA; (2) as an extension of the βγ system; (3) as an extension of the abelian +WZWk ⊗ WZW−k by its bimodules. +The nonabelian GC NLSM is interacting, which makes the non-perturbative +corrections to the perturbative answer Dk[GC] much more challenging to quan- +tify. We are certain from the previous discussion, however, that such corrections +must be present. The known instanton effects in 2D N = (0, 2) theories are vor- +tices that are best understood in the gauged linear sigma model case [SW95; +BS03; BP15] (see also [LNS00] for the N = (2, 2) case). In the NLSMs they +are captured by the worldsheet wrapping compact holomorphic curves in the +target [Din+86; Din+87; BW06], which is notoriously hard to compute except +for the simplest models [TY08]. In our case, however, GC does not support any +compact holomorphic curves, which basically implies that there must be some +novel nonperturbative corrections at play. They correspond to the boundary +monopoles that exist in the parent 3D gauge theory, and can be described as new +“noncompact” vortices in 2D. Indeed, monopoles are labeled by the cocharacters +eibϕ : U(1) → G, up to conjugation, which are complexified to zb : C∗ → GC. +This allows to define a natural vortex singularity for the NLSM field φ(z, z): +φ ∼ zb, +as z → 0. +(1.2) +Since it is meromorphic, it will define a half-BPS defect. +Dynamics in the +presence of such defects is expected to modify the chiral algebra Dt[GC] ap- +propriately. We will explore this elsewhere, while here we only focus on the +perturbative aspects when G is non-abelian. +Another subtle point is that the presence of such disorder operators, and +hence the non-perturbative corrections, in principle, depends on the UV com- +pletion. +We will assume that the 3D gauge theory with compact G admits +monopoles.4 Without them, the Dirichlet boundary on the “positive” end would +appear in tension with unitarity [DGP18; CDG20] (the same issue is not present +on the “negative” boundary since it supports a non-compact CFT). Thus the +NLSM originating from the gauge theory on the interval must admit the vortex- +type disorder operators and the corresponding non-perturbative phenomena. On +the other hand, if one completes the GC NLSM in the UV into the LG model as +in [DL22], there is no room for any vortex defects. In this case the Dt[GC] is con- +jectured to give the exact chiral algebra. Purely from the viewpoint of NLSM, +it is conceivable that the knowledge of metric on the target (which is usually +ignored in the BPS calculations, but which was computed in [DL22],) allows one +4Which has far-reaching consequences for its dynamics, as, e.g., in Polyakov’s argument +for confinement in 3D [Pol77]. +6 + +to tell apart the cases with and without the nonperturbative corrections. This +question is likewise left for the future. +Motivation via VOA[M4]. +One of the motivations behind this work is to de- +velop a toolkit for computing VOA[M4] [DGP17].5 Recall that the VOA[M4], or +more precisely VOA[M4, g], is defined as the chiral algebra in the Q+-cohomology +of the 2D N = (0, 2) theory T[M4, g], which is obtained by the twisted compact- +ification of the 6d (2, 0) SCFT (of type g) on the four-manifold M4 [GGP16]. +Let us consider a class of four-manifolds M4 admitting a metric with S1 +isometry, such that the S1 action is not free. Note that when the S1 action +is free, the four-manifold is an S1 bundle over some smooth three-manifold +M3 (which is a relatively tame class of four-manifolds), and it is conceptually +clear how the dimensional reduction simplifies (first reduce on S1 and then +on M3). Of course this is still an interesting and nontrivial problem, but our +motivation comes from the opposite case, when the S1 action is not free. Some +examples of such four-manifolds include but are not limited to: (1) Σg × S2, +where Σg is an arbitrary genus-g surface, and the S1 action rotates S2, which +is equipped with an S1-invariant “sausage” metric; (2) the unique nontrivial +sphere bundle over the Riamann surface of genus g, denoted as Σg �×S2; (3) +CP2 with the standard Fubini-Study metric; (4) Hirzebruch surface, given by +the connected sum CP2#CP +2, which is actually isomorphic to the nontrivial +sphere bundle over a sphere, S2 �×S2; (5) four-sphere S4. In fact, the general +class of such four-manifolds is quite well understood, at least in the simply +connected compact case. It was shown by Fintushel [Fin77; Fin78] that if a +simply connected M4 admits an S1 action (not necessarily an isometry), it must +be a connected sum of some number of S2 × S2, S4, CP2 and CP +2. If S1 is an +isometry and the corresponding simply-connected four-manifold is, in addition, +non-negatively curved, [SY94] proved (see also [HK89]) that it belongs to the +list {S4, CP2, S2 × S2, CP2#CP +2, CP2#CP2}. If we only allow codimension-2 +fixed loci, then the list is even shorter: {S2 × S2, CP2#CP +2}. Such lists may +appear utterly specialized, however, these manifolds present certain interest to +us in view of conjectures in [FG20], which we aim to check in the future work. +We may consider the twisted compactification on a manifold with S1 isom- +etry in two steps: (1) first reduce the 6D theory along the S1 orbits; (2) then +perform reduction along the remaining quotient space M4/U(1). The advantage +of this procedure is that the first step yields a relatively simple and concrete re- +sult – the maximal 5D SYM (MSYM) with gauge group G (the simply-connected +Lie group whose algebra is g), albeit placed on some curved space with bound- +aries and, possibly, defects. For example, for M4 = Σg × S2, reducing along +the parallels of S2 gives the Σg × I geometry, where I is an interval with the +principal Nahm pole boundary conditions at both ends [CDT13]. Further re- +duction of the 5D MSYM along Σg with the topological twist simply gives a 3D +N = 4 SYM with g adjoint hypermultiplets. Thus we end up with the former +3D theory on the interval, with the (0, 4) Nahm pole boundary conditions at +both ends. In this case, the resulting effective 2D theory in the IR has (0, 4) +SUSY [PSY16]. Other examples would lead to the interval reductions of vari- +5A few recent appearances of the interval compactifications in related contexts include +[GR19; PR18; DP19]. +7 + +ous 3D N = 2 gauge theories with matter and CS levels, which would flow to +N = (0, 2) theories in the 2D limit. +We will explore various such examples in the future work [DL23]. However, +it is natural to start with the most basic 3D N = 2 gauge theory, that is the pure +SYM, and study the interval VOA in this case. This is one of the underlying +motivations for the current paper. +The rest of this paper is structured as follows. +In Section 2 we review +the necessary background material. Then we move on to computing the interval +compactification chiral algebra in the 3D N = 2 SYM theory with N = (0, 2) +Dirichlet boundaries in Section 3. In Section 4 we compute the same chiral +algebra from the 2D perspective and discuss some issues. In the abelian case, we +end up with three different presentations of the same VOA. In the nonabelian +case, we make general statements when possible, but mostly work with the +G = SU(2) example. Then we finish with some open questions and speculations, +and conclude in Section 5. +2 +Basics +In this section, we set up the conventions and briefly review the background +material, including the holomorphic-topological (HT) twist. In particular, we +discuss the 3D N = 2 supersymmetric theory and its twisted content. +We +will describe protected sectors, namely, the cohomology of a Q+ supercharge in +3D and 2D theories. A βγ system will be briefly discussed as well. Also the +connection between the cohomology of N = (0, 2) theories and the Čech coho- +mology of the βγ system is expounded upon, as it will be one of the important +computational tools later. +Conventions: +We consider R2 × I with Euclidean signature and with coor- +dinates xµ on R2 and t ∈ [0, L] on I. We choose γi +αβ matrices to be the Pauli +matrices σi, and the antisymmetric symbol ϵ12 = ϵ21 = 1 to lower and raise +indices [IS13]. +2.1 +Basic Supersymmetry +In Euclidean 3D space spinors lie in a 2-dimensional complex representation of +SU (2) and the N = 2 supersymmetry algebra takes the following form: +� +QI, QJ� += δIJγµPµ. +(2.1) +By defining a new combination of supercharges Q = Q1+iQ2 and Q = Q1−iQ2, +one can obtain the following conventional form of the superalgebra: +� +Q, Q +� += 2γµPµ. +(2.2) +Note that the supercharges are not conjugate to each other in Euclidean sig- +nature contrary to Minkowski space, where minimal representations are real +(Majorana). This algebra admits a U (1)R-charge, which is an automorphism +of this algebra and acts by rotating Q-charges. Operators also have a charge J0 +8 + +with respect to Spin(2)E rotation parallel to boundaries. Let us also define the +combination J := R +2 − J0, then all charges can be summarized by the following +table: +Q+ +Q+ +Q− +Q− +dz +dz +U(1)R +1 +−1 +1 +−1 +0 +0 +Spin(2)E +1 +2 +1 +2 +− 1 +2 +− 1 +2 +1 +−1 +U(1)J +0 +−1 +1 +0 +−1 +1 +In what follows, we will be considering the cohomology of Q = +def Q+. The Pz +is the only Pµ which is not Q-exact. It makes our algebra into an algebra with +only holomorphic dependence on the coordinates. We would consider boundary +conditions that preserve a (0, 2)-part of the supersymmetry algebra generated +by Q+ and Q+. We also want to leave U (1)R unbroken in the bulk and on the +boundary. +The only relevant 3D N = 2 multiplet for this paper is a vector multiplet +for some gauge group G: +V3D = θσmθAm + iθθσ − iθ2θλ − iθ +2θλ + 1 +2θ2θ +2D3d +(WZ gauge) , +where all the fields lie in the Lie algebra g = Lie(G). It consists of a connection +Am, a real scalar σ, a complex fermion λα, and a real auxiliary field D3d. We +can also define a covariant superfield +Σ3D = − i +2ϵαβDαDβV3d = σ − θλ + θλ + θγµθϵµνρF νρ + iθθD + . . . +that satisfies DαDαΣ3D = D +αDαΣ3D = 0. +2.2 +Holomorphic-topological twist +In this section, we review some formulas of the HT-twisted formalism [Aga+17; +CDG20]. The Q-cohomology of operators of the twisted theory and physical +theory are the same. The convenience of this formalism is that some calculations +have only a finite number of Feynman diagrams. +The twisted formalism is reviewed nicely in [CDG20, Section 3.2]. Let us +consider a dg-algebra: +Ω• = C∞ � +R3� +[dt, dz] , +(2.3) +where the multiplication is the multiplication of differential forms. One also +needs to consider forms with values in the k-th power of the canonical line +bundle in the z-direction: +Ω•,k = Ω• ⊗ Kk = C∞ � +R3� +[dt, dz] dzk. +(2.4) +There is cohomological charge R, which is related to the original R-charge +in the physical theory by adding a ghost charge to it, and a twisted spin charge +J. In the holomorphic-topological twisted 3D N = 2 theory the fields can be +organized into the following BV superfields: +A = c + (At dt + Az dz) + B∗ +zt dz dt ∈ Ω• ⊗ g [1] , +B = +� +B + A∗ +µ dxµ + c∗ dz dt +� +dz ∈ Ω•,1 ⊗ g∗, +(2.5) +9 + +where the superfields A and B are obtained from the vector multiplet, and we +also introduced ghosts. For example, the field +A := Az dz + At dt +is just a connection with complexified At = At − iσ and ordinary Az. The +field B ≡ Bz is identified with +1 +g2 Fzt = +1 +g2 Fzt + . . . on shell. +The bracket +[1] indicates a shift of cohomological degree by one. The forms dt and dz are +treated as Grassmann variables, so they anticommute with fermionic fields, and +superfields can be regarded as either bosonic or fermionic. +The action of the Q-charge in the twisted formalism can be written as follows: +QA = F(A), +QB = dAB − +k +2π∂A. +(2.6) +Here the differentials are defined as follows: +dA = d′ − iA, +d′ = dt ∂t + dz ∂z, +∂ = dz ∂z +(2.7) +and the curvature is +F(A) = id2 +A = d′A − iA2. +(2.8) +It will be also useful to keep in mind the following tables of the R and J charges +of the operators: +c +At +Az +R +1 +0 +0 +J +0 +0 +-1 +B +A⋆ +t +A⋆ +z +c⋆ +R +0 +-1 +-1 +-2 +J +1 +1 +0 +0 +Table 1: +The charges of the fields. +The following charges are assigned to the differential forms: +dt +dz +dz +R +1 +1 +0 +J +0 +1 +-1 +Table 2: +The charges of the differential forms. +2.3 +βγ System +In this section we review the βγ system, or as it is usually called in mathematical +literature, a sheaf of chiral differential operators. All formulas can be found in +[GMS99; GMS01; Nek05; Wit07] +Classically, consider a complex manifold M, a map γ : Σ → M, and a (1, 0)- +form β on Σ with values in the pullback γ∗(T ∗M), governed by the following +action: +� +Σ +βi∂γi, +(2.9) +where γi and βi are the holomorphic components of γ and β, respectively. +10 + +Quantum mechanically, the situation is more interesting as we want to pre- +serve the OPE’s locally. On each patch we have the usual βγ system with the +OPE: +γi (z) βj (w) ∼ δi +j +dw +z − w, +(2.10) +which in the physics notation yields: +� +γi +n, βj k +� += δi +jδn+k,0. +(2.11) +The normal ordering prescription for polynomials is defined by the point-splitting +procedure and depends on a chosen complex structure. +To get a global theory, we need to learn how to glue fields on different patches +together. First, let us choose two sets of local coordinates γi and �γb on some +open set. The gluing is done by a local automorphisms and γ is transformed as +in the classical theory. As we mentioned before, β is transformed classically as +β �→ �β = f ∗β, where f is a local holomorphic diffeomorphism. The quantum +version of this transformation law is given by the following general formula +[Nek05]: +�βa = βi +∂γi +∂�γa − 1 +2 +� +∂jgi +a∂igj +b +� ∂�γb +∂γk ∂γk +� +�� +� +quantum part ++ +1 +2µab∂�γb +� +�� +� +moduli parameter +, +(2.12) +where the Jacobian of the transformation is gi +a = ∂γi +∂�γa . The “quantum part” +appears because we want to keep the right OPE on both patches after gluing. +There is an intrinsic ambiguity associated to solving for the OPE equations. +Moreover, the moduli space of the βγ system is parametrized by µ or, stating +it simply, different ways of gluing our system globally are in one to one corre- +spondence with the possible choices of µ. The parameter µ takes values in the +first Čech cohomology group with coefficients in the sheaf of closed holomorphic +two-forms, i.e. H1 � +Ω2,cl, M +� +. +This algebra becomes VOA if c1(M) = 0. The global stress energy tensor is +T = −βi∂γi − 1 +2(log w)′′, +(2.13) +where w is the coefficient of the holomorphic top form ω = wdγ1 ∧ . . . ∧ dγn. +2.4 +(0,2) Cohomology And βγ System +One of the physics applications of the curved βγ system is in the realm of (0, 2) +theories. As discussed in [Wit07] and will be reviewed shortly, the βγ system +describes the perturbative cohomology of half-twisted (0, 2) theories. +Let us first discuss a general (0, 2) sigma model. The Lagrangian is con- +structed locally by introducing a (1, 0)-form K = Ki dφi, with complex conju- +gate K = Ki dφ +i, and setting +I = +� ��d2z +�� dθ ++ dθ+ +� +− i +2Ki(Φ, Φ)∂zΦi + i +2Ki(Φ, Φ)∂zΦ +i� +, +11 + +where Φi is a chiral superfield whose bottom component φi defines a map from +a Riemann surface Σ to a target complex manifold X. +The cohomology of +the supercharge Q+ can be deformed by H = 2i∂ω ∈ H1 � +M, Ω2,cl� +, where +ω = i +2 +� +∂K − ∂K +� +. Not only that but H must be of type (2, 1) to preserve (0, 2) +supersymmetry. Note that it is the same class that parametrizes the βγ system +moduli. +If we set αi = − +√ +2ψ +i ++, ρi = −iψi ++/ +√ +2 and twist the theory then ρ is an +element of Ω0,1 (Σ) ⊗ φ∗ (TX) and α is from φ∗ � +TX +� +. Both α and ρ are Grass- +mann variables. After the twisting, Q+ becomes a worldsheet scalar with the +following action on the fields: +Qφi = 0, +Qφ +i = αi, +Qρi +z = −∂zφi, +Qαi = 0, +(2.14) +and the action is given by: +I = +� +d2z +� +gij∂zφi∂zφ +j + gijρi +z∂zαj − gij,kαkρi +z∂zφ +j� ++ ST , +(2.15) +where ST = − +� +d2z +� +Tij∂zφi∂zφj − Tij,kαkρi∂zφj� +and H = dT should be of +the type described above. We also note that T is not a 2-form but a 2-gauge +field. +Locally, the structure of the Q-cohomology can be understood easily with +the help of the βγ system. Consider an open ball Uα: +I = 1 +2π +� +Uα +��d2z +�� � +i,j +δi,j +� +∂zφi∂zφ +j + ρi∂zαj +� +. +(2.16) +All the sections in the cohomology can be written as (for details refer to [Wit07]): +F +� +φ, ∂zφ, . . . ; ∂zφ, ∂2 +zφ, . . . +� +∈ H0 � +Ops2d, Q +2d ++ +� +. +(2.17) +If we set βi = δij∂zφ +j, which is an operator of dimension (1, 0), and γi = φi of +dimension (0, 0), the bosonic part of the action can be rewritten as: +Iβγ +Uα = 1 +2π +� ��d2z +�� � +i +βi∂zγi +(2.18) +and the space of all sections of this theory is +F +� +γ, ∂zγ, ∂2 +zγ, . . . ; β, ∂zβ, ∂2 +zβ . . . +� +. +(2.19) +So, locally, the space of sections of the βγ system and the Q-cohomology of +the (0, 2) sigma model coincide. Globally, things are a little more complicated +and we are required to consider Čech cohomology to find the operators with all +possible R-charges. The R-charge in the sigma model description is matched +with the cohomological degree: +H• � +Ops2d, Q +2d ++ +� +∼= H• +ˇCech(X, �A), +(2.20) +where �A is a sheaf of free βγ systems. +12 + +3 +3D Perspective +In this section, we discuss the Q-cohomology from the 3D N = 2 point of view. +There are a few constructions one could consider. Firstly, the Q-cohomology of +local operators in the bulk is a commutative vertex algebra (VA) V intrinsic to +the theory [CDG20; OY20]. Secondly, the Q-cohomology of local operators at +the boundary preserving (0, 2) SUSY (explored in the same reference) is, gener- +ally speaking, a noncommutative VA. Thirdly, — and this is the new structure +that we study here, — one can define the Q-cohomology on the interval, or +the chiral algebra of the interval compactification. If both the 3D theory and +its (0, 2) boundary conditions preserve the R-symmetry, this is a vertex op- +erator algebra (VOA), as opposed to just VA, i.e., it necessarily contains the +stress energy tensor. This is obvious since in the IR limit, the theory becomes +effectively two-dimensional [DL22], and the chiral algebra of an R-symmetric +2D N = (0, 2) theory always has the Virasoro element, as can be seen from +the general R-multiplet structure [Ded15]. In fact, one can also prove this by +constructing the (0, 2) R-multiplet from the integrated currents directly in 3D +[BST19]. +Intuitively, the interval VOA contains all 3D observables that look like local +operators in the 2D limit. These includes 3D local operators and lines, thus +effectively enhancing the Q-cohomology of local operators by the line operators +stretched between the boundaries. The line operators can additionally be dec- +orated by local operators in the Q-cohomology. We are allowed to move them +to the boundaries, as follows from the properties of Q [CDG20]. Additionally, +the two ends of the line operator can support some other boundary operators +that are stuck there and cannot be shifted into the bulk. Thus the most gen- +eral configuration in the Q-cohomology consists of a line stretched between the +boundaries with some local operators sitting at its two endpoints. +This in- +cludes the possibility of colliding a boundary operator from the boundary VA +mentioned earlier with the endpoint of a line. +On the half-space, the latter +implies that lines ending at the boundary engineer modules for the boundary +VA [CCG19]. In our case, i.e. on the interval, this similarly means that the line +operators give bi-modules of the pair of boundary VAs supported at the two +ends of the interval. +Examples of line operators that appear here include descendants of the Q- +closed local operators integrated over the interval [CDG20]. Things like Wilson +and vortex lines or their generalizations [Dim+20] may appear as well (the +Wilson line can be also viewed as a descendent of the ghost field). We are striving +to compute the OPE involving such operators. In fact, we will compute the exact +chiral algebra in the abelian case and the perturbative one in the nonabelian +case, that is the OPE of both local and line operators, for gauge theories with +the Dirichlet boundary conditions preserving (0, 2) supersymetry. We will find +that the order line operators, i.e. Wilson lines, create representations for the +boundary operator algebras. They are naturally included into the perturbative +interval VOA. The disorder or vortex lines (when allowed), on the other hand, +together with the boundary monopoles should be viewed as manifestation of the +non-perturbative phenomena. We claim to fully understand them in the abelian +case but only briefly discuss in the nonabelian setting. +Generally speaking, we have chiral algebras on the left and right boundaries +denoted by Vℓ and Vr, respectively. There is also the bulk algebra (commutative +13 + +VA) V, which includes only local operators. Moreover, V maps naturally into +the left and right algebras via the bulk-boundary maps, allowing to define their +tensor product over V. There are two maps, which are defined by pushing the +local operators from V to the two boundaries: +ρℓ,r : V → Vℓ,r. +(3.1) +Let us first define the algebra that only includes the local operators in 3D: +Vℓ ⊗V Vr. +(3.2) +The tensor product over V involves the identification of operators that can be ob- +tained from the same operator in the bulk. The next step is to extend this alge- +bra by modules that correspond to the Q-closed line operators stretched between +the boundaries. We will denote the resulting 3D cohomology as H•(Ops3d, Q). +Last but not least, let us note explicitly that in the non-abelian case, we +will be mostly discussing the CS level k > h∨. The IR physics of a 3D N = 2 +YM-CS is known for all values of k [Bas+18], and for 0 < |k| < h∨ it exhibits +spontaneous SUSY breaking [Ber+99; Oht99], and runaway for k = 0 [AHW82]. +What happens on the interval in the range 0 ≤ |k| < h∨ will be addressed +elsewhere, while the k ≥ h∨ case is more straightforward. Yet, it is interesting +enough, as we see in this work. +3.1 +Vector Multiplet +Consider a vector multiplet sandwiched between the Dirichlet boundary condi- +tions. In the twisted formalism this amounts to choosing A +�� = 0[CDG20] at +both ends. This, in turn, is equivalent to setting c| = 0 and Az| = 0 +The transformation rules for c, A, B, and A∗ in the bulk follow from (2.6): +Q (c) = −ic2, +Q (A) = dAc, +QB = −i[c, B] − +k +2π∂zc, +QA∗ = d′B − i [A, B] − +k +2π∂zA. +(3.3) +where dA = d′ − iA. +Before diving deeper, we review what is known about the perturbative alge- +bra on the boundary. Recall that the action in the HT twist takes the following +form: +� +BF (A) + k +4π +� +A∂A. +(3.4) +In the twisted formalism, the propagator connects A with B, as follows from +the kinetic energy B d′A. The rest of terms, including the Chern-Simons, are +treated as interactions, which induces the bivalent and the trivalent vertices: +1. the vertex connecting two A, +2. the vertex connecting two A and one B. +This form of Feynmann rules is very restrictive, and there can only be a finite +number of diagrams for a given number of external legs [GW19; CDG20]. +For the group G the field B lies in g∗. The gauge group is broken on the +boundary and becomes a global symmetry there. There is also a non-trivial +14 + +boundary anomaly due to the bulk Chern-Simons term and the fermions in +the gauge multiplet. The former contributes ±k to the anomaly and the latter +contributes −h∨. +Thus, we expect to get two boundary affine algebras, one for each bound- +ary global symmetry, with levels dictated by the anomaly. From (3.3), on the +boundary we have QB = 0. So, B is in the cohomology and its OPE with itself +was obtained in [CDG20, section 7.1]: +Ba (z) Bb (w) ∼ (−h∨ ± k) κab +(z − w)2 ++ +ifabc +(z − w) Bc (w) , +(3.5) +where Ba are the components of B in some basis, κab is the standard bilinear +form equal to +1 +2h∨ times the Killing form in that basis, and h∨ is the dual +Coxeter number. This expression can be obtained from the charge conservation +and anomalies alone. The J charge of B is 1 (see Table 1). Thus, only z up to +the second power can contribute. The first term is the anomaly term explained +above. +All half-BPS line operators hitting the boundaries are expected to create +modules for the boundary algebras, and we will show that it is true perturba- +tively for the Wilson line momentarily. A Wilson line can be written as Pe +� +t A. +Observe that it is indeed Q-invariant in the usual formalism, or in the twisted +formalism by invoking Q (A) = dAc and c| = 0. The kinetic term for B and A +can be written as: +Tr B (∂zAt − ∂tAz) . +(3.6) +There is a gauge symmetry associated to this term: +At → At + ∂tη, +Az → Az + ∂zη. +(3.7) +The propagator for B and At with the appropriate gauge fixing [CDG20] is just +G(z, z; t) ∝ +z +|x2| +3 +2 , +where +x2 = zz + t2, +(3.8) +where we do not keep track of a proportionality constant. +Next, we can calculate the OPE of the boundary operator B(0) with the +Wilson line segment W (λ)(z) in the irreducible representation of g labeled by +the dominant weight λ. Expanding the Wilson line in a Taylor series, one finds +that there is only one diagram that can possibly contribute, where a single A +from the Wilson line is directly connected to the operator B at the boundary, +see Fig. 2. This is proven simply by looking at the two vertices mentioned +earlier and realizing that they cannot contribute. The diagram evaluates to +� +Aa +t (t, z, z)Bb (0) dt ∝ δa +b +� L +0 +z dt +(t2 + zz) +3 +2 = δa +b +L +z +√ +L2 + zz += δa +b +1 +z + . . . , +(3.9) +where we keep only the singular term in the z → 0 expansion on the right. This +computation already includes corrections to the propagator in the presence of +15 + +W +B +Figure 2: The diagram connecting single A in the Wilson line to the operator +B at the boundary. +boundaries. Indeed, such corrections can be accounted for using the method of +images. Since we are interested in the singular term in the OPE, it is enough +to only include the first image At(−t, z, z) = At(t, z, z), as the other ones never +get close to B(0). This simply doubles the contribution of the original insertion +At(t, z, z). Combining everything together, this computation shows that the +OPE with the Wilson line is +Ba (z) W (λ) (w) ∼ TaW (λ) (w) +z − w +, +(3.10) +where Ta denotes the Lie algebra generator in the same representation λ as the +Wilson line, and TaW (λ)(w) means the matrix product. Looking at the Table 1, +we immediately see that this is the only possible OPE as W (λ) is not charged. +More precisely, there are two copies of the affine generators on the interval, +denoted Bℓ +a and Br +a for the left and right boundaries, respectively. Assuming +that the Wilson line performs parallel transport from the right to the left, we +find the following OPE’s on the interval: +Bℓ +a (z) W (λ) (w) ∼ TaW (λ) (w) +z − w +, +Br +a (z) W (λ) (w) ∼ W (λ) (w) Ta +z − w +. +(3.11) +For completeness, note that the OPE of Wilson line’s matrix elements is regular: +W (λ) +ij (z)W (µ) +kl (w) ∼ 0, +(3.12) +simply because no Feynmann diagram can connect two At’s. +Let us pause and contemplate on what we have obtained so far. We found +that Bℓ,r and W (λ) are elements of the extended cohomology, and we claim +that they generate the perturbative chiral algebra. The boundary B’s satisfy +the OPE relations of the affine Kac-Moody vertex algebras V−h∨±k(g). They +also act on W (λ) as on a primary field of the highest weight representation +of the affine algebra (i.e., a Weyl module Vλ,−h∨±k = Ind�g +�bVλ, where Vλ is a +finite-dimensional module for the underlying Lie algebra g). Thus, we obtain +the following result for the perturbative chiral algebra: +Ck[GC] := +� +λ∈P+ +Vλ,−h∨+k ⊗ Vλ∗,−h∨−k, +(3.13) +16 + +where λ runs over the set P+ of dominant weights, and λ∗ = −w(λ), where +w is the longest element of the Weyl group of G. It is not a coincidence that +Ck[GC] looks like Dk[GC] from the equation (1.1) in the Introduction. +This +object is well known in the mathematical literature [AG02; FS06; Zhu08; CG20; +CKM22; Mor22], and away from the rational values of k, Ck[GC] is a simple VOA +isomorphic to the VOA Dk[GC] of chiral differential operators on GC, with the +deformation parameter (perturbative B-field flux) k ∈ C = H3(G, C). At the +rational points k ∈ Q, we can encounter singular vectors, and life is getting +much more interesting, e.g., Ck[GC] and Dk[GC] are no longer the same [Zhu08]. +We can also ask what happens to the stress-energy tensor in our setup. We +know that it does not exist as a local operator in the bulk chiral algebra [CDG20, +section 2.2], and generally, the boundary VA does not have to possess a stress- +energy tensor as well. At the same time, we have the current that generates +holomorphic translations. It acts on the boundary operators as: +∂wO (w) = +� +HS2 ∗(Tzµ dxµ)O(w). +(3.14) +There is no boundary part in this expression as we do not introduce any non- +trivial degrees of freedom at the boundary. We can create a line stress-energy +operator by stretching the integration surface HS2 to a cylinder in a way that +is shown in Fig. 3. This allows to act with the holomorphic translations also on += +O1(x) +O2(x) +O2(x) +O1(x) +Figure 3: Possible codimension one surfaces over which Tzν is integrated to +generate holomorphic translations along the boundary. +line operators by enclosing them with such a cylinder. The integration over this +tube can be separated into two parts. The integral over dt defines the integrated +stress-energy tensor, +T int +zz = +� L +0 +dt Tzz, +T int +zz = +� L +0 +dt Tzz, +(3.15) +which behaves as a 2D stress tensor generating the holomorphic translations. +The remaining integration over the contour in the boundary plane is reminiscent +of the 2D CFT setup. In fact, it was shown in [BST19] that such integrated 3D +currents (the stress tensor, the R-current and the supercurrents) fit precisely in +the 2D N = (0, 2) R-multiplet. The presence of this multiplet automatically +17 + +implies existence of the stress-energy tensor in the cohomology [Ded15]. In the +IR regime, as t collapses, T int of course becomes the 2D stress tensor, and the +2D N = (0, 2) arguments are applicable directly. In either case, we see again +that the interval chiral algebra has the stress-energy tensor that follows from +the integrated currents. +Outside of critical levels, the boundary algebras are VOAs and have well- +defined Sugawara stress tensors. Physically, we expect that the interval stress- +energy tensor becomes the sum of T sug +ℓ +and T sug +r +as an element in the 2D chiral +algebra. It is clear that they act in the same way on the boundary operators. +It indeed turns out to be true as we will argue in the next section using the 2D +perspective. +3.2 +The non-perturbative corrections +What are the possible non-perturbative corrections to the above? One comes +from the boundary monopole operators discussed in [Bul+16; DGP18; CDG20]. +Another possibility is the vortex line connecting the two boundaries. General +gauge vortices discussed in [KWY13; DOP14; HS21] are characterized by a +singular gauge background A = b dϕ close to the vortex locus, where ϕ is an +angular coordinate in the plane orthogonal to the line, and b is from the Cartan +subalgebra. This defines a line defect that is local in the plane orthogonal to +the vortex, at least away from the boundary, because gauge invariant objects +do not feel the gauge holonomy. This still holds near the Neumann boundary, +where the gauge symmetry is unbroken. However, if such a vortex ends at the +Dirichlet boundary, it creates a non-trivial monodromy e2πib for the boundary +global symmetry. Hence for generic b, it is not a local operator from the 2D +boundary point of view as it can be detected far away from the insertion point. +On the interval with the Dirichlet boundaries, such vortices do not lead to local +operators in the IR, they become the twist fields that are not included in the +VOA. +A less general possibility is a vortex characterised by a nontrivial background +A = b dϕ, yet its monodromy is trivial, e2πib = 1. The latter means that b is +a co-weight (in fact, a co-character, because the gauge symmetry forces us to +consider the Weyl group orbit of b). Thus such a vortex has its magnetic charge +labeled by a cocharacter of G, or a subgroup U(1) �→ G taken up to conjugation. +This is the same as magnetic charges of the monopoles, and we can think of the +vortex as an infinitesimal tube of magnetic flux, with the same amount of flux as +created by the charge-b magnetic monopole. This also suggests that monopoles +can be located at the endpoints or at the junctions of vortices. +Such vortex lines, however, are not expected to be independent line operators +in the IR, at least for a non-zero CS level there. When k > h∨, our 3D theory +becomes the level k − h∨ CS theory at large distances. It has been argued in +[MS89] that such vortex lines are equivalent to Wilson lines in a CS theory +(for the abelian case, see the argument in [KWY13].) Thinking of the CS level +as a pairing K : Γ × Γ → Z, where Γ ⊂ t is the co-weight lattice of G, the +representation of the Wilson line is determined precisely by the weight K(·, b). +This is also consistent with the well-known fact (at least in the abelian case +[KS11]) that in the presence of CS level, monopoles develop electric charges, +and so Wilson lines can end on them. In particular, [KS11] used this to argue +that the Wilson lines whose charges differ by a multiple of k are isomorphic, +18 + +thus showing that the finite spectrum of Wilson lines in a CS theory is a non- +perturbative effect manifested via the monopoles. Note that all the statements +we referred to here are about the non-SUSY CS theory, but they extend verbatim +to the half-BPS lines in the N = 2 case. +To apply these observations to the interval theory, it is convenient to assume +that the interval is long enough, such that we flow to the CS first and only then +to 2D. At least for k > h∨ this appears to be a harmless assumption, since +SUSY suggests that the BPS sector is not sensitive to the interval length, and +the IR physics is also more straightforward in this case [Bas+18]. +It is therefore natural to conjecture that the non-perturbative effects on the +interval with Dirichlet boundaries are captured by the monopoles. In the bulk +they ensure that there are only finitely many inequivalent Wilson lines, and the +boundary monopoles modify the boundary VAs. The details depend strongly +on whether G is abelian or non-abelian. +In the non-abelian case, the monopoles at the level k−h∨ boundary, accord- +ing to the conjecture in [CDG20], turn the perturbative affine VOA Vk−h∨(g) +into its simple quotient Lk−h∨(g). The finitely many bulk Wilson lines cor- +respond to the integrable representations of the latter. +As for the bound- +ary monopoles at the opposite end, it seems unlikely that they can modify +V−k−h∨(g), which is already simple. The total nonperturbative interval alge- +bra in this case appears to be some modification of the CDO that contains +Lk−h∨(g) rather than Vk−h∨(g). We do not know its structure yet, and will +explore it elsewhere. +The abelian case will be studied in the next sections, where we consider +G = U (1) in detail. It is a little bit different as there is no abelian WZ term +in 2D, and the level is encoded in the periodicity of the compact boson. The +monopole corrections extend the boundary affine u(1) to the lattice VOA, also +known as the abelian WZW. The non-isomorphic Wilson lines correspond to +the finite set of modules of the lattice VOA. +3.3 +U(1) +We restrict k to lie in 2Z and consider the k ̸= 0 case first. Bℓ, Br, +� +At dt +are the possible candidates for elements of the perturbative interval VOA. The +analysis of the OPE of B’s still holds, and B’s on the left and right boundaries +commute with each other (have the regular OPE). So, the full set of OPEs again: +Br (z) Br (w) ∼ +k +(z − w)2 , +Bℓ (z) Bℓ (w) ∼ +−k +(z − w)2 , +Br(z)Bℓ(w) ∼ 0. +(3.16) +Surprisingly, there is also one relation connecting the left and right B’s to the +Wilson line, which follows from the following transformation: +Q +� +A∗ +t dt = Br − Bℓ − k +2π ∂z +� +At dt . +(3.17) +Thus, we see that the derivatives of +� +A are not independent operators in the +Q-cohomology. We will encounter similar phenomena when we consider sin- +gular vectors for affine algebras on the boundaries later. We can choose any +19 + +two operators out of {Br, Bℓ, ∂ +� +A} as the independent generators, and to be +consistent with the previous section, we take {Bℓ,r}. The stress-energy tensor +is also included, but for k ̸= 0 it should be expressed in terms of Bℓ,r as we +discussed before. In this particular case, T = +1 +2kB2 +r − +1 +2kB2 +ℓ . +We also expect that this perturbative algebra is extended by the boundary +monopole operators [DGP18]. The boundary monopole Mp is obtained from +the usual monopole by cutting it in half and restricting to a half-space in such +a way that the integral over the half sphere is +� +HS2 F = 2πp, +p ∈ Z. +(3.18) +Due to the CS term, the monopole operator Mp develops an electric charge, +as was mentioned before. Hence this operator can only exist by itself on the +boundary where its electric charge is global. Under the global boundary U(1) +action by eiα it transforms as: +Mp → e−ipkαMp. +(3.19) +To insert it in the bulk, we need to consider a composite operator with a Wilson +line attached to a monopole eikp +� t +0 AMp(t) to cancel the anomalous transfor- +mation. In fact, we can pull a boundary monopole Mp off the boundary while +extending a Wilson line of charge kp between the monopole and the boundary +to respect gauge invariance, as shown in Fig. 4. Recall that the twisted the- +Mp +Wkp +⇐⇒ +Mp +Figure 4: The bulk monopole Mp is connected by a Wilson line of charge kp to +the Dirichlet boundary. In the Q-cohomology, due to the topological invariance +in the t direction, this is equivalent to the boundary monopole Mp. +ory is topological in the t direction [CDG20], meaning that the t translations +are Q-exact. Thus the length of the Wilson line in Fig. 4 is irrelevant, and +pulling a monopole off the boundary is an identity operation in the cohomology. +Using this observation, we can easily find the OPE of a boundary monopole +Mp with the boundary current B. Pulling Mp far away from the boundary, it +can no longer contribute to such an OPE. Essentially, for the purpose of com- +puting OPEs with the boundary operators (in the cohomology), the boundary +monopole Mp is equivalent to the Wilson line of charge kp ending at the bound- +ary. We have already computed the OPE of B with the Wilson line in earlier +sections, so the answer follows immediately. Recalling that there is also a sec- +ond boundary (and operators on the opposite boundaries have regular OPE), +we thus find: +Bℓ(z)Mp(0) ∼ kp +z Mp(0), +Br(z)Mp(0) ∼ 0, +(3.20) +20 + +for the monopole on the left boundary. A more semi-classical way to derive this +is by computing the on-shell value of B in the twisted formalism. One can easily +check that the monopole singularity implies B ∼ kp +z (where the factors of 2π +were scaled away). Similar results hold for monopoles on the right boundary. +In fact, repeating our argument, a monopole on the right boundary is equiva- +lent to the same monopole on the left boundary connected by the Wilson line to +the right boundary (and vice versa), see Fig. 5. Thus, one can express the right +monopoles as M ′ +p(t = L) = e−ikp +� L +0 AMp(t = 0), and they are not independent +generators. +Via the semiclassical analysis of the monopole operator, one can show that +its OPE with the Wilson line is +Mp(z)eiq +� L +0 A(0,t) ∼ zqp : Mp(z)eiq +� L +0 A(0,t) :, +(3.21) +where :: means the normal ordering. From this equation we see that the normal +ordering for e−ikp +� L +0 AMp(t = 0) is not required: The leading OPE term scales +as zp2. To compute the missing Mp1(z)Mp2(0) OPE, we again use the trick of +replacing the left boundary monopole Mp2 by the Wilson line attached to the +right monopole eikp2 +� L +0 AtdtM ′ +p2, +Mp1(z, 0)Mp2(0, 0) ∼ Mp1(z, 0)eikp2 +� L +0 At(0,t)dtM ′ +p2(0, L). +(3.22) +In this equation, the monopole M ′ +p2 is separated in the t direction from +the rest of the operators. Hence the singular terms only come from the OPE +between the Wilson line and the monopole Mp1. This results in the following: +Mp1(z, 0)Mp2(0, 0) ∼ zkp1p2 : Mp1(z, 0)eikp2 +� L +0 At(0,t)dt : M ′ +p2(0, L). +(3.23) +It remains to bring M ′ +p2 back to the left boundary, colliding with it along the +t direction, which produces no further singularities due to the topological in- +variance. Finally, we observe that the magnetic charges are additive under the +collision, and conclude: +Mp1(z)Mp2(0) ∼ zkp1p2 : Mp1(z)Mp2(0) := zkp1p2Mp1+p2(0) + . . . . +(3.24) +The full set of strong generators can be chosen to be +� +Bℓ,r, eip +� +A, Mp|p ∈ Z +� +. +(3.25) +Looking closer at (3.16), (3.20) and (3.24), we recognize the rank one lattice +VOA setting, with the compact boson of radius +√ +k. We can extend the U (1)k +current B by the monopoles (vertex operators) Mp to the Z[ +√ +k] lattice VOA, +also called the U(1)k WZW. This can be done individually on the left and right +boundaries, giving the abelian WZWk and WZW−k, respectively. +Then the +number of Wilson lines is rendered finite, and the algebra is generated as an +extension of +WZWk ⊗ WZW−k +(3.26) +by the bimodules corresponding to the Wilson lines e−i k−2 +2 +� +A, . . . , ei k +2 +� +A. In- +deed, it is consistent with the limit of the large interval. The theory in this limit +reduces to the Chern-Simons at level k in the IR, which only admits a finite +21 + +number (k, to be more precise) of Wilson lines in the bulk, and supports WZW +at the boundaries. +Now, let us turn to the k = 0 case. One can easily see that we no longer +have two separate operators Bℓ,r, as B is in the bulk cohomology. Thus, we can +pull it off the boundary and bring it all the way to the opposite one without +changing the cohomology class. Technically, this follows from the relation (3.17) +involving the descendant line of B. Monopole operators no longer have electric +charge, so they can be freely moved into the bulk, and they commute (have +regular OPE) with all local operators. In other words, monopoles are naturally +elements of the bulk VA V. The stress-energy tensor is no longer a sum of the +two boundary terms but an independent operator. So, the algebra ceases to +be generated as a bimodule of the left and right algebras, and we propose the +following set of strong generators: +� +B, T, ein +� +A, Mn|n ∈ Z +� +. +(3.27) +4 +2D Perspective or βγ System +The IR limit of a 3D N = 2 YM-CS theory on a slab is in general controlled +by the dimensionless parameter γ = Le2 +3d. +When L ≫ +1 +e2 +3d or γ ≫ 1, the +bulk first flows as a 3D theory to the Chern-Simons TFT with keff = k − h∨ +(for k ≥ h∨). The boundaries flow to some 2D relative CFTs matching the CS +boundary anomalies. The interval VOA in this limit was studied in the previous +section. We do expect, however, that the answer does not depend on γ, at least +for k > h∨ (when the SUSY is unbroken). +In the opposite limit γ ≪ 1, the system behaves as a 2D sigma model. It was +shown in [DL22] that in this regime, the theory is described by an N = (0, 2) +NLSM into the complexified group GC ≈ T ∗G at level k. The 2D coupling is +related to the 3D coupling as +L +e2 +3d = +1 +e2 +2d . The gauge degrees of freedom are +integrated out, except for the complex Wilson line along the interval, which is +left as an effective degree of freedom. Its compact part is valued in G, and +the vector multiplet scalars lie in the cotangent space. The Chern-Simons term +reduces to a non-trivial B-field, or the Wess-Zumino term, necessary for anomaly +matching. In this section, we explore our problem from such a 2D viewpoint, +as well as study connections between the 3D and 2D setups. +As reviewed in Section 2, the perturbative chiral algebra of 2D N = (0, 2) +NLSM into the target X is captured by the βγ system into X [Wit07; Nek05]. +More precisely, it is given by the cohomology of a sheaf of βγ systems on X, +also called the sheaf of chiral differential operators (CDO) [GMS99; GMS01; +GMS04]. +It is conveniently computed using the Čech cohomology, and the +resulting vector space with the VA structure on it is denoted Dk[X], where +k is the B-field. +In our case, the target is a simple complex group GC, so +k ∈ C = H3(GC, C). This parameter is the well-known modulus of the CDO +valued in H1(X, Ω2,cl(X)), which in general is not quantized, however, in our +models k is an integer originating as the CS level in 3D. Since GC has a trivial +tangent bundle, c1(GC) = 0 and p1(GC) = 0, so the sigma model anomalies +[MN85] vanish. As reviewed before, c1(GC) = 0 implies that Dk[GC] is a VOA, +i.e., it has a well-defined Virasoro element. +22 + +Notably, Dk[GC] containes the affine sub-VOAs Vk−h∨(g) and V−k−h∨(g) +corresponding to the left and right G-actions on GC [GMS01]. These clearly +originate as the perturbative baoudnary VOAs in 3D. Additionally, Dk[GC] +contains holomorphic functions on the group (functions of γ in the βγ lan- +guage). +These originate from the Wilson lines stretched across the interval. +For generic k ∈ C \ Q, functions on the group together with the pair of affine +VOAs V±k−h∨(g) generate the whole Dk[GC]. For k ∈ Q this is no longer true, +and Dk[GC] as a Vk−h∨(g) ⊗ V−k−h∨(g)–bimodule has a very intricate structure +[Zhu08]. Nonetheless, Dk[GC] remains a simple VOA for all values of k. +Below we will find a full non-perturbative answer in the abelian case and +comment on what is known in the non-abelian case. By comparing the 3D and +2D answers when possible, we will veryfy that the interpolation between γ ≫ 1 +and γ ≪ 1 determines an isomorphism of the respective chiral algebras: +H• � +Ops3d, Q+ +� ∼ +−→ H• � +Ops2d, Q +2d ++ +� +. +(4.1) +4.1 +U(1) +For G = U(1), the target space of our model is U (1)C ∼= C∗, which is a cylinder. +The N = (0, 2) sigma model into C∗ was considered in [DGP17], and we will +make contact with it later. For now, let us follow the βγ approach first. The +CDO only have zeroth cohomology in this case, which are the global sections +of this sheaf forming the perturbative VOA. Then we will identify its non- +perturbative extension. We work with the even Chern-Simons level k in what +follows. +We introduce two cooridnate systems on the cylinder: One is just γ ∈ C∗, +and another has �γ as a periodic complex boson. Its periodicity is what encodes +the “level” in the abelian case, descending from the CS level in 3D: +�γ ∼ �γ + iπ +√ +2k, +(4.2) +where the conventions are adjusted to match [Wit07]. The relation between the +two is, naturally, +γ = e +√ +2 +k �γ. +(4.3) +The quantum transformation between β and �β is +�β = +� +2 +k +� +βγ − 1 +2γ−1∂γ +� +, +β = +� +k +2 +� +�βe−√ +2 +k �γ − 1 +k e−√ +2 +k �γ∂�γ +� +. +(4.4) +Let us define two currents: +Jℓ = +1 +√ +2 +� +�β + ∂�γ +� +, +Jr = +1 +√ +2 +� +�β − ∂�γ +� +, +(4.5) +where �γ ∝ +� +t A is a (log of a) Wilson line from the 3D perspective, and the +difference is +√ +2∂�γ, which is proportional to ∂ +� +t A as in 3.17. One can easily +see that these operators commute and have the following OPEs: +Jℓ(z)Jℓ(0) ∼ −1 +z2 , +Jr(z)Jr(0) ∼ 1 +z2 . +(4.6) +23 + +We observe that they only differ from the 3D boundary currents Bℓ,r by the +normalization factor +√ +k. We find such conventions useful for this section. The +stress-energy tensor is −�β∂�γ and the central charge is equal to 2. The stress- +energy tensor does not require any modifications. The single-valued operator +corresponding to the charge p Wilson line is Wp = γp. The conformal dimensions +of Wp can be easily found to be equal to zero. +The operators β and γp, p ∈ Z, are already global sections of the CDO, and +they generate the full perturbative VOA, which is most concisely described as +the C∗-valued βγ system. Note that this implies that γ can be inverted. In +practice, it is convenient to do so by inverting the zero mode of γ only: +γ−1(z) = +1 +γ0 + ∆γ = 1 +γ0 +∞ +� +n=1 +(−1)n � +γ−1 +0 ∆γ +�n , +(4.7) +where +∆γ = +� +n∈Z\0 +γn +zn . +(4.8) +So, what are the non-perturbative effects that we are missing here? +3D +physics suggests that they are boundary monopoles. Imagine inserting an im- +properly quantized monopole on the boundary with +� +HS2 F = 2πα at the origin +z = 0. We know that γ is related to the connection γ ∝ ei +� +t A. Now we can +consider moving γ around the insertion of this monopole, as in γ +� +eiφz +� +, and let +us track the phase that is acquired in the process. It is clear that the Wilson +line sweeps the entire magnetic flux of the monopole in the end, and we obtain: +γ +� +e2πiz +� += ei +� +M F γ (z) = e2iπαγ (z) , +(4.9) +where M is a tube stretched between the two boundaries. +Thus, γ in the +presence of the improperly quantized monopole behaves as: +γ = +� +n +γn +zn−α . +(4.10) +Now let us go back to the actual monopole that has α = p ∈ Z. Equation (4.9) +shows that γ winds p times around the origin as we go once around z = 0, and +we expect the same shift as in (4.10) with α = p. The actual answer is a little +bit trickier, but this gives us a good starting point. +An operator that “shifts the vacuum” in the βγ system by p units will be +called Mp. The module that it creates is known as the spectrally flowed module.6 +We can look at the Hilbert space interpretation of these modules. If we place +an Mp operator at the origin, then the mode expansions of β and γ take the +following form: +β = +� +n +βn +zn+p+1 , +γ = +� +n +γn +zn−p . +(4.11) +Here the modes obey the usual commutation relations: +[βn, γm] = δn+m,0, +(4.12) +6In superstrings such operators are often called the picture changing operators. +24 + +and the lowest state |p⟩ corresponding to Mp via the state-operator map is +defined by +γn+1|p⟩ = βn|p⟩ = 0, +for n ≥ 0. +(4.13) +Essentially, we shifted the modes of the vacuum module by p positions and then +relabeled them. +The inverse of γ in the presence of Mp is defined in the similar manner: +γ−1 (z) = +1 +γ0zp + ∆γ = z−p +γ0 +∞ +� +n=0 +(−1)n � +zpγ−1 +0 ∆γ +�n . +(4.14) +Let us compute charges and the conformal dimension of Mp. The currents +Jℓ, Jr in the γ coordinate system are +� +Jℓ +Jr +� += +� +� +� +1 +√ +k +� +βγ + (k−1) +2 +γ−1∂γ +� +, +1 +√ +k +� +βγ + (−k−1) +2 +γ−1∂γ +� +. +(4.15) +The following product can be computed using the mode expansions: +β (z) γ (w) |p⟩ = − +�w +z +�p +1 +z − w |p⟩ . +(4.16) +Subtracting the singularity +1 +z−w, we are getting: +: βγ : |p⟩ = p +z |p⟩ . +(4.17) +Let us denote : βγ : as J0 in what follows. The J0 charge of Mp is p, and the +charges of β and γ are 1 and −1 respectively. The difference of Jℓ and Jr is +proportional to γ−1∂γ, which is itself a current measuring the winding number. +In the presence of Mp, the expression γ−1∂γ can be computed directly from the +definition: +γ−1∂γ = p +z + . . . +(4.18) +Thus the winding charge is equal to p for Mp and 0 for β and γ. The U (1)ℓ,r +charges for Mp are then computed from (4.15) and are +p +√ +k( 1+k +2 , 1−k +2 ). +The stress energy tensor written in terms of βγ is −β∂γ + 1 +2 (log γ)′′ as the +holomorphic top form in this case is w ∝ dγ +γ . Moreover, it can be written as +Tβγ = 1 +2J2 +r − 1 +2J2 +ℓ , which is indeed what was expected. Computing the conformal +dimensions of Mp requires again the normal ordering prescription: +lim +z→w +� +−β (z) ∂γ (w) |p⟩ − +1 +(z − w)2 |p⟩ +� += +�p − p2 +2w2 ++ β−1γ0 +w ++ . . . +� +|p⟩ , +1 +2(log γ)′′ |p⟩ = +� +− 1 +w2 +p +2 + . . . +� +|p⟩ , +(4.19) +where we again subtracted all singular terms. It follows that the dimension of +the operator Mp is − p2 +2 . +25 + +≈ +Figure 5: Moving a monopole of magnetic charge p from one boundary to an- +other creates a Wilson line of electric charge kp. +Now that we understand the operators Mp fairly well, we need to iden- +tify precisely those that should be added to the βγ system to obtain our non- +perturbative VOA. They correspond to the boundary monopoles in 3D. For that +we need to find operators that are only charged under the left or right affine +currents Jℓ,r. If we had k = 1 for a moment, the left monopole is just the flowed +module Mp, as can be seen from its charges (Jℓ, Jr) = (p, 0). The mathematical +perspective on such modules was given in [RW14; AW22], and in their notations, +Mp generates σpW+ +0 , where σ denotes the spectral flow automorphism. +To tackle the general case, we just need to take a composite operator that +has the correct charge. The Mp by itself is charged as +p +√ +k +� k+1 +2 , −k+1 +2 +� +. So, if +we take M ′ +p = (γ +p(k−1) +2 +0 +|p⟩)(z), we get an operator with the charges p( +√ +k, 0), +as required, and the conformal dimension ∆M ′p = p2 k +2. +This M ′ +p generates +the spectrally flowed module σpW p(k−1) +2 +in the notations of [AW22], which is +just σpW+ +0 for p(k − 1) even and σpW 1 +2 for p(k − 1) odd.7 +In order to get +monopoles on the other boundary, we need to consider the composite operator +with the Wilson line WkM ′ +1 (Fig. +5), which has charges p(0, − +√ +k) and the +same conformal dimension. Thus we do not need a separate generator for that +monopole. The generator content of our algebra matches the eqn. (3.25). +Let us summarize the result that we have obtained so far for the full non- +perturbative VOA. It is given by the C∗-valued βγ VOA (i.e., γ is invertible,) +extended8 by its modules σ2pW+ +0 and σ2p+1W 1 +2 for all p ∈ Z, in the notations +of [RW14; AW22]. +One can also easily compute a character over the full chiral algebra: +Z = TrH(qL0− c +24 xJℓyJr) = +1 +η(q)2 +� +n,m∈Z +qnmx +n +√ +k +m +√ +k +2 y +n +√ +k −m +√ +k +2 , +(4.20) +which is clearly not a meromorphic function and can only be understood as a +formal power series. The obvious non-convergence of the character is expected, +as characters on the boundary with the positive level are convergent when |q| < 1 +7Since we consider even k, this is determined by the parity of p only. +8In fact, it is slightly redundant to say that γ is invertible. The module W+ +0 of the usual βγ +system coincides with the vacuum module of such a C∗-valued βγ system. Thus the extension +by W+ +0 automatically inverts γ. +26 + +[DGP18], and on the opposite boundary the convergence is at |q| > 1. This +trace, of course, can be reinterpreted from the 3D perspective as an index on +the interval (see also [SY20]): +ZI×T 2 = TrH(−1)F e−2πRH � +e2πiJizi, +(4.21) +where Ji are generators of the maximal tori of the boundary symmetries. +The k = 0 situation is different, and does not appear to be particularly +interesting and well behaved from the 2D viewpoint, so we skip it. +Dual boson +One can also calculate the same algebra directly in the C∗ sigma model, without +going to the βγ-description. The calculation was first done in [DGP17]. Let us +connect it with our formulas for completeness. Let σ be the radial coordinate +and X = XL(z) + XR(z) be an angular coordinate on C∗. Then �β and �γ are +related to the free boson as in Sec. 2.4: +∂�γ = ∂σ + i∂X, +�β = ∂σ − i∂X. +(4.22) +In the γ coordinate system one gets from (4.4): +γ = e +√ +2 +√ +k (σ+i(XL+XR)), +β = +√ +k +√ +2 +� +∂(σ − iX)e− +√ +2 +√ +k (σ+iX) − 1 +k e− +√ +2 +√ +k (σ+iX)∂(σ + iX) +� += − +√ +k +√ +2 +� +(1 − 1 +k )∂σ + i(1 + 1 +k )∂X +� +e− +√ +2 +√ +k (σ+iX). +(4.23) +Redefine both X and σ by +√ +2 to match the notations of [DGP17], so we find: +γ = e +1 +√ +k (σ+i(XL+XR)), +β = − +√ +k +2 +� +(1 − 1 +k )∂σ + i(1 + 1 +k )∂X +� +e− +1 +√ +k (σ+iX). +(4.24) +The BPS vertex operators in this description take the form: +eikℓXL+kr(σ+iXR), +(4.25) +with +(kℓ, kr) = +� n +R + wR +2 , n +R − wR +2 +� +, +n, w ∈ Z. +(4.26) +The radius R is related to the Chern-Simons level as R2 = k. +We can see +that these operators form the same lattice as we found in the βγ system with +the special shifted modules included. +In particular, γn here is the vertex +operator with kℓ = kr = +n +R. +At the same time the left monopoles have +(kℓ, kr) = p( +√ +k, 0), which means n = wk/2 = pk/2, and the right monopoles +have (kℓ, kr) = p(0, − +√ +k), which corresponds to n = −wk/2 = −pk/2. +27 + +No Mercy +This final presentation allows us to identify the nonperturbative VOA even +more explicitly in terms of the known VOAs. Namely, let us denote the vertex +operator representing the Q-cohomology class with the momentum and winding +charges (n, w) ∈ Z2 by Vn,w(z). Then computing the OPE of vertex operators +defined in (4.25), we easily find the following: +Vn1,w1(z)Vn2,w2(0) ∼ zn1w2+n2w1 : Vn1,w1(z)Vn2,w2(0) : , +(4.27) +which identifies our VOA as a lattice VOA for the smallest Narain lattice [Nar86; +NSW87], namely Z2 ⊂ R2 with the scalar product: +(n1, w1) ◦ (n2, w2) = n1w2 + n2w1. +(4.28) +Note that the two U(1) currents can be obtained as V−1,0∂V1,0 and V0,−1∂V0,1: +J1 = 1 +R (i∂XL + i∂XR + ∂σ) , +J2 = R +2 (i∂XL − i∂XR − ∂σ) . +(4.29) +Also notice a curious fact: While many of the steps in our analysis involved +the CS level, the final answer does not depend on it. This in fact serves as a +consistency check for the following reason. From the N = (0, 2) point of view, +the compact boson radius +√ +k only enters the Kähler potential, thus it cannot +affect the chiral algebra structure. +Together with the other two results in the earlier sections, we thus find three +presentations for the nonperturbative VOA in the abelian case: +Narain lattice VOA +of rank two +∼= +βγ extended by +σ2pW+ +0 and σ2p+1W 1 +2 +∼= +WZWk ⊗ WZW−k +extended by bimodules +(4.30) +4.2 +SU(2) +Let us now turn to a less-trivial example and discuss G = SU(2), that is +GC =SL(2, C). +The computation is more involved in this case, as we need +to define everything on patches and consider a non-trivial gluing. We will first +find the global theory and discuss the moduli space, and then will turn to the +non-trivial modules for boundary VOAs. +SL(2, C) can be covered by two patches, the coordinates on which will be +denoted as γi and �γi : +�a +b +c +d +� +, ad − bc = 1 +a̸=0 +�−−→ +�γ1 +γ2 +γ3 +� += +�a +b +c +� +∈ C3 \ {γ1 = 0}, +� +a +b +c +d +� +, ad − bc = 1 +b̸=0 +�−−→ +� +�γ1 +�γ2 +�γ3 +� += +� +a +b +d +� +∈ C3 \ {�γ2 = 0}. +28 + +Thus, the coordinate transformations have the following form: +γ1 = �γ1, γ2 = �γ2, γ3 = �γ1�γ3 − 1 +�γ2 +or +�γ1 = γ1, �γ2 = γ2, �γ3 = 1 + γ2γ3 +γ1 +. +(4.31) +The Jacobian matrix and its inverse can be computed to be +gi +j ≡ ∂γi +∂�γj = +� +� +1 +0 +0 +0 +1 +0 +1+γ2γ3 +γ1γ2 +− γ3 +γ2 +γ1 +γ2 +� +� , +∂�γi +∂γj = +� +� +1 +0 +0 +0 +1 +0 +− 1+γ2γ3 +(γ1)2 +γ3 +γ1 +γ2 +γ1 +� +� . +∂jgi +a∂igj +b = ∂3g3 +a∂3g3 +b = +� +� +1 +(γ1)2 +− +1 +γ1γ2 +0 +− +1 +γ1γ2 +1 +(γ2)2 +0 +0 +0 +0 +� +� . +As we mentioned earlier, for each left- and right-invariant vector fields there +exists a corresponding VOA sub-algebra [GMS01] that saturates boundary anoma- +lies. The boundary with the negative anomaly usually corresponds to a rela- +tive CFT with the anti-holomorphic dependence on the coordinates, and it +transforms into a chiral algebra with the negative level after passing to the +Q-cohomology: kcoh = kℓ − kr [Wit07]. +Let us find these algebras and the CDO sections explicitly. Note that there is +nothing left except global sections as the manifold is Stein and does not support +geometric objects that can form a higher degree cohomology. So, the only fields +that can contribute are in H0(SL(2), �A). +Let us first write out all classical vector fields. To do this, we will use the +following well-known form of basis at the identity point of the group: +e = +�0 +1 +0 +0 +� +f = +�0 +0 +1 +0 +� +h = +�1 +0 +0 +−1 +� +(4.32) +and carry it over the whole manifold by L∗ +gV µ∂µ|1 = (gV )µ∂µ|g. +Local sections, corresponding to the left-invariant vector fields, then have +the following form in both patches: +eℓ = γ1β2 +�eℓ = �γ1 �β2 + �γ−1 +2 (�γ1�γ3 − 1)�β3 +fℓ = γ2β1 + γ−1 +1 (1 + γ2γ3)β3 +�fℓ = �γ2 �β1 +hℓ = γ1β1 − γ2β2 + γ3β3 +�hℓ = �γ1 �β1 − �γ2 �β2 − �γ3 �β3. +(4.33) +Local sections corresponding to the right-invariant vector fields in both patches +are +er = γ1β3 +�er = �γ2 �β3 +fr = γ3β1 + γ−1 +1 (1 + γ2γ3)β2 +�fr = �γ−1 +2 (�γ1�γ3 − 1)�β1 + �γ3 �β2 +hr = γ1β1 + γ2β2 − γ3β3 +�hr = �γ1 �β1 + �γ2 �β2 − �γ3 �β3. +(4.34) +The normal ordering for these fields is chosen exactly in the way they are written, +abc = +def: a : bc ::, and will be omitted from this point on to unclutter notations. +29 + +By using (2.12), one can easily obtain the following transformation formulas: +�β1 = β1 + β3 +�γ3 +γ1 + +1 +γ1γ2 +� +− 1 +2 +� +1 +(γ1)2 ∂γ1 − +1 +γ1γ2 ∂γ2 +� +, +�β2 = β2 − β3 +γ3 +γ2 − 1 +2 +� +− +1 +γ1γ2 ∂γ1 + +1 +(γ2)2 ∂γ2 +� +, +�β3 = β3 +γ1 +γ2 . +(4.35) +Note that here we choose to set the moduli parameter µab to zero. Now we will +combine (4.31) and (4.35) to find the corrected version of these fields. After +either doing a tedious calculation or applying the Mathematica tool attached, +one can obtain the corrected version of the left- and right-invariant vector fields, +respectively: +−Hℓ ≡ hℓ = �hℓ +−Hr ≡ hr + ∂γ1 +γ1 += �hr + ∂γ2 +γ2 +Eℓ ≡ eℓ = �eℓ + 1 +2∂ +��γ1 +�γ2 +� +Er ≡ er = �er +Fℓ ≡ fℓ + 1 +2∂ +�γ2 +γ1 +� += �fℓ +Fr ≡ fr + 1 +2∂ +�γ3 +γ1 +� ++ ∂γ3 +γ1 += �fr + 1 +2∂ +��γ3 +�γ2 +� ++ ∂�γ3 +�γ2 +. +(4.36) +As one can see, we regrouped terms in the expression, so the sections are actually +smooth and well-defined on the whole patch. For example, the term ∂γ1 +γ1 would +have a pole on the second patch, where γ1 can be equal to zero. Thus, it is only +defined smoothly on the first patch. +The OPEs, as expected, constitute the affine Kac-Moody vertex algebra. +For example, the left OPEs are: +Hℓ(z)Eℓ(w) ∼ 2Eℓ(w) +z − w , +Hℓ(z)Hℓ(w) ∼ +−3 +(z − w)2 , +Hℓ(z)Fℓ(w) ∼ −2Fℓ(w) +z − w +, +Eℓ(z)Fℓ(w) ∼ +−3/2 +(z − w)2 + Hℓ(w) +z − w , +which make it into V−3/2 (sl(2, C)). From the anomaly inflow argument we know +that the total level should be kℓ + kr = −2h∨ = −4. Thus, the right algebra +is the affine algebra V−5/2 (sl (2, C)), which we could again check by a direct +computation. +Operator products between all the left and right global sections are non- +singular. The level is defined with respect to the standard bilinear form (, ) = +(2h∨)−1(, )K, where (, )K is the Killing form. In the case of sl(2, C) it is given +by (h, h) = 2, (e, f) = (f, e) = 1. +General k +To find the most general form of this algebra, we need to find the moduli +space of this CDO. By section 2 we know that it is equivalent to finding +30 + +H1 � +Ω2,cl, SL (2, C) +� +. We want to show that +µ = 2tdγ1 ∧ dγ2 +γ1γ2 +, +t ∈ C, +(4.37) +is the only generator of that cohomology. We show it in three steps. First, we +use that H1 � +SL (2, C) , Ω2,cl� +→ H3 +dR (SL (2, C) , C) is injective (see Appendix +A). Second, it is known that the 3rd de Rham cohomology for simple Lie groups +is generated by Tr +� +g−1 dg +�3, i.e., H3 +dR (SL (2, C)) ∼= C. Third, the form above +is well-defined on U12 = U1 ∩ U2 ∼= C∗ × C∗ × C and has a non-trivial pe- +riod over a non-contractible cycle on the intersection. +Thus, it means that +it represents a non-trivial class in H1(SL(2, C), Ω2,cl). +So, we showed that +H1 � +Ω2,cl, SL (2, C) +� ∼= H3 +dR (SL (2, C) , C) and 4.37 is the non-trivial element. +The coefficient t there is thus the only CDO modulus. +After all these preparations, we can finally redo the calculations with the +transformation law shifted by µ (2.12). One gets the following sections: +−Hℓ ≡ hℓ + tγ′ +1 +γ1 += �hℓ − tγ′ +2 +γ2 +−Hr ≡ hr + (1 − t)∂γ1 +γ1 += �hr + (1 − t) ∂γ2 +γ2 +Eℓ ≡ eℓ = tγ′ +1 +γ2 ++ �eℓ + 1 +2∂ +�γ1 +γ2 +� +Er ≡ er = �er +Fℓ ≡ fℓ + 1 +2∂ +�γ2 +γ1 +� ++ tγ′ +2 +γ1 += �fℓ +Fr ≡ fr + 1 +2∂ +�γ3 +γ1 +� ++ (1 − t)∂γ3 +γ1 += �fr + 1 +2∂ +��γ3 +�γ2 +� ++ (1 − t)∂�γ3 +�γ2 +. +(4.38) +Effectively, we observe that introducing the form (4.37) leads to a shift of +levels kℓ → kℓ − t and kr → kr + t. We will set t = 1 +2 + k for convenience, where +k now is the actual Chern-Simons level. The levels of the boundary algebras +are then −2−k and −2+k. The quantization condition is not necessary in this +approach, but is necessary from the 3D perspective. In this context it means +that k ∈ Z. +Thus, the affine algebras of the global left and right G-action +sections are V−2±k (sl (2, C)). The explicit form of these sections is one of the +key technical results of this chapter. +Module Structure +To reveal the module structure of Dk[SL(2, C)] with respect to V−2±k (sl (2, C)), +let us consider: +Eℓ(z)γ1(w) ∼ 0 +Eℓ(z)(−γ2)(w) ∼ γ1(w) +z − w +Hℓ(z)γ1(w) ∼ γ1(w) +z − w +Hℓ(z)(−γ2)(w) ∼ γ2(w) +z − w +Fℓ(z)γ1(w) ∼ −γ2(w) +z − w +Fℓ(z)(−γ2)(w) ∼ 0. +(4.39) +Thus, as before, γ’s generate modules for our boundary algebras and are iden- +tified with the Wilson lines in the 3D description. One finds that we quotient +out the singular vector of the underlying sl2 algebra, i.e. (F 0 +ℓ )2γ1 = 0. +31 + +One also finds that the vectors (γ1)0 |0⟩, −(γ2)0 |0⟩, and all vectors obtained +from them by the action of the negative modes of Eℓ, Hℓ, and Fℓ span a module +over the left current algebra, with the vector (γ1)0 |0⟩ being the highest weight +vector of weight 19. There is an isomorphic module over this subalgebra, “gener- +ated” by γ3 and γ−1 +1 (1+γ2γ3), with the first field giving the highest vector. One +also finds analogous modules over the right current algebra, “generated” by pairs +of global sections γ1, γ3 and γ2, γ−1 +1 (1+γ2γ3), where again the first field in each +pair defines the highest weight vector of weight 1. Note that these expressions +indeed define global sections due to (4.31). All global functions depend only on +(γ1, γ2, γ3, γ−1 +1 (1 + γ2γ3)) and are modules for zero modes of our currents. +Let us put these building blocks together and consider the vector space: +(γ1)0 |0⟩ +(γ2)0 |0⟩ +(γ3)0 |0⟩ +� +1+γ2γ3 +γ1 +� +0 |0⟩ +This is (1, 1) representation for sl(2)ℓ ⊗ sl(2)r. We can act on this vector space +by negative modes of Ja +ℓ and Ja +r . The vector (γ1)0 |0⟩ is the highest weight +vector of weight (1, 1) in the representations of the corresponding �gℓ ⊗ �gr affine +Kac-Moody algebras. In order to obtain other representation one can act with +higher powers of γn +1 on the vacuum and this yields representation (n, n). So, +the answer at the generic point is +Dk[SU(2)C] = +� +λ∈Z+ +Vλ,−2+k (g) ⊗ Vλ,−2−k (g) , +(4.40) +where again Vλ,−2±k are Weyl modules. +Two points require important clarifications. First, what happens with the +stress-energy tensor of the βγ system −βi∂γi? +It is guaranteed to exist by +[GMS99] as the canonical bundle on a Lie group is trivial. The holomorphic +top form can be written as w = dγ1dγ2dγ3 +γ1 +on the first patch. Thus, the stress- +energy tensor gets corrected to +T (z) = − +� +βi∂γi + 1 +2 (log γ1)′′ , +(4.41) +where the correction is a derivative of the coefficient of the holomorphic top +form. One can show by a direct computation that outside of the critical levels +of the boundary algebras, +Tβγ = Tℓ + Tr, +(4.42) +where Tℓ,r = +1 +2(kℓ,r+h∨) +� +ef + fe + hh +2 +� +are the Sugawara stress-energy tensors. +This result was expected from the general discussion in 3. +9We assumed the mathematical notation, where representations of sl2 are labeled by inte- +gers, not half-integers. +32 + +The second point is that the modules that we are considering are reducible +for the physical values of k ∈ Z. +Not only that, but those singular vectors +are singular for both left and right algebras [Zhu08]. To see this, let us set +the right algebra level kr to be 0. It is an obvious limiting case, but it will +nevertheless show the important feature that is carried over to other values of +k. The simplest singular vector for this module can be found to be +(Er)−1 |0⟩ . +(4.43) +We need to find the form of this vector in terms of the left-invariant fields. +Classically, the vector fields are related by the following change of basis: +� +� +er +fr +hr +� +� = +� +� +� +−γ2 +2 +γ2 +1 +−γ1γ2 +(1+γ2γ3)2 +γ2 +−γ2 +3 +γ3(1+γ2γ3) +γ1 +2 γ2(1+γ2γ3) +γ1 +−2γ2γ3 +1 + 2γ2γ3 +� +� +� +� +� +eℓ +fℓ +hℓ +� +� = S · Vℓ +(4.44) +Of course, at the quantum level the relation is corrected, and for the e field the +correct answer is found to be +Er = V i +ℓ S1i + (−2 + k)(γ1∂γ2 − γ2∂γ1). +(4.45) +So, we see that for the special value k = 2, the correction term disappears, +and the vector Er +−1 can now be obtained both from the left and right algebras. +It is actually lying inside the γ2 +1 representation for the left algebra. Thus for +discrete values of k, different modules start to intersect. Moreover, now there is +no way to obtain this correction term ωf = γ1∂γ2 − γ2∂γ1 from a Wilson line +by the action of �gℓ ⊗ �gr. Note that this form is actually dual to the f-vector +field < ωf, f >= 1. This phenomenon happens for all singular vectors [Zhu08]. +One could ask what happens when we include monopoles. We do not have +a definitive answer, but as mentioned in the introduction and in Section 3.2, +we have conjectures as to what the answer might look like. One expects to get +some sort of truncation of the CDO that contains simple quotients L−h∨±k(g) +rather than V−h∨±k(g). We will look into this issue elsewhere. +4.3 +Open Questions +We have already emphasized many times that determining the non-perturbative +modification of Dk[GC] is an interesting problem. It requires, perhaps, improved +understanding of the non-compact models in 2D, of which our theory is an +example. The usual arguments with quotienting out singular vectors based on +the unitarity of the Hilbert space do not work in such theories. But we expect +that monopoles on the boundary with positive k − h∨ are still required, as in +the opposite limit γ ≫ 1 this boundary is a relative CFT with a normalizable +vacuum. The problems lie on the other boundary which has a non-compact +mode [DL22] that effectively renders our theory non-compact. +Another intriguing question arises from an alternative UV completion of the +GC NLSM via the 2D Landau-Ginzburg (LG) model described in [DL22]. The +simplest example is for G = SU (N). The UV completion is chosen to be the +N = (0, 2) LG model with the following field content: +33 + +1. M i +j are chiral multiplets valued in complex matrices Mat(N, C); Φa +i is +a chiral multiplet that is bifundamental under U (k) × G, where i, j ∈ +1, . . . , N and a ∈ 1, . . . , (k = anomaly). +2. Fermi multiplets Γ and Λj +b. +3. Superpotential W = Γ (det M − 1) + µΛj +aM i +jΦa +i . +This model has the same anomalies as our theory, and the superpotential is +engineered in such a way that it flows to the SL(N, C) NLSM. It is generally +believed that LG models do not carry any non-perturbative physics. Thus one +could hope that the perturbative chiral algebra in this model could provide some +useful information. It is captured by the βγ systems (V j +i , M i +j) (here V is the +“beta” for M), (Ri +a, Φa +i ), and the bc systems (Γ, Γ) and (Λ +a +i , Λi +a). The chiral +algebra is defined in the cohomology of Q that acts according to: +QΓ = det M − 1, +QΛ = MΦ, +QV j +i = Γ∂ det M +∂M i +j ++ µΛj +aΦa +i , +QRi +a = µΛj +aM i +j, +(4.46) +and by zeros on the rest of fields. +All these βγ and bc systems are already +globally defined, so one simply computes the cohomology of such Q. The answer +appears to be just Dk[SL(2, C)], which would be interesting to prove. But more +importantly, this, supposedly exact, answer in the LG model is the same as the +perturbative VOA we find in the interval theory. This suggests that the exact +non-perturbative physics in these models may depend on the UV completion. +Other open questions involve applications to the VOA[M4], which requires +computing the interval reductions of more complicated gauge theories, and +which we will study in the future works. +5 +Conclusion +In this paper we considered the chiral algebra of a 3D N = 2 YM on R2 × [0, L] +with the N = (0, 2) Dirichlet boundary conditions. The algebra was computed +both from the 3D and 2D perspectives. We analyzed this protected sector using +the holomorphic-topological twist of the 3D theory and, among other things, +the holomorphic twist of the 2D theory. +From the 3D perspective, the perturbative algebra was found to be an en- +hancement of two affine vertex algebras living at the boundaries by their bimod- +ules realized via the Wilson lines. The boundary monopoles seem to modify the +answer non-perturbatively, on which we proposed some conjectures. +The two-dimensional system after reduction in the right regime is an N = +(0, 2) NLSM into GC. +The compactification algebra is the chiral algebra of +this 2D model, and the beta-gamma system is the main tool to compute its +perturbative approximation. The global sections corresponding to the left and +right actions of the group on itself were explicitly found for G = SU(2). +In the abelian case, we find that the spectrally flowed modules of the βγ +system are required to get the full result for the algebra. Combining the latter +perspective with the 3D analysis and with the known results on the sigma model +into C∗, we obtain three different presentations of the chiral algebra (also called +34 + +the interval VOA) in (4.30). We also saw that the stress-energy tensor is de- +composed in terms of the Sugawara stress tensors for the boundary symmetries, +both in the abelian and the non-abelian cases. The answers, when available, +fully agree between the 2D and 3D calculations. Some puzzles and speculations +are discussed towards the end and in Section 3.2. +Acknowledgements +We benefited from the useful discussions and/or correspondence with: A. Abanov, +T. Creutzig, T. Dimofte, D. Gaiotto, Z. Komargodski, I. Melnikov, N. Nekrasov, +W. Niu, M. Roček. +A +De Rham cohomology +In this appendix we will show that H1 � +SL (2, C) , Ω2,cl� +→ H3 +dR (SL(2, C)) is in- +jective. H1 � +SL (2, C) , Ω2,cl� +is isomorphic to Zd +� +Ω3,0 ⊕ Ω2,1�/dΩ2,0 [Wit07]. There +is an obvious map from Zd +� +Ω3,0 ⊕ Ω2,1�/dΩ2,0 to the third de Rham cohomology +group H3 +dR(SL(2, C)) given by [α] �→ [α] for any closed 2-form α ∈ Ω3,0 ⊕ Ω2,1. +Proposition 1. For any [α] ∈ Zd +� +Ω3,0 ⊕ Ω2,1�/dΩ2,0 there exists β ∈ Zd +� +Ω3,0� +such that +[α] = [β], +(A.1) +so there is an isomorphism: +A ≡ Zd +� +Ω3,0 ⊕ Ω2,1� +⧸dΩ2,0 ∼= Zd +� +Ω3,0� +⧸dΩ2,0, +(A.2) +where we have made use of a slight abuse of notation, and dΩ2,0 in the last +quotient should be understood as dΩ2,0 ∩ Ω3,0. +Proof. A general form from A has the form α+β for some α ∈ Ω3,0 and β ∈ Ω2,1. +The closeness conditions are +∂α + ∂β = 0, +∂β = 0. +(A.3) +The second condition says that β ∈ Z∂ +� +Ω2,1� +, and using the fact that SL (2, C) is +a Stein manifold with all positive degree Dolbeault cohomology groups vanishing +H·,·≥1 +∂ += 0, one gets that the form β is in fact exact: β = ∂γ for some γ ∈ Ω2,0. +So, shifting α + β by −dγ, we get the desired representative in Ω3,0. +■ +Proposition 2. The map from A to H3 +dR defined above is injective. +Proof. Suppose we have a closed (3,0)-form ω that goes under the map to zero +in H3 +dR(SL(2, C)), i.e. ω = dα for some α ∈ Ω2,0 ⊕ Ω1,1 ⊕ Ω0,2. Let us represent +α as α(2,0) + α(1,1) + α(0,2), where each α(p,q) ∈ Ωp,q. Thus, +ω = dα(2,0) + dα(1,1) + dα(0,2), +(A.4) +and as ω ∈ Ω3,0 we find that ∂α(0,2) = 0. Recalling that SL(2, C) has trivial +non-zero Dolbeault cohomology groups as a Stein manifold, one obtains that +35 + +α(0,2) = ∂γ(0,1) for some γ(0,1) ∈ Ω0,1. +It means, however, that dα(0,2) ≡ +∂∂γ(0,1) + ∂ +2γ(0,1) = −∂∂γ(0,1) = ∂β(1,1). Redefining α(1,1), +ω = dα(2,0) + dα(1,1). +(A.5) +Repeating the same argument with α(1,1), we obtain that ω = dα(2,0), meaning +that it was trivial in A, which proves the statement. +■ +Let us show that the only generator of H3 +dR(SL(2, C)) (which is Tr(g−1dg)3) +after mapping to A indeed corresponds to the closed holomorphic (2,0)-form µ +in H1 � +SL (2, C) , Ω2,cl� +from the Eq. 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In: Advances in Mathe- +matics 219.5 (2008), pp. 1513–1547. issn: 0001-8708. doi: 10.1016/ +j.aim.2008.07.005. arXiv: math/0611517 [math.QA]. +45 + diff --git a/4dAyT4oBgHgl3EQfP_bc/content/tmp_files/load_file.txt b/4dAyT4oBgHgl3EQfP_bc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1f49ce021d0842af3dc29f49486e31396428cd1 --- /dev/null +++ b/4dAyT4oBgHgl3EQfP_bc/content/tmp_files/load_file.txt @@ -0,0 +1,2381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf,len=2380 +page_content='Chiral life on a slab Sergey Alekseev1, Mykola Dedushenko2, and Mikhail Litvinov1 1Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA 2Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3636, USA January 3, 2023 Abstract We study chiral algebra in the reduction of 3D N = 2 supersymmetric gauge theories on an interval with the N = (0, 2) Dirichlet boundary conditions on both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' By invoking the 3D “twisted formalism” and the 2D βγ-description we explicitly find the perturbative Q+ cohomology of the reduced theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is shown that the vertex algebras of boundary operators are enhanced by the line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A full non-perturbative result is found in the abelian case, where the chiral algebra is given by the rank two Narain lattice VOA, and two more equivalent descriptions are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Conjectures and speculations on the nonperturbative answer in the non-abelian case are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='00038v1 [hep-th] 30 Dec 2022 Contents 1 Introduction 2 2 Basics 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1 Basic Supersymmetry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2 Holomorphic-topological twist .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 33 5 Conclusion 34 A De Rham cohomology 35 1 Introduction The role of vertex operator algebras (VOA) in theoretical physics and math- ematics has vastly expanded over the recent decades, way beyond their origi- nal scope [BPZ84;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Bor86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FLM88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This includes, among others, applications of VOAs in differential topology of four- and three-manifolds [DGP17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FG20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Che+22], and in higher-dimensional QFT [Bee+15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' BRR15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CG19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CCG19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' While some of these topics are more developed [Bee+15], the existing literature only scratches the surface of the topological applications and some other top- ics, such as boundary algebras in [CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Our main interest in the current paper is an interplay between the older role of VOAs as chiral algebras in 2D N = (0, 2) theories [Wit94;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' SW94], and their modern appearance as boundary algebras supported by the (0, 2) boundary conditions in 3D N = 2 theories [CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As will be explained later, this is also motivated by applications to the four-manifolds, following [DGP17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FG20], see also earlier works [GGP16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ASW16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The main subject of this work is a 3D N = 2 theory placed on an interval with N = (0, 2) boundary conditions on both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This basic setup is also explored in a companion paper [DL22] from the more physical perspective, where we compute the effective 2D N = (0, 2) action in the infrared (IR) limit of the interval-reduced 3D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is known that 2D N = (0, 2) theories contain VOAs as their chiral algebras in the Q+-cohomology of local observables [Wit94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Being invariant under the renormalization group (RG) flow [Ded15], such a chiral algebra must admit a UV realization in a 3D theory on the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It consists of a pair of (possibly different) VOAs, associated to the boundaries, 2 extended by a category of their bi-modules associated to the appropriate line operators stretched between the boundaries (see an illustration on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Bℓ Br L Figure 1: Two boundaries with a line operator connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Generally speaking, one expects various types of line operators to appear, including the descendant lines compatible with our supercharge [CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the simplest case of pure N = 2 super Yang-Mills (SYM) with Chern-Simons (CS) level, which will be our main focus in the bulk of this paper, we will fully describe the spectrum of relevant lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In particular, Wilson lines (which are descendants of the ghost field) play an important part in this story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This connects us to another topic that has seen a lot of interest recently, – line defects and their role in related or similar constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, holomorphic boundary conditions in 3D topological QFT (TQFT) may sup- port non-trivial (relative) rational CFT (RCFT), and such TQFT/RCFT cor- respondence [Wit89;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FK89] has been studied in great detail [Fel+00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FRS02a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FRS02b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FRS04a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FRS04b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Frö+04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Bulk topological lines that are parallel to the boundary lead in this case to the boundary topological lines, while the bulk topological lines piercing the boundary give modules of the boundary chiral algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Fusion of such lines corresponds to fusion of the VOA modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that in this story one also naturally encounters interval reductions: A TQFT on an interval (with the appropriate boundary conditions) leads to the full (non- chiral) RCFT, and segments of line operators stretched between the boundaries generate its primary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A more sophisticated class of examples comes from the topologically twisted 3D N = 4 theories, whose categories of line defects were recently studied in [Dim+20], see also [Cre+21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Gar22a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Gar22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Such theories may also possess holomorphic boundary conditions [CG19] supporting certain boundary VOAs, and the bulk lines piercing boundaries generate their modules as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This leads to interesting questions of determining categories of VOA modules corresponding to the bulk lines [BN22], and it has also been argued that moduli spaces of vacua of the underlying physical theory can be re- covered from the knowledge of boundary VOAs and their categories of modules [CCG19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We are interested in similar constructions applied to three-dimensional the- ories with N = 2 supersymmetry (including 3D N = 4 viewed as N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The N = (0, 2) boundary conditions [GGP14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' OY13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' YS20] in such theories support non-trivial boundary VOAs in the Q+-cohomology, which is a direct 3 analog of the 2D chiral algebra construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The N = (0, 2) boundary condi- tions are compatible with the holomorphic-topological (HT) twist in 3D N = 2 [Aga+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, one may focus entirely on such HT-twisted theo- ries, which often simplifies the VOA-related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The HT twist is also compatible with complexified Wilson lines, vortex lines in abelian [DOP12] and non-abelian [HS21] cases and their generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Unlike in the topologically twisted theories, these line operators are not fully topological and cannot have arbitrary shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' They are supported along a straight line in the topological direction and are point-like in the holomorphic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In general, one could choose different left and right boundary conditions Bℓ and Br, leading to the left and right boundary chiral algebras Vℓ and Vr and relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, often Bℓ = B admits a counterpart Br = B⊥, such that the interval theory is trivially gapped in the IR, with the trivial chiral algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This would imply an interesting “duality” between the boundary chiral algebras, perhaps of relevance to some of the questions studied in [Che+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The only pairs of N = (0, 2) boundary conditions on the interval that appear in the literature so far include: Neumann-Neumann for gauge theories in [SY20], and Dirichlet-Neumann in [DN21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' BZ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this paper we will focus on the simplest nontrivial case of 3D N = 2 pure SYM with group G and CS level k, which is placed on the interval with N = (0, 2) Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One of the main goals of this paper is to elucidate both the structure of chiral algebra and the underlying physics in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Perturbatively, each boundary supports the affine VOA Vn(g), where g = Lie(G) and the level is n = −h∨ ± k for the left/right boundary, respectively, with h∨ being the dual Coxeter number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The full perturbative VOA (that one may compare to the IR one) is given by V−h∨−k(g)⊗V−h∨+k(g) extended by a series of bimodules, realized in this case via SUSY Wilson lines (in all representations) stretched between the boundaries (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Non- perturbatively, this answer is significantly modified, and we have a complete understanding for the abelian G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Partial results, conjectures, and challenges of the non-abelian case will be discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The IR description of the interval-reduced theory is identified in the com- panion paper [DL22] as an N = (0, 2) non-linear sigma-model (NLSM) into the complexification of the gauge group GC ≡ exp(g ⊗ C), with the Wess-Zumino (WZ) level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is essentially a non-compact N = (0, 2) version of the Wess- Zumino-Witten (WZW) model, which exists for arbitrary Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 1 The HT twist in 3d reduces to the purely holomorphic twist in 2d N = (0, 2) theories, sometimes called the half-twist2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The perturbative chiral algebra in such models is related to the theory of chiral differential operators (CDO) [GMS99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GMS01] on the target or, synonymously, curved βγ systems [Nek05], as explored in de- tail in [Wit07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The VOA extracted from the sheaf of CDO on GC is denoted Dk[GC] and is the perturbative chiral algebra in the IR sigma-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Here k is the modulus of the CDO corresponding to the B-field on GC (proportional to the connection on the canonical WZ gerbe on GC, with k appearing as the 1Note that this differs from the compact N = (0, 2) WZW models that were previously con- structed for even-dimensional target groups, such as U(1) × SU(2), which all possess complex structure [Spi+88a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Spi+88b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Sev+88;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' RSS91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Roc+91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Ang+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2However, more often than not, the term refers to something else, usually in the context of N = (0, 2) deformations of the N = (2, 2) theories, see [KS06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Sha09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' MS08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GS09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' MM09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Mel09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Kre+11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' MP11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' MSS12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GJS15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GS17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GGS21] 4 proportionality factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From the CDO perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' in perturbation theory, k does not have to be quantized, and indeed it labels the complex B-field flux in H3(GC, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In our interval model, however, the B-flux is non-generic and related to the CS level in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For convenience, in this paper the B-flux k is parameter- ized in such a way that it is precisely equal to the CS level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is a mathematical fact that V−h∨−k(g) ⊗ V−h∨+k(g) is a sub-VOA of Dk[GC] [GMS01], thus the latter is expected to be an extension of the former by bi-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For generic k (which here means k ∈ C\\Q) such an extension is indeed known to hold [AG02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FS06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Zhu08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CG20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CKM22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Mor22]: Dk[GC] ∼= � λ∈P+ Vλ,−h∨+k (g) ⊗ Vλ∗,−h∨−k (g) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) where one sums over the dominant weights λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Physically, the bi-modules in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) come from the Wilson lines connecting the boundaries, as stated earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) makes perfect sense in perturbation theory, where we are allowed to treat the CS level as a generic complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' At physi- cal integer values of k, however, the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) no longer holds, since the structure of Dk[GC] as a V−h∨−k(g)⊗V−h∨+k(g) bi-module becomes much more intricate [Zhu08] (for non-abelian G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Not only that, but also the nonperturba- tive corrections are expected to significantly modify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Indeed, Dk[GC] contains universal affine sub-VOAs V−h∨−k(g) and V−h∨+k(g) [GMS01], not their simple quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Now, the algebra V−h∨−k(g) for k > 0, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' at the below-critical level,) is already simple, see [KL93, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12] in the simply-laced case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' However, for k > h∨, the affine VOA V−h∨+k(g) living on the “positive” bound- ary is not simple: It is very well understood [KK79], has the proper maximal ideal, and the simple quotient denoted L−h∨+k(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It was argued in [CDG20] that the nonperturbative boundary monopoles implement this simple quotient, at least on the “positive” boundary, turning V−h∨+k(g) into L−h∨+k(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus we expect the exact chiral algebra (for k > h∨) to contain V−h∨−k(g)⊗ L−h∨+k(g), not V−h∨−k(g) ⊗ V−h∨+k(g), which already differs from Dk[GC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Furthermore, the 3D bulk is expected to admit only finitely many line operators in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Indeed, it is gapped (at least if the interval is long enough) and given by the Gk−h∨ CS for levels3 k > h∨, which only admits finitely many Wilson lines labeled by the integrable representations of L−h∨+k(g) [Wit89;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Kac95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We, therefore, expect the exact VOA to be, roughly, V−h∨−k(g) ⊗ L−h∨+k(g) extended by such a finite set of bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This appears to be significantly different from Dt[GC] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', the latter is an infinite extension for generic t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this paper, we fully address the abelian case, G = U(1), and give par- tial results on the nonabelian G, including the perturbative treatment outlined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We also speculate on the kind of nonperturbative corrections in the non- abelian GC NLSM that are expected to modify Dk[GC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We leave the detailed study of such nonperturbative effects for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the abelian case, GC = C∗, so the IR regime is described by the NLSM into C∗, which was analyzed previously in [DGP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This theory of course has a C∗-valued βγ system for its perturbative chiral algebra Dt[C∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We think of it ei- ther multiplicatively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', γ is C∗-valued, or additively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', �γ ∼ �γ +2πiR, where R is the radius of the compact boson (to be determined by the CS level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Since the theory is free, the distinction between perturbative and non-perturbative 3The IR behavior at arbitrary k is described in [Bas+18], see also [AHW82;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Oht99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GKS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 5 is slightly formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the case of the C∗ model, the natural nonperturbative completion amounts to including twisted sectors into the theory, which corre- spond to windings around the nontrivial one-cycle in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The twist fields are vortex-like disorder operators, whose 3D origin is, expectedly, the boundary monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The presence of such sectors extends the βγ system by the so-called spectrally flowed modules [RW14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' AW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We show that this results in the lat- tice VOA for the simplest Narain lattice [Nar86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' NSW87], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', Z2 ⊂ R2 with the scalar product whose Gram matrix is (0 1 1 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In addition to showing this, we also provide the 3D perspective, where each boundary supports a 1D lattice VOA (abelian WZW [DGP18]), and Wilson lines extend them by the bi-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In total, we find three presentation of the same VOA: (1) as a Narain lattice VOA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2) as an extension of the βγ system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3) as an extension of the abelian WZWk ⊗ WZW−k by its bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The nonabelian GC NLSM is interacting, which makes the non-perturbative corrections to the perturbative answer Dk[GC] much more challenging to quan- tify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We are certain from the previous discussion, however, that such corrections must be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The known instanton effects in 2D N = (0, 2) theories are vor- tices that are best understood in the gauged linear sigma model case [SW95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' BS03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' BP15] (see also [LNS00] for the N = (2, 2) case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the NLSMs they are captured by the worldsheet wrapping compact holomorphic curves in the target [Din+86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Din+87;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' BW06], which is notoriously hard to compute except for the simplest models [TY08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In our case, however, GC does not support any compact holomorphic curves, which basically implies that there must be some novel nonperturbative corrections at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' They correspond to the boundary monopoles that exist in the parent 3D gauge theory, and can be described as new “noncompact” vortices in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Indeed, monopoles are labeled by the cocharacters eibϕ : U(1) → G, up to conjugation, which are complexified to zb : C∗ → GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This allows to define a natural vortex singularity for the NLSM field φ(z, z): φ ∼ zb, as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2) Since it is meromorphic, it will define a half-BPS defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Dynamics in the presence of such defects is expected to modify the chiral algebra Dt[GC] ap- propriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will explore this elsewhere, while here we only focus on the perturbative aspects when G is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Another subtle point is that the presence of such disorder operators, and hence the non-perturbative corrections, in principle, depends on the UV com- pletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will assume that the 3D gauge theory with compact G admits monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4 Without them, the Dirichlet boundary on the “positive” end would appear in tension with unitarity [DGP18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CDG20] (the same issue is not present on the “negative” boundary since it supports a non-compact CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus the NLSM originating from the gauge theory on the interval must admit the vortex- type disorder operators and the corresponding non-perturbative phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' On the other hand, if one completes the GC NLSM in the UV into the LG model as in [DL22], there is no room for any vortex defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this case the Dt[GC] is con- jectured to give the exact chiral algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Purely from the viewpoint of NLSM, it is conceivable that the knowledge of metric on the target (which is usually ignored in the BPS calculations, but which was computed in [DL22],) allows one 4Which has far-reaching consequences for its dynamics, as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', in Polyakov’s argument for confinement in 3D [Pol77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 6 to tell apart the cases with and without the nonperturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This question is likewise left for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Motivation via VOA[M4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One of the motivations behind this work is to de- velop a toolkit for computing VOA[M4] [DGP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='5 Recall that the VOA[M4], or more precisely VOA[M4, g], is defined as the chiral algebra in the Q+-cohomology of the 2D N = (0, 2) theory T[M4, g], which is obtained by the twisted compact- ification of the 6d (2, 0) SCFT (of type g) on the four-manifold M4 [GGP16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us consider a class of four-manifolds M4 admitting a metric with S1 isometry, such that the S1 action is not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that when the S1 action is free, the four-manifold is an S1 bundle over some smooth three-manifold M3 (which is a relatively tame class of four-manifolds), and it is conceptually clear how the dimensional reduction simplifies (first reduce on S1 and then on M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Of course this is still an interesting and nontrivial problem, but our motivation comes from the opposite case, when the S1 action is not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Some examples of such four-manifolds include but are not limited to: (1) Σg × S2, where Σg is an arbitrary genus-g surface, and the S1 action rotates S2, which is equipped with an S1-invariant “sausage” metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2) the unique nontrivial sphere bundle over the Riamann surface of genus g, denoted as Σg �×S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3) CP2 with the standard Fubini-Study metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4) Hirzebruch surface, given by the connected sum CP2#CP 2, which is actually isomorphic to the nontrivial sphere bundle over a sphere, S2 �×S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (5) four-sphere S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, the general class of such four-manifolds is quite well understood, at least in the simply connected compact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It was shown by Fintushel [Fin77;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Fin78] that if a simply connected M4 admits an S1 action (not necessarily an isometry), it must be a connected sum of some number of S2 × S2, S4, CP2 and CP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If S1 is an isometry and the corresponding simply-connected four-manifold is, in addition, non-negatively curved, [SY94] proved (see also [HK89]) that it belongs to the list {S4, CP2, S2 × S2, CP2#CP 2, CP2#CP2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If we only allow codimension-2 fixed loci, then the list is even shorter: {S2 × S2, CP2#CP 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Such lists may appear utterly specialized, however, these manifolds present certain interest to us in view of conjectures in [FG20], which we aim to check in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We may consider the twisted compactification on a manifold with S1 isom- etry in two steps: (1) first reduce the 6D theory along the S1 orbits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2) then perform reduction along the remaining quotient space M4/U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The advantage of this procedure is that the first step yields a relatively simple and concrete re- sult – the maximal 5D SYM (MSYM) with gauge group G (the simply-connected Lie group whose algebra is g), albeit placed on some curved space with bound- aries and, possibly, defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, for M4 = Σg × S2, reducing along the parallels of S2 gives the Σg × I geometry, where I is an interval with the principal Nahm pole boundary conditions at both ends [CDT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Further re- duction of the 5D MSYM along Σg with the topological twist simply gives a 3D N = 4 SYM with g adjoint hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus we end up with the former 3D theory on the interval, with the (0, 4) Nahm pole boundary conditions at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this case, the resulting effective 2D theory in the IR has (0, 4) SUSY [PSY16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Other examples would lead to the interval reductions of vari- 5A few recent appearances of the interval compactifications in related contexts include [GR19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' PR18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' DP19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 7 ous 3D N = 2 gauge theories with matter and CS levels, which would flow to N = (0, 2) theories in the 2D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will explore various such examples in the future work [DL23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' However, it is natural to start with the most basic 3D N = 2 gauge theory, that is the pure SYM, and study the interval VOA in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is one of the underlying motivations for the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In Section 2 we review the necessary background material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then we move on to computing the interval compactification chiral algebra in the 3D N = 2 SYM theory with N = (0, 2) Dirichlet boundaries in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In Section 4 we compute the same chiral algebra from the 2D perspective and discuss some issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the abelian case, we end up with three different presentations of the same VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the nonabelian case, we make general statements when possible, but mostly work with the G = SU(2) example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then we finish with some open questions and speculations, and conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2 Basics In this section, we set up the conventions and briefly review the background material, including the holomorphic-topological (HT) twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In particular, we discuss the 3D N = 2 supersymmetric theory and its twisted content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will describe protected sectors, namely, the cohomology of a Q+ supercharge in 3D and 2D theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A βγ system will be briefly discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Also the connection between the cohomology of N = (0, 2) theories and the Čech coho- mology of the βγ system is expounded upon, as it will be one of the important computational tools later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Conventions: We consider R2 × I with Euclidean signature and with coor- dinates xµ on R2 and t ∈ [0, L] on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We choose γi αβ matrices to be the Pauli matrices σi, and the antisymmetric symbol ϵ12 = ϵ21 = 1 to lower and raise indices [IS13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1 Basic Supersymmetry In Euclidean 3D space spinors lie in a 2-dimensional complex representation of SU (2) and the N = 2 supersymmetry algebra takes the following form: � QI, QJ� = δIJγµPµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) By defining a new combination of supercharges Q = Q1+iQ2 and Q = Q1−iQ2, one can obtain the following conventional form of the superalgebra: � Q, Q � = 2γµPµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2) Note that the supercharges are not conjugate to each other in Euclidean sig- nature contrary to Minkowski space, where minimal representations are real (Majorana).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This algebra admits a U (1)R-charge, which is an automorphism of this algebra and acts by rotating Q-charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Operators also have a charge J0 8 with respect to Spin(2)E rotation parallel to boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us also define the combination J := R 2 − J0, then all charges can be summarized by the following table: Q+ Q+ Q− Q− dz dz U(1)R 1 −1 1 −1 0 0 Spin(2)E 1 2 1 2 − 1 2 − 1 2 1 −1 U(1)J 0 −1 1 0 −1 1 In what follows, we will be considering the cohomology of Q = def Q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The Pz is the only Pµ which is not Q-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It makes our algebra into an algebra with only holomorphic dependence on the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We would consider boundary conditions that preserve a (0, 2)-part of the supersymmetry algebra generated by Q+ and Q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We also want to leave U (1)R unbroken in the bulk and on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The only relevant 3D N = 2 multiplet for this paper is a vector multiplet for some gauge group G: V3D = θσmθAm + iθθσ − iθ2θλ − iθ 2θλ + 1 2θ2θ 2D3d (WZ gauge) , where all the fields lie in the Lie algebra g = Lie(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It consists of a connection Am, a real scalar σ, a complex fermion λα, and a real auxiliary field D3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can also define a covariant superfield Σ3D = − i 2ϵαβDαDβV3d = σ − θλ + θλ + θγµθϵµνρF νρ + iθθD + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' that satisfies DαDαΣ3D = D αDαΣ3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2 Holomorphic-topological twist In this section, we review some formulas of the HT-twisted formalism [Aga+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The Q-cohomology of operators of the twisted theory and physical theory are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The convenience of this formalism is that some calculations have only a finite number of Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The twisted formalism is reviewed nicely in [CDG20, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us consider a dg-algebra: Ω• = C∞ � R3� [dt, dz] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3) where the multiplication is the multiplication of differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One also needs to consider forms with values in the k-th power of the canonical line bundle in the z-direction: Ω•,k = Ω• ⊗ Kk = C∞ � R3� [dt, dz] dzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4) There is cohomological charge R, which is related to the original R-charge in the physical theory by adding a ghost charge to it, and a twisted spin charge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the holomorphic-topological twisted 3D N = 2 theory the fields can be organized into the following BV superfields: A = c + (At dt + Az dz) + B∗ zt dz dt ∈ Ω• ⊗ g [1] , B = � B + A∗ µ dxµ + c∗ dz dt � dz ∈ Ω•,1 ⊗ g∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='5) 9 where the superfields A and B are obtained from the vector multiplet, and we also introduced ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, the field A := Az dz + At dt is just a connection with complexified At = At − iσ and ordinary Az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The field B ≡ Bz is identified with 1 g2 Fzt = 1 g2 Fzt + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' on shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The bracket [1] indicates a shift of cohomological degree by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The forms dt and dz are treated as Grassmann variables, so they anticommute with fermionic fields, and superfields can be regarded as either bosonic or fermionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The action of the Q-charge in the twisted formalism can be written as follows: QA = F(A), QB = dAB − k 2π∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6) Here the differentials are defined as follows: dA = d′ − iA, d′ = dt ∂t + dz ∂z, ∂ = dz ∂z (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='7) and the curvature is F(A) = id2 A = d′A − iA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='8) It will be also useful to keep in mind the following tables of the R and J charges of the operators: c At Az R 1 0 0 J 0 0 1 B A⋆ t A⋆ z c⋆ R 0 1 1 2 J 1 1 0 0 Table 1: The charges of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The following charges are assigned to the differential forms: dt dz dz R 1 1 0 J 0 1 1 Table 2: The charges of the differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3 βγ System In this section we review the βγ system, or as it is usually called in mathematical literature, a sheaf of chiral differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' All formulas can be found in [GMS99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GMS01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Nek05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Wit07] Classically, consider a complex manifold M, a map γ : Σ → M, and a (1, 0)- form β on Σ with values in the pullback γ∗(T ∗M), governed by the following action: � Σ βi∂γi, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='9) where γi and βi are the holomorphic components of γ and β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 10 Quantum mechanically, the situation is more interesting as we want to pre- serve the OPE’s locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' On each patch we have the usual βγ system with the OPE: γi (z) βj (w) ∼ δi j dw z − w, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='10) which in the physics notation yields: � γi n, βj k � = δi jδn+k,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='11) The normal ordering prescription for polynomials is defined by the point-splitting procedure and depends on a chosen complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To get a global theory, we need to learn how to glue fields on different patches together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' First, let us choose two sets of local coordinates γi and �γb on some open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The gluing is done by a local automorphisms and γ is transformed as in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As we mentioned before, β is transformed classically as β �→ �β = f ∗β, where f is a local holomorphic diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The quantum version of this transformation law is given by the following general formula [Nek05]: �βa = βi ∂γi ∂�γa − 1 2 � ∂jgi a∂igj b � ∂�γb ∂γk ∂γk � �� � quantum part + 1 2µab∂�γb � �� � moduli parameter , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12) where the Jacobian of the transformation is gi a = ∂γi ∂�γa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The “quantum part” appears because we want to keep the right OPE on both patches after gluing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There is an intrinsic ambiguity associated to solving for the OPE equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Moreover, the moduli space of the βγ system is parametrized by µ or, stating it simply, different ways of gluing our system globally are in one to one corre- spondence with the possible choices of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The parameter µ takes values in the first Čech cohomology group with coefficients in the sheaf of closed holomorphic two-forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' H1 � Ω2,cl, M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This algebra becomes VOA if c1(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The global stress energy tensor is T = −βi∂γi − 1 2(log w)′′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='13) where w is the coefficient of the holomorphic top form ω = wdγ1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ∧ dγn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4 (0,2) Cohomology And βγ System One of the physics applications of the curved βγ system is in the realm of (0, 2) theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As discussed in [Wit07] and will be reviewed shortly, the βγ system describes the perturbative cohomology of half-twisted (0, 2) theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us first discuss a general (0, 2) sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The Lagrangian is con- structed locally by introducing a (1, 0)-form K = Ki dφi, with complex conju- gate K = Ki dφ i, and setting I = � ��d2z �� dθ + dθ+ � − i 2Ki(Φ, Φ)∂zΦi + i 2Ki(Φ, Φ)∂zΦ i� , 11 where Φi is a chiral superfield whose bottom component φi defines a map from a Riemann surface Σ to a target complex manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The cohomology of the supercharge Q+ can be deformed by H = 2i∂ω ∈ H1 � M, Ω2,cl� , where ω = i 2 � ∂K − ∂K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Not only that but H must be of type (2, 1) to preserve (0, 2) supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that it is the same class that parametrizes the βγ system moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If we set αi = − √ 2ψ i +, ρi = −iψi +/ √ 2 and twist the theory then ρ is an element of Ω0,1 (Σ) ⊗ φ∗ (TX) and α is from φ∗ � TX � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Both α and ρ are Grass- mann variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' After the twisting, Q+ becomes a worldsheet scalar with the following action on the fields: Qφi = 0, Qφ i = αi, Qρi z = −∂zφi, Qαi = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='14) and the action is given by: I = � d2z � gij∂zφi∂zφ j + gijρi z∂zαj − gij,kαkρi z∂zφ j� + ST , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='15) where ST = − � d2z � Tij∂zφi∂zφj − Tij,kαkρi∂zφj� and H = dT should be of the type described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We also note that T is not a 2-form but a 2-gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Locally, the structure of the Q-cohomology can be understood easily with the help of the βγ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Consider an open ball Uα: I = 1 2π � Uα ��d2z �� � i,j δi,j � ∂zφi∂zφ j + ρi∂zαj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='16) All the sections in the cohomology can be written as (for details refer to [Wit07]): F � φ, ∂zφ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ∂zφ, ∂2 zφ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' � ∈ H0 � Ops2d, Q 2d + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='17) If we set βi = δij∂zφ j, which is an operator of dimension (1, 0), and γi = φi of dimension (0, 0), the bosonic part of the action can be rewritten as: Iβγ Uα = 1 2π � ��d2z �� � i βi∂zγi (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='18) and the space of all sections of this theory is F � γ, ∂zγ, ∂2 zγ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' β, ∂zβ, ∂2 zβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='19) So, locally, the space of sections of the βγ system and the Q-cohomology of the (0, 2) sigma model coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Globally, things are a little more complicated and we are required to consider Čech cohomology to find the operators with all possible R-charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The R-charge in the sigma model description is matched with the cohomological degree: H• � Ops2d, Q 2d + � ∼= H• ˇCech(X, �A), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='20) where �A is a sheaf of free βγ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 12 3 3D Perspective In this section, we discuss the Q-cohomology from the 3D N = 2 point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There are a few constructions one could consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Firstly, the Q-cohomology of local operators in the bulk is a commutative vertex algebra (VA) V intrinsic to the theory [CDG20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' OY20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Secondly, the Q-cohomology of local operators at the boundary preserving (0, 2) SUSY (explored in the same reference) is, gener- ally speaking, a noncommutative VA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thirdly, — and this is the new structure that we study here, — one can define the Q-cohomology on the interval, or the chiral algebra of the interval compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If both the 3D theory and its (0, 2) boundary conditions preserve the R-symmetry, this is a vertex op- erator algebra (VOA), as opposed to just VA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', it necessarily contains the stress energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is obvious since in the IR limit, the theory becomes effectively two-dimensional [DL22], and the chiral algebra of an R-symmetric 2D N = (0, 2) theory always has the Virasoro element, as can be seen from the general R-multiplet structure [Ded15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, one can also prove this by constructing the (0, 2) R-multiplet from the integrated currents directly in 3D [BST19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Intuitively, the interval VOA contains all 3D observables that look like local operators in the 2D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' These includes 3D local operators and lines, thus effectively enhancing the Q-cohomology of local operators by the line operators stretched between the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The line operators can additionally be dec- orated by local operators in the Q-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We are allowed to move them to the boundaries, as follows from the properties of Q [CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Additionally, the two ends of the line operator can support some other boundary operators that are stuck there and cannot be shifted into the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus the most gen- eral configuration in the Q-cohomology consists of a line stretched between the boundaries with some local operators sitting at its two endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This in- cludes the possibility of colliding a boundary operator from the boundary VA mentioned earlier with the endpoint of a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' On the half-space, the latter implies that lines ending at the boundary engineer modules for the boundary VA [CCG19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In our case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' on the interval, this similarly means that the line operators give bi-modules of the pair of boundary VAs supported at the two ends of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Examples of line operators that appear here include descendants of the Q- closed local operators integrated over the interval [CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Things like Wilson and vortex lines or their generalizations [Dim+20] may appear as well (the Wilson line can be also viewed as a descendent of the ghost field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We are striving to compute the OPE involving such operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, we will compute the exact chiral algebra in the abelian case and the perturbative one in the nonabelian case, that is the OPE of both local and line operators, for gauge theories with the Dirichlet boundary conditions preserving (0, 2) supersymetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will find that the order line operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Wilson lines, create representations for the boundary operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' They are naturally included into the perturbative interval VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The disorder or vortex lines (when allowed), on the other hand, together with the boundary monopoles should be viewed as manifestation of the non-perturbative phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We claim to fully understand them in the abelian case but only briefly discuss in the nonabelian setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Generally speaking, we have chiral algebras on the left and right boundaries denoted by Vℓ and Vr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There is also the bulk algebra (commutative 13 VA) V, which includes only local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Moreover, V maps naturally into the left and right algebras via the bulk-boundary maps, allowing to define their tensor product over V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There are two maps, which are defined by pushing the local operators from V to the two boundaries: ρℓ,r : V → Vℓ,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) Let us first define the algebra that only includes the local operators in 3D: Vℓ ⊗V Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2) The tensor product over V involves the identification of operators that can be ob- tained from the same operator in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The next step is to extend this alge- bra by modules that correspond to the Q-closed line operators stretched between the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will denote the resulting 3D cohomology as H•(Ops3d, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Last but not least, let us note explicitly that in the non-abelian case, we will be mostly discussing the CS level k > h∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The IR physics of a 3D N = 2 YM-CS is known for all values of k [Bas+18], and for 0 < |k| < h∨ it exhibits spontaneous SUSY breaking [Ber+99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Oht99], and runaway for k = 0 [AHW82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' What happens on the interval in the range 0 ≤ |k| < h∨ will be addressed elsewhere, while the k ≥ h∨ case is more straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Yet, it is interesting enough, as we see in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1 Vector Multiplet Consider a vector multiplet sandwiched between the Dirichlet boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the twisted formalism this amounts to choosing A �� = 0[CDG20] at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This, in turn, is equivalent to setting c| = 0 and Az| = 0 The transformation rules for c, A, B, and A∗ in the bulk follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6): Q (c) = −ic2, Q (A) = dAc, QB = −i[c, B] − k 2π∂zc, QA∗ = d′B − i [A, B] − k 2π∂zA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3) where dA = d′ − iA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Before diving deeper, we review what is known about the perturbative alge- bra on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Recall that the action in the HT twist takes the following form: � BF (A) + k 4π � A∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4) In the twisted formalism, the propagator connects A with B, as follows from the kinetic energy B d′A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The rest of terms, including the Chern-Simons, are treated as interactions, which induces the bivalent and the trivalent vertices: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' the vertex connecting two A, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' the vertex connecting two A and one B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This form of Feynmann rules is very restrictive, and there can only be a finite number of diagrams for a given number of external legs [GW19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For the group G the field B lies in g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The gauge group is broken on the boundary and becomes a global symmetry there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There is also a non-trivial 14 boundary anomaly due to the bulk Chern-Simons term and the fermions in the gauge multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The former contributes ±k to the anomaly and the latter contributes −h∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, we expect to get two boundary affine algebras, one for each bound- ary global symmetry, with levels dictated by the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3), on the boundary we have QB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, B is in the cohomology and its OPE with itself was obtained in [CDG20, section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1]: Ba (z) Bb (w) ∼ (−h∨ ± k) κab (z − w)2 + ifabc (z − w) Bc (w) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='5) where Ba are the components of B in some basis, κab is the standard bilinear form equal to 1 2h∨ times the Killing form in that basis, and h∨ is the dual Coxeter number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This expression can be obtained from the charge conservation and anomalies alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The J charge of B is 1 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, only z up to the second power can contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The first term is the anomaly term explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' All half-BPS line operators hitting the boundaries are expected to create modules for the boundary algebras, and we will show that it is true perturba- tively for the Wilson line momentarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A Wilson line can be written as Pe � t A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Observe that it is indeed Q-invariant in the usual formalism, or in the twisted formalism by invoking Q (A) = dAc and c| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The kinetic term for B and A can be written as: Tr B (∂zAt − ∂tAz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6) There is a gauge symmetry associated to this term: At → At + ∂tη, Az → Az + ∂zη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='7) The propagator for B and At with the appropriate gauge fixing [CDG20] is just G(z, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' t) ∝ z |x2| 3 2 , where x2 = zz + t2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='8) where we do not keep track of a proportionality constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Next, we can calculate the OPE of the boundary operator B(0) with the Wilson line segment W (λ)(z) in the irreducible representation of g labeled by the dominant weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Expanding the Wilson line in a Taylor series, one finds that there is only one diagram that can possibly contribute, where a single A from the Wilson line is directly connected to the operator B at the boundary, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is proven simply by looking at the two vertices mentioned earlier and realizing that they cannot contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The diagram evaluates to � Aa t (t, z, z)Bb (0) dt ∝ δa b � L 0 z dt (t2 + zz) 3 2 = δa b L z √ L2 + zz = δa b 1 z + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='9) where we keep only the singular term in the z → 0 expansion on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This computation already includes corrections to the propagator in the presence of 15 W B Figure 2: The diagram connecting single A in the Wilson line to the operator B at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Indeed, such corrections can be accounted for using the method of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Since we are interested in the singular term in the OPE, it is enough to only include the first image At(−t, z, z) = At(t, z, z), as the other ones never get close to B(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This simply doubles the contribution of the original insertion At(t, z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Combining everything together, this computation shows that the OPE with the Wilson line is Ba (z) W (λ) (w) ∼ TaW (λ) (w) z − w , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='10) where Ta denotes the Lie algebra generator in the same representation λ as the Wilson line, and TaW (λ)(w) means the matrix product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Looking at the Table 1, we immediately see that this is the only possible OPE as W (λ) is not charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' More precisely, there are two copies of the affine generators on the interval, denoted Bℓ a and Br a for the left and right boundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Assuming that the Wilson line performs parallel transport from the right to the left, we find the following OPE’s on the interval: Bℓ a (z) W (λ) (w) ∼ TaW (λ) (w) z − w , Br a (z) W (λ) (w) ∼ W (λ) (w) Ta z − w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='11) For completeness, note that the OPE of Wilson line’s matrix elements is regular: W (λ) ij (z)W (µ) kl (w) ∼ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12) simply because no Feynmann diagram can connect two At’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us pause and contemplate on what we have obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We found that Bℓ,r and W (λ) are elements of the extended cohomology, and we claim that they generate the perturbative chiral algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The boundary B’s satisfy the OPE relations of the affine Kac-Moody vertex algebras V−h∨±k(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' They also act on W (λ) as on a primary field of the highest weight representation of the affine algebra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', a Weyl module Vλ,−h∨±k = Ind�g �bVλ, where Vλ is a finite-dimensional module for the underlying Lie algebra g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, we obtain the following result for the perturbative chiral algebra: Ck[GC] := � λ∈P+ Vλ,−h∨+k ⊗ Vλ∗,−h∨−k, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='13) 16 where λ runs over the set P+ of dominant weights, and λ∗ = −w(λ), where w is the longest element of the Weyl group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is not a coincidence that Ck[GC] looks like Dk[GC] from the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This object is well known in the mathematical literature [AG02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' FS06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Zhu08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CG20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CKM22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Mor22], and away from the rational values of k, Ck[GC] is a simple VOA isomorphic to the VOA Dk[GC] of chiral differential operators on GC, with the deformation parameter (perturbative B-field flux) k ∈ C = H3(G, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' At the rational points k ∈ Q, we can encounter singular vectors, and life is getting much more interesting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', Ck[GC] and Dk[GC] are no longer the same [Zhu08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can also ask what happens to the stress-energy tensor in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We know that it does not exist as a local operator in the bulk chiral algebra [CDG20, section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2], and generally, the boundary VA does not have to possess a stress- energy tensor as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' At the same time, we have the current that generates holomorphic translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It acts on the boundary operators as: ∂wO (w) = � HS2 ∗(Tzµ dxµ)O(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='14) There is no boundary part in this expression as we do not introduce any non- trivial degrees of freedom at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can create a line stress-energy operator by stretching the integration surface HS2 to a cylinder in a way that is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This allows to act with the holomorphic translations also on = O1(x) O2(x) O2(x) O1(x) Figure 3: Possible codimension one surfaces over which Tzν is integrated to generate holomorphic translations along the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' line operators by enclosing them with such a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The integration over this tube can be separated into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The integral over dt defines the integrated stress-energy tensor, T int zz = � L 0 dt Tzz, T int zz = � L 0 dt Tzz, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='15) which behaves as a 2D stress tensor generating the holomorphic translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The remaining integration over the contour in the boundary plane is reminiscent of the 2D CFT setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, it was shown in [BST19] that such integrated 3D currents (the stress tensor, the R-current and the supercurrents) fit precisely in the 2D N = (0, 2) R-multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The presence of this multiplet automatically 17 implies existence of the stress-energy tensor in the cohomology [Ded15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the IR regime, as t collapses, T int of course becomes the 2D stress tensor, and the 2D N = (0, 2) arguments are applicable directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In either case, we see again that the interval chiral algebra has the stress-energy tensor that follows from the integrated currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Outside of critical levels, the boundary algebras are VOAs and have well- defined Sugawara stress tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Physically, we expect that the interval stress- energy tensor becomes the sum of T sug ℓ and T sug r as an element in the 2D chiral algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is clear that they act in the same way on the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It indeed turns out to be true as we will argue in the next section using the 2D perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2 The non-perturbative corrections What are the possible non-perturbative corrections to the above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One comes from the boundary monopole operators discussed in [Bul+16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' DGP18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' CDG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Another possibility is the vortex line connecting the two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' General gauge vortices discussed in [KWY13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' DOP14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' HS21] are characterized by a singular gauge background A = b dϕ close to the vortex locus, where ϕ is an angular coordinate in the plane orthogonal to the line, and b is from the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This defines a line defect that is local in the plane orthogonal to the vortex, at least away from the boundary, because gauge invariant objects do not feel the gauge holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This still holds near the Neumann boundary, where the gauge symmetry is unbroken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' However, if such a vortex ends at the Dirichlet boundary, it creates a non-trivial monodromy e2πib for the boundary global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Hence for generic b, it is not a local operator from the 2D boundary point of view as it can be detected far away from the insertion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' On the interval with the Dirichlet boundaries, such vortices do not lead to local operators in the IR, they become the twist fields that are not included in the VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A less general possibility is a vortex characterised by a nontrivial background A = b dϕ, yet its monodromy is trivial, e2πib = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The latter means that b is a co-weight (in fact, a co-character, because the gauge symmetry forces us to consider the Weyl group orbit of b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus such a vortex has its magnetic charge labeled by a cocharacter of G, or a subgroup U(1) �→ G taken up to conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is the same as magnetic charges of the monopoles, and we can think of the vortex as an infinitesimal tube of magnetic flux, with the same amount of flux as created by the charge-b magnetic monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This also suggests that monopoles can be located at the endpoints or at the junctions of vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Such vortex lines, however, are not expected to be independent line operators in the IR, at least for a non-zero CS level there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' When k > h∨, our 3D theory becomes the level k − h∨ CS theory at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It has been argued in [MS89] that such vortex lines are equivalent to Wilson lines in a CS theory (for the abelian case, see the argument in [KWY13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=') Thinking of the CS level as a pairing K : Γ × Γ → Z, where Γ ⊂ t is the co-weight lattice of G, the representation of the Wilson line is determined precisely by the weight K(·, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This is also consistent with the well-known fact (at least in the abelian case [KS11]) that in the presence of CS level, monopoles develop electric charges, and so Wilson lines can end on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In particular, [KS11] used this to argue that the Wilson lines whose charges differ by a multiple of k are isomorphic, 18 thus showing that the finite spectrum of Wilson lines in a CS theory is a non- perturbative effect manifested via the monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that all the statements we referred to here are about the non-SUSY CS theory, but they extend verbatim to the half-BPS lines in the N = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To apply these observations to the interval theory, it is convenient to assume that the interval is long enough, such that we flow to the CS first and only then to 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' At least for k > h∨ this appears to be a harmless assumption, since SUSY suggests that the BPS sector is not sensitive to the interval length, and the IR physics is also more straightforward in this case [Bas+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is therefore natural to conjecture that the non-perturbative effects on the interval with Dirichlet boundaries are captured by the monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the bulk they ensure that there are only finitely many inequivalent Wilson lines, and the boundary monopoles modify the boundary VAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The details depend strongly on whether G is abelian or non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the non-abelian case, the monopoles at the level k−h∨ boundary, accord- ing to the conjecture in [CDG20], turn the perturbative affine VOA Vk−h∨(g) into its simple quotient Lk−h∨(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The finitely many bulk Wilson lines cor- respond to the integrable representations of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As for the bound- ary monopoles at the opposite end, it seems unlikely that they can modify V−k−h∨(g), which is already simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The total nonperturbative interval alge- bra in this case appears to be some modification of the CDO that contains Lk−h∨(g) rather than Vk−h∨(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We do not know its structure yet, and will explore it elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The abelian case will be studied in the next sections, where we consider G = U (1) in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is a little bit different as there is no abelian WZ term in 2D, and the level is encoded in the periodicity of the compact boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The monopole corrections extend the boundary affine u(1) to the lattice VOA, also known as the abelian WZW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The non-isomorphic Wilson lines correspond to the finite set of modules of the lattice VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3 U(1) We restrict k to lie in 2Z and consider the k ̸= 0 case first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Bℓ, Br, � At dt are the possible candidates for elements of the perturbative interval VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The analysis of the OPE of B’s still holds, and B’s on the left and right boundaries commute with each other (have the regular OPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, the full set of OPEs again: Br (z) Br (w) ∼ k (z − w)2 , Bℓ (z) Bℓ (w) ∼ −k (z − w)2 , Br(z)Bℓ(w) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='16) Surprisingly, there is also one relation connecting the left and right B’s to the Wilson line, which follows from the following transformation: Q � A∗ t dt = Br − Bℓ − k 2π ∂z � At dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='17) Thus, we see that the derivatives of � A are not independent operators in the Q-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will encounter similar phenomena when we consider sin- gular vectors for affine algebras on the boundaries later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can choose any 19 two operators out of {Br, Bℓ, ∂ � A} as the independent generators, and to be consistent with the previous section, we take {Bℓ,r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The stress-energy tensor is also included, but for k ̸= 0 it should be expressed in terms of Bℓ,r as we discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this particular case, T = 1 2kB2 r − 1 2kB2 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We also expect that this perturbative algebra is extended by the boundary monopole operators [DGP18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The boundary monopole Mp is obtained from the usual monopole by cutting it in half and restricting to a half-space in such a way that the integral over the half sphere is � HS2 F = 2πp, p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='18) Due to the CS term, the monopole operator Mp develops an electric charge, as was mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Hence this operator can only exist by itself on the boundary where its electric charge is global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Under the global boundary U(1) action by eiα it transforms as: Mp → e−ipkαMp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='19) To insert it in the bulk, we need to consider a composite operator with a Wilson line attached to a monopole eikp � t 0 AMp(t) to cancel the anomalous transfor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, we can pull a boundary monopole Mp off the boundary while extending a Wilson line of charge kp between the monopole and the boundary to respect gauge invariance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Recall that the twisted the- Mp Wkp ⇐⇒ Mp Figure 4: The bulk monopole Mp is connected by a Wilson line of charge kp to the Dirichlet boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the Q-cohomology, due to the topological invariance in the t direction, this is equivalent to the boundary monopole Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ory is topological in the t direction [CDG20], meaning that the t translations are Q-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus the length of the Wilson line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 4 is irrelevant, and pulling a monopole off the boundary is an identity operation in the cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Using this observation, we can easily find the OPE of a boundary monopole Mp with the boundary current B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Pulling Mp far away from the boundary, it can no longer contribute to such an OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Essentially, for the purpose of com- puting OPEs with the boundary operators (in the cohomology), the boundary monopole Mp is equivalent to the Wilson line of charge kp ending at the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We have already computed the OPE of B with the Wilson line in earlier sections, so the answer follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Recalling that there is also a sec- ond boundary (and operators on the opposite boundaries have regular OPE), we thus find: Bℓ(z)Mp(0) ∼ kp z Mp(0), Br(z)Mp(0) ∼ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='20) 20 for the monopole on the left boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A more semi-classical way to derive this is by computing the on-shell value of B in the twisted formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One can easily check that the monopole singularity implies B ∼ kp z (where the factors of 2π were scaled away).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Similar results hold for monopoles on the right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In fact, repeating our argument, a monopole on the right boundary is equiva- lent to the same monopole on the left boundary connected by the Wilson line to the right boundary (and vice versa), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, one can express the right monopoles as M ′ p(t = L) = e−ikp � L 0 AMp(t = 0), and they are not independent generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Via the semiclassical analysis of the monopole operator, one can show that its OPE with the Wilson line is Mp(z)eiq � L 0 A(0,t) ∼ zqp : Mp(z)eiq � L 0 A(0,t) :, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='21) where :: means the normal ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From this equation we see that the normal ordering for e−ikp � L 0 AMp(t = 0) is not required: The leading OPE term scales as zp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To compute the missing Mp1(z)Mp2(0) OPE, we again use the trick of replacing the left boundary monopole Mp2 by the Wilson line attached to the right monopole eikp2 � L 0 AtdtM ′ p2, Mp1(z, 0)Mp2(0, 0) ∼ Mp1(z, 0)eikp2 � L 0 At(0,t)dtM ′ p2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='22) In this equation, the monopole M ′ p2 is separated in the t direction from the rest of the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Hence the singular terms only come from the OPE between the Wilson line and the monopole Mp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This results in the following: Mp1(z, 0)Mp2(0, 0) ∼ zkp1p2 : Mp1(z, 0)eikp2 � L 0 At(0,t)dt : M ′ p2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='23) It remains to bring M ′ p2 back to the left boundary, colliding with it along the t direction, which produces no further singularities due to the topological in- variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Finally, we observe that the magnetic charges are additive under the collision, and conclude: Mp1(z)Mp2(0) ∼ zkp1p2 : Mp1(z)Mp2(0) := zkp1p2Mp1+p2(0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='24) The full set of strong generators can be chosen to be � Bℓ,r, eip � A, Mp|p ∈ Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='25) Looking closer at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='24), we recognize the rank one lattice VOA setting, with the compact boson of radius √ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can extend the U (1)k current B by the monopoles (vertex operators) Mp to the Z[ √ k] lattice VOA, also called the U(1)k WZW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This can be done individually on the left and right boundaries, giving the abelian WZWk and WZW−k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then the number of Wilson lines is rendered finite, and the algebra is generated as an extension of WZWk ⊗ WZW−k (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='26) by the bimodules corresponding to the Wilson lines e−i k−2 2 � A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' , ei k 2 � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In- deed, it is consistent with the limit of the large interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The theory in this limit reduces to the Chern-Simons at level k in the IR, which only admits a finite 21 number (k, to be more precise) of Wilson lines in the bulk, and supports WZW at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Now, let us turn to the k = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One can easily see that we no longer have two separate operators Bℓ,r, as B is in the bulk cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, we can pull it off the boundary and bring it all the way to the opposite one without changing the cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Technically, this follows from the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='17) involving the descendant line of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Monopole operators no longer have electric charge, so they can be freely moved into the bulk, and they commute (have regular OPE) with all local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In other words, monopoles are naturally elements of the bulk VA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The stress-energy tensor is no longer a sum of the two boundary terms but an independent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, the algebra ceases to be generated as a bimodule of the left and right algebras, and we propose the following set of strong generators: � B, T, ein � A, Mn|n ∈ Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='27) 4 2D Perspective or βγ System The IR limit of a 3D N = 2 YM-CS theory on a slab is in general controlled by the dimensionless parameter γ = Le2 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' When L ≫ 1 e2 3d or γ ≫ 1, the bulk first flows as a 3D theory to the Chern-Simons TFT with keff = k − h∨ (for k ≥ h∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The boundaries flow to some 2D relative CFTs matching the CS boundary anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The interval VOA in this limit was studied in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We do expect, however, that the answer does not depend on γ, at least for k > h∨ (when the SUSY is unbroken).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the opposite limit γ ≪ 1, the system behaves as a 2D sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It was shown in [DL22] that in this regime, the theory is described by an N = (0, 2) NLSM into the complexified group GC ≈ T ∗G at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The 2D coupling is related to the 3D coupling as L e2 3d = 1 e2 2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The gauge degrees of freedom are integrated out, except for the complex Wilson line along the interval, which is left as an effective degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Its compact part is valued in G, and the vector multiplet scalars lie in the cotangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The Chern-Simons term reduces to a non-trivial B-field, or the Wess-Zumino term, necessary for anomaly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this section, we explore our problem from such a 2D viewpoint, as well as study connections between the 3D and 2D setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As reviewed in Section 2, the perturbative chiral algebra of 2D N = (0, 2) NLSM into the target X is captured by the βγ system into X [Wit07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Nek05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' More precisely, it is given by the cohomology of a sheaf of βγ systems on X, also called the sheaf of chiral differential operators (CDO) [GMS99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GMS01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' GMS04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is conveniently computed using the Čech cohomology, and the resulting vector space with the VA structure on it is denoted Dk[X], where k is the B-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In our case, the target is a simple complex group GC, so k ∈ C = H3(GC, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This parameter is the well-known modulus of the CDO valued in H1(X, Ω2,cl(X)), which in general is not quantized, however, in our models k is an integer originating as the CS level in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Since GC has a trivial tangent bundle, c1(GC) = 0 and p1(GC) = 0, so the sigma model anomalies [MN85] vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As reviewed before, c1(GC) = 0 implies that Dk[GC] is a VOA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', it has a well-defined Virasoro element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 22 Notably, Dk[GC] containes the affine sub-VOAs Vk−h∨(g) and V−k−h∨(g) corresponding to the left and right G-actions on GC [GMS01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' These clearly originate as the perturbative baoudnary VOAs in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Additionally, Dk[GC] contains holomorphic functions on the group (functions of γ in the βγ lan- guage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' These originate from the Wilson lines stretched across the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For generic k ∈ C \\ Q, functions on the group together with the pair of affine VOAs V±k−h∨(g) generate the whole Dk[GC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For k ∈ Q this is no longer true, and Dk[GC] as a Vk−h∨(g) ⊗ V−k−h∨(g)–bimodule has a very intricate structure [Zhu08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Nonetheless, Dk[GC] remains a simple VOA for all values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Below we will find a full non-perturbative answer in the abelian case and comment on what is known in the non-abelian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' By comparing the 3D and 2D answers when possible, we will veryfy that the interpolation between γ ≫ 1 and γ ≪ 1 determines an isomorphism of the respective chiral algebras: H• � Ops3d, Q+ � ∼ −→ H• � Ops2d, Q 2d + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1 U(1) For G = U(1), the target space of our model is U (1)C ∼= C∗, which is a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The N = (0, 2) sigma model into C∗ was considered in [DGP17], and we will make contact with it later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For now, let us follow the βγ approach first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The CDO only have zeroth cohomology in this case, which are the global sections of this sheaf forming the perturbative VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then we will identify its non- perturbative extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We work with the even Chern-Simons level k in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We introduce two cooridnate systems on the cylinder: One is just γ ∈ C∗, and another has �γ as a periodic complex boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Its periodicity is what encodes the “level” in the abelian case, descending from the CS level in 3D: �γ ∼ �γ + iπ √ 2k, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2) where the conventions are adjusted to match [Wit07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The relation between the two is, naturally, γ = e √ 2 k �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3) The quantum transformation between β and �β is �β = � 2 k � βγ − 1 2γ−1∂γ � , β = � k 2 � �βe−√ 2 k �γ − 1 k e−√ 2 k �γ∂�γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4) Let us define two currents: Jℓ = 1 √ 2 � �β + ∂�γ � , Jr = 1 √ 2 � �β − ∂�γ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='5) where �γ ∝ � t A is a (log of a) Wilson line from the 3D perspective, and the difference is √ 2∂�γ, which is proportional to ∂ � t A as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One can easily see that these operators commute and have the following OPEs: Jℓ(z)Jℓ(0) ∼ −1 z2 , Jr(z)Jr(0) ∼ 1 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6) 23 We observe that they only differ from the 3D boundary currents Bℓ,r by the normalization factor √ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We find such conventions useful for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The stress-energy tensor is −�β∂�γ and the central charge is equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The stress- energy tensor does not require any modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The single-valued operator corresponding to the charge p Wilson line is Wp = γp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The conformal dimensions of Wp can be easily found to be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The operators β and γp, p ∈ Z, are already global sections of the CDO, and they generate the full perturbative VOA, which is most concisely described as the C∗-valued βγ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that this implies that γ can be inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In practice, it is convenient to do so by inverting the zero mode of γ only: γ−1(z) = 1 γ0 + ∆γ = 1 γ0 ∞ � n=1 (−1)n � γ−1 0 ∆γ �n , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='7) where ∆γ = � n∈Z\\0 γn zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='8) So, what are the non-perturbative effects that we are missing here?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3D physics suggests that they are boundary monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Imagine inserting an im- properly quantized monopole on the boundary with � HS2 F = 2πα at the origin z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We know that γ is related to the connection γ ∝ ei � t A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Now we can consider moving γ around the insertion of this monopole, as in γ � eiφz � , and let us track the phase that is acquired in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is clear that the Wilson line sweeps the entire magnetic flux of the monopole in the end, and we obtain: γ � e2πiz � = ei � M F γ (z) = e2iπαγ (z) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='9) where M is a tube stretched between the two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, γ in the presence of the improperly quantized monopole behaves as: γ = � n γn zn−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='10) Now let us go back to the actual monopole that has α = p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='9) shows that γ winds p times around the origin as we go once around z = 0, and we expect the same shift as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='10) with α = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The actual answer is a little bit trickier, but this gives us a good starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' An operator that “shifts the vacuum” in the βγ system by p units will be called Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The module that it creates is known as the spectrally flowed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6 We can look at the Hilbert space interpretation of these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If we place an Mp operator at the origin, then the mode expansions of β and γ take the following form: β = � n βn zn+p+1 , γ = � n γn zn−p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='11) Here the modes obey the usual commutation relations: [βn, γm] = δn+m,0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12) 6In superstrings such operators are often called the picture changing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 24 and the lowest state |p⟩ corresponding to Mp via the state-operator map is defined by γn+1|p⟩ = βn|p⟩ = 0, for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='13) Essentially, we shifted the modes of the vacuum module by p positions and then relabeled them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The inverse of γ in the presence of Mp is defined in the similar manner: γ−1 (z) = 1 γ0zp + ∆γ = z−p γ0 ∞ � n=0 (−1)n � zpγ−1 0 ∆γ �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='14) Let us compute charges and the conformal dimension of Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The currents Jℓ, Jr in the γ coordinate system are � Jℓ Jr � = � � � 1 √ k � βγ + (k−1) 2 γ−1∂γ � , 1 √ k � βγ + (−k−1) 2 γ−1∂γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='15) The following product can be computed using the mode expansions: β (z) γ (w) |p⟩ = − �w z �p 1 z − w |p⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='16) Subtracting the singularity 1 z−w, we are getting: : βγ : |p⟩ = p z |p⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='17) Let us denote : βγ : as J0 in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The J0 charge of Mp is p, and the charges of β and γ are 1 and −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The difference of Jℓ and Jr is proportional to γ−1∂γ, which is itself a current measuring the winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the presence of Mp, the expression γ−1∂γ can be computed directly from the definition: γ−1∂γ = p z + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='18) Thus the winding charge is equal to p for Mp and 0 for β and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The U (1)ℓ,r charges for Mp are then computed from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='15) and are p √ k( 1+k 2 , 1−k 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The stress energy tensor written in terms of βγ is −β∂γ + 1 2 (log γ)′′ as the holomorphic top form in this case is w ∝ dγ γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Moreover, it can be written as Tβγ = 1 2J2 r − 1 2J2 ℓ , which is indeed what was expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Computing the conformal dimensions of Mp requires again the normal ordering prescription: lim z→w � −β (z) ∂γ (w) |p⟩ − 1 (z − w)2 |p⟩ � = �p − p2 2w2 + β−1γ0 w + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' � |p⟩ , 1 2(log γ)′′ |p⟩ = � − 1 w2 p 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' � |p⟩ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='19) where we again subtracted all singular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It follows that the dimension of the operator Mp is − p2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 25 ≈ Figure 5: Moving a monopole of magnetic charge p from one boundary to an- other creates a Wilson line of electric charge kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Now that we understand the operators Mp fairly well, we need to iden- tify precisely those that should be added to the βγ system to obtain our non- perturbative VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' They correspond to the boundary monopoles in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For that we need to find operators that are only charged under the left or right affine currents Jℓ,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' If we had k = 1 for a moment, the left monopole is just the flowed module Mp, as can be seen from its charges (Jℓ, Jr) = (p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The mathematical perspective on such modules was given in [RW14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' AW22], and in their notations, Mp generates σpW+ 0 , where σ denotes the spectral flow automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To tackle the general case, we just need to take a composite operator that has the correct charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The Mp by itself is charged as p √ k � k+1 2 , −k+1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, if we take M ′ p = (γ p(k−1) 2 0 |p⟩)(z), we get an operator with the charges p( √ k, 0), as required, and the conformal dimension ∆M ′p = p2 k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This M ′ p generates the spectrally flowed module σpW p(k−1) 2 in the notations of [AW22], which is just σpW+ 0 for p(k − 1) even and σpW 1 2 for p(k − 1) odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='7 In order to get monopoles on the other boundary, we need to consider the composite operator with the Wilson line WkM ′ 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 5), which has charges p(0, − √ k) and the same conformal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus we do not need a separate generator for that monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The generator content of our algebra matches the eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us summarize the result that we have obtained so far for the full non- perturbative VOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is given by the C∗-valued βγ VOA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', γ is invertible,) extended8 by its modules σ2pW+ 0 and σ2p+1W 1 2 for all p ∈ Z, in the notations of [RW14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' AW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One can also easily compute a character over the full chiral algebra: Z = TrH(qL0− c 24 xJℓyJr) = 1 η(q)2 � n,m∈Z qnmx n √ k +m √ k 2 y n √ k −m √ k 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='20) which is clearly not a meromorphic function and can only be understood as a formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The obvious non-convergence of the character is expected, as characters on the boundary with the positive level are convergent when |q| < 1 7Since we consider even k, this is determined by the parity of p only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 8In fact, it is slightly redundant to say that γ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The module W+ 0 of the usual βγ system coincides with the vacuum module of such a C∗-valued βγ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus the extension by W+ 0 automatically inverts γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 26 [DGP18], and on the opposite boundary the convergence is at |q| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This trace, of course, can be reinterpreted from the 3D perspective as an index on the interval (see also [SY20]): ZI×T 2 = TrH(−1)F e−2πRH � e2πiJizi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='21) where Ji are generators of the maximal tori of the boundary symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The k = 0 situation is different, and does not appear to be particularly interesting and well behaved from the 2D viewpoint, so we skip it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Dual boson One can also calculate the same algebra directly in the C∗ sigma model, without going to the βγ-description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The calculation was first done in [DGP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us connect it with our formulas for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let σ be the radial coordinate and X = XL(z) + XR(z) be an angular coordinate on C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then �β and �γ are related to the free boson as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4: ∂�γ = ∂σ + i∂X, �β = ∂σ − i∂X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='22) In the γ coordinate system one gets from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4): γ = e √ 2 √ k (σ+i(XL+XR)), β = √ k √ 2 � ∂(σ − iX)e− √ 2 √ k (σ+iX) − 1 k e− √ 2 √ k (σ+iX)∂(σ + iX) � = − √ k √ 2 � (1 − 1 k )∂σ + i(1 + 1 k )∂X � e− √ 2 √ k (σ+iX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='23) Redefine both X and σ by √ 2 to match the notations of [DGP17], so we find: γ = e 1 √ k (σ+i(XL+XR)), β = − √ k 2 � (1 − 1 k )∂σ + i(1 + 1 k )∂X � e− 1 √ k (σ+iX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='24) The BPS vertex operators in this description take the form: eikℓXL+kr(σ+iXR), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='25) with (kℓ, kr) = � n R + wR 2 , n R − wR 2 � , n, w ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='26) The radius R is related to the Chern-Simons level as R2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can see that these operators form the same lattice as we found in the βγ system with the special shifted modules included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In particular, γn here is the vertex operator with kℓ = kr = n R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' At the same time the left monopoles have (kℓ, kr) = p( √ k, 0), which means n = wk/2 = pk/2, and the right monopoles have (kℓ, kr) = p(0, − √ k), which corresponds to n = −wk/2 = −pk/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 27 No Mercy This final presentation allows us to identify the nonperturbative VOA even more explicitly in terms of the known VOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Namely, let us denote the vertex operator representing the Q-cohomology class with the momentum and winding charges (n, w) ∈ Z2 by Vn,w(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Then computing the OPE of vertex operators defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='25), we easily find the following: Vn1,w1(z)Vn2,w2(0) ∼ zn1w2+n2w1 : Vn1,w1(z)Vn2,w2(0) : , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='27) which identifies our VOA as a lattice VOA for the smallest Narain lattice [Nar86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' NSW87], namely Z2 ⊂ R2 with the scalar product: (n1, w1) ◦ (n2, w2) = n1w2 + n2w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='28) Note that the two U(1) currents can be obtained as V−1,0∂V1,0 and V0,−1∂V0,1: J1 = 1 R (i∂XL + i∂XR + ∂σ) , J2 = R 2 (i∂XL − i∂XR − ∂σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='29) Also notice a curious fact: While many of the steps in our analysis involved the CS level, the final answer does not depend on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This in fact serves as a consistency check for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From the N = (0, 2) point of view, the compact boson radius √ k only enters the Kähler potential, thus it cannot affect the chiral algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Together with the other two results in the earlier sections, we thus find three presentations for the nonperturbative VOA in the abelian case: Narain lattice VOA of rank two ∼= βγ extended by σ2pW+ 0 and σ2p+1W 1 2 ∼= WZWk ⊗ WZW−k extended by bimodules (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='30) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2 SU(2) Let us now turn to a less-trivial example and discuss G = SU(2), that is GC =SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The computation is more involved in this case, as we need to define everything on patches and consider a non-trivial gluing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will first find the global theory and discuss the moduli space, and then will turn to the non-trivial modules for boundary VOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' SL(2, C) can be covered by two patches, the coordinates on which will be denoted as γi and �γi : �a b c d � , ad − bc = 1 a̸=0 �−−→ �γ1 γ2 γ3 � = �a b c � ∈ C3 \\ {γ1 = 0}, � a b c d � , ad − bc = 1 b̸=0 �−−→ � �γ1 �γ2 �γ3 � = � a b d � ∈ C3 \\ {�γ2 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 28 Thus, the coordinate transformations have the following form: γ1 = �γ1, γ2 = �γ2, γ3 = �γ1�γ3 − 1 �γ2 or �γ1 = γ1, �γ2 = γ2, �γ3 = 1 + γ2γ3 γ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='31) The Jacobian matrix and its inverse can be computed to be gi j ≡ ∂γi ∂�γj = � � 1 0 0 0 1 0 1+γ2γ3 γ1γ2 − γ3 γ2 γ1 γ2 � � , ∂�γi ∂γj = � � 1 0 0 0 1 0 − 1+γ2γ3 (γ1)2 γ3 γ1 γ2 γ1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ∂jgi a∂igj b = ∂3g3 a∂3g3 b = � � 1 (γ1)2 − 1 γ1γ2 0 − 1 γ1γ2 1 (γ2)2 0 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' As we mentioned earlier, for each left- and right-invariant vector fields there exists a corresponding VOA sub-algebra [GMS01] that saturates boundary anoma- lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The boundary with the negative anomaly usually corresponds to a rela- tive CFT with the anti-holomorphic dependence on the coordinates, and it transforms into a chiral algebra with the negative level after passing to the Q-cohomology: kcoh = kℓ − kr [Wit07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us find these algebras and the CDO sections explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that there is nothing left except global sections as the manifold is Stein and does not support geometric objects that can form a higher degree cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, the only fields that can contribute are in H0(SL(2), �A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us first write out all classical vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To do this, we will use the following well-known form of basis at the identity point of the group: e = �0 1 0 0 � f = �0 0 1 0 � h = �1 0 0 −1 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='32) and carry it over the whole manifold by L∗ gV µ∂µ|1 = (gV )µ∂µ|g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Local sections, corresponding to the left-invariant vector fields, then have the following form in both patches: eℓ = γ1β2 �eℓ = �γ1 �β2 + �γ−1 2 (�γ1�γ3 − 1)�β3 fℓ = γ2β1 + γ−1 1 (1 + γ2γ3)β3 �fℓ = �γ2 �β1 hℓ = γ1β1 − γ2β2 + γ3β3 �hℓ = �γ1 �β1 − �γ2 �β2 − �γ3 �β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='33) Local sections corresponding to the right-invariant vector fields in both patches are er = γ1β3 �er = �γ2 �β3 fr = γ3β1 + γ−1 1 (1 + γ2γ3)β2 �fr = �γ−1 2 (�γ1�γ3 − 1)�β1 + �γ3 �β2 hr = γ1β1 + γ2β2 − γ3β3 �hr = �γ1 �β1 + �γ2 �β2 − �γ3 �β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='34) The normal ordering for these fields is chosen exactly in the way they are written, abc = def: a : bc ::, and will be omitted from this point on to unclutter notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 29 By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12), one can easily obtain the following transformation formulas: �β1 = β1 + β3 �γ3 γ1 + 1 γ1γ2 � − 1 2 � 1 (γ1)2 ∂γ1 − 1 γ1γ2 ∂γ2 � , �β2 = β2 − β3 γ3 γ2 − 1 2 � − 1 γ1γ2 ∂γ1 + 1 (γ2)2 ∂γ2 � , �β3 = β3 γ1 γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='35) Note that here we choose to set the moduli parameter µab to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Now we will combine (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='31) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='35) to find the corrected version of these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' After either doing a tedious calculation or applying the Mathematica tool attached, one can obtain the corrected version of the left- and right-invariant vector fields, respectively: −Hℓ ≡ hℓ = �hℓ −Hr ≡ hr + ∂γ1 γ1 = �hr + ∂γ2 γ2 Eℓ ≡ eℓ = �eℓ + 1 2∂ ��γ1 �γ2 � Er ≡ er = �er Fℓ ≡ fℓ + 1 2∂ �γ2 γ1 � = �fℓ Fr ≡ fr + 1 2∂ �γ3 γ1 � + ∂γ3 γ1 = �fr + 1 2∂ ��γ3 �γ2 � + ∂�γ3 �γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='36) As one can see, we regrouped terms in the expression, so the sections are actually smooth and well-defined on the whole patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, the term ∂γ1 γ1 would have a pole on the second patch, where γ1 can be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, it is only defined smoothly on the first patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The OPEs, as expected, constitute the affine Kac-Moody vertex algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For example, the left OPEs are: Hℓ(z)Eℓ(w) ∼ 2Eℓ(w) z − w , Hℓ(z)Hℓ(w) ∼ −3 (z − w)2 , Hℓ(z)Fℓ(w) ∼ −2Fℓ(w) z − w , Eℓ(z)Fℓ(w) ∼ −3/2 (z − w)2 + Hℓ(w) z − w , which make it into V−3/2 (sl(2, C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From the anomaly inflow argument we know that the total level should be kℓ + kr = −2h∨ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, the right algebra is the affine algebra V−5/2 (sl (2, C)), which we could again check by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Operator products between all the left and right global sections are non- singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The level is defined with respect to the standard bilinear form (, ) = (2h∨)−1(, )K, where (, )K is the Killing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the case of sl(2, C) it is given by (h, h) = 2, (e, f) = (f, e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' General k To find the most general form of this algebra, we need to find the moduli space of this CDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' By section 2 we know that it is equivalent to finding 30 H1 � Ω2,cl, SL (2, C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We want to show that µ = 2tdγ1 ∧ dγ2 γ1γ2 , t ∈ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='37) is the only generator of that cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We show it in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' First, we use that H1 � SL (2, C) , Ω2,cl� → H3 dR (SL (2, C) , C) is injective (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Second, it is known that the 3rd de Rham cohomology for simple Lie groups is generated by Tr � g−1 dg �3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=', H3 dR (SL (2, C)) ∼= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Third, the form above is well-defined on U12 = U1 ∩ U2 ∼= C∗ × C∗ × C and has a non-trivial pe- riod over a non-contractible cycle on the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, it means that it represents a non-trivial class in H1(SL(2, C), Ω2,cl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, we showed that H1 � Ω2,cl, SL (2, C) � ∼= H3 dR (SL (2, C) , C) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='37 is the non-trivial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The coefficient t there is thus the only CDO modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' After all these preparations, we can finally redo the calculations with the transformation law shifted by µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One gets the following sections: −Hℓ ≡ hℓ + tγ′ 1 γ1 = �hℓ − tγ′ 2 γ2 −Hr ≡ hr + (1 − t)∂γ1 γ1 = �hr + (1 − t) ∂γ2 γ2 Eℓ ≡ eℓ = tγ′ 1 γ2 + �eℓ + 1 2∂ �γ1 γ2 � Er ≡ er = �er Fℓ ≡ fℓ + 1 2∂ �γ2 γ1 � + tγ′ 2 γ1 = �fℓ Fr ≡ fr + 1 2∂ �γ3 γ1 � + (1 − t)∂γ3 γ1 = �fr + 1 2∂ ��γ3 �γ2 � + (1 − t)∂�γ3 �γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='38) Effectively, we observe that introducing the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='37) leads to a shift of levels kℓ → kℓ − t and kr → kr + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will set t = 1 2 + k for convenience, where k now is the actual Chern-Simons level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The levels of the boundary algebras are then −2−k and −2+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The quantization condition is not necessary in this approach, but is necessary from the 3D perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In this context it means that k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, the affine algebras of the global left and right G-action sections are V−2±k (sl (2, C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The explicit form of these sections is one of the key technical results of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Module Structure To reveal the module structure of Dk[SL(2, C)] with respect to V−2±k (sl (2, C)), let us consider: Eℓ(z)γ1(w) ∼ 0 Eℓ(z)(−γ2)(w) ∼ γ1(w) z − w Hℓ(z)γ1(w) ∼ γ1(w) z − w Hℓ(z)(−γ2)(w) ∼ γ2(w) z − w Fℓ(z)γ1(w) ∼ −γ2(w) z − w Fℓ(z)(−γ2)(w) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='39) Thus, as before, γ’s generate modules for our boundary algebras and are iden- tified with the Wilson lines in the 3D description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One finds that we quotient out the singular vector of the underlying sl2 algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (F 0 ℓ )2γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 31 One also finds that the vectors (γ1)0 |0⟩, −(γ2)0 |0⟩, and all vectors obtained from them by the action of the negative modes of Eℓ, Hℓ, and Fℓ span a module over the left current algebra, with the vector (γ1)0 |0⟩ being the highest weight vector of weight 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There is an isomorphic module over this subalgebra, “gener- ated” by γ3 and γ−1 1 (1+γ2γ3), with the first field giving the highest vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One also finds analogous modules over the right current algebra, “generated” by pairs of global sections γ1, γ3 and γ2, γ−1 1 (1+γ2γ3), where again the first field in each pair defines the highest weight vector of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that these expressions indeed define global sections due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' All global functions depend only on (γ1, γ2, γ3, γ−1 1 (1 + γ2γ3)) and are modules for zero modes of our currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us put these building blocks together and consider the vector space: (γ1)0 |0⟩ (γ2)0 |0⟩ (γ3)0 |0⟩ � 1+γ2γ3 γ1 � 0 |0⟩ This is (1, 1) representation for sl(2)ℓ ⊗ sl(2)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We can act on this vector space by negative modes of Ja ℓ and Ja r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The vector (γ1)0 |0⟩ is the highest weight vector of weight (1, 1) in the representations of the corresponding �gℓ ⊗ �gr affine Kac-Moody algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In order to obtain other representation one can act with higher powers of γn 1 on the vacuum and this yields representation (n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, the answer at the generic point is Dk[SU(2)C] = � λ∈Z+ Vλ,−2+k (g) ⊗ Vλ,−2−k (g) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='40) where again Vλ,−2±k are Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Two points require important clarifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' First, what happens with the stress-energy tensor of the βγ system −βi∂γi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is guaranteed to exist by [GMS99] as the canonical bundle on a Lie group is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The holomorphic top form can be written as w = dγ1dγ2dγ3 γ1 on the first patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, the stress- energy tensor gets corrected to T (z) = − � βi∂γi + 1 2 (log γ1)′′ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='41) where the correction is a derivative of the coefficient of the holomorphic top form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One can show by a direct computation that outside of the critical levels of the boundary algebras, Tβγ = Tℓ + Tr, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='42) where Tℓ,r = 1 2(kℓ,r+h∨) � ef + fe + hh 2 � are the Sugawara stress-energy tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This result was expected from the general discussion in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 9We assumed the mathematical notation, where representations of sl2 are labeled by inte- gers, not half-integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 32 The second point is that the modules that we are considering are reducible for the physical values of k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Not only that, but those singular vectors are singular for both left and right algebras [Zhu08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' To see this, let us set the right algebra level kr to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is an obvious limiting case, but it will nevertheless show the important feature that is carried over to other values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The simplest singular vector for this module can be found to be (Er)−1 |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='43) We need to find the form of this vector in terms of the left-invariant fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Classically, the vector fields are related by the following change of basis: � � er fr hr � � = � � � −γ2 2 γ2 1 −γ1γ2 (1+γ2γ3)2 γ2 −γ2 3 γ3(1+γ2γ3) γ1 2 γ2(1+γ2γ3) γ1 −2γ2γ3 1 + 2γ2γ3 � � � � � eℓ fℓ hℓ � � = S · Vℓ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='44) Of course, at the quantum level the relation is corrected, and for the e field the correct answer is found to be Er = V i ℓ S1i + (−2 + k)(γ1∂γ2 − γ2∂γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='45) So, we see that for the special value k = 2, the correction term disappears, and the vector Er −1 can now be obtained both from the left and right algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is actually lying inside the γ2 1 representation for the left algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus for discrete values of k, different modules start to intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Moreover, now there is no way to obtain this correction term ωf = γ1∂γ2 − γ2∂γ1 from a Wilson line by the action of �gℓ ⊗ �gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Note that this form is actually dual to the f-vector field < ωf, f >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This phenomenon happens for all singular vectors [Zhu08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One could ask what happens when we include monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We do not have a definitive answer, but as mentioned in the introduction and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2, we have conjectures as to what the answer might look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' One expects to get some sort of truncation of the CDO that contains simple quotients L−h∨±k(g) rather than V−h∨±k(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We will look into this issue elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3 Open Questions We have already emphasized many times that determining the non-perturbative modification of Dk[GC] is an interesting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It requires, perhaps, improved understanding of the non-compact models in 2D, of which our theory is an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The usual arguments with quotienting out singular vectors based on the unitarity of the Hilbert space do not work in such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' But we expect that monopoles on the boundary with positive k − h∨ are still required, as in the opposite limit γ ≫ 1 this boundary is a relative CFT with a normalizable vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The problems lie on the other boundary which has a non-compact mode [DL22] that effectively renders our theory non-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Another intriguing question arises from an alternative UV completion of the GC NLSM via the 2D Landau-Ginzburg (LG) model described in [DL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The simplest example is for G = SU (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The UV completion is chosen to be the N = (0, 2) LG model with the following field content: 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' M i j are chiral multiplets valued in complex matrices Mat(N, C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Φa i is a chiral multiplet that is bifundamental under U (k) × G, where i, j ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' , N and a ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' , (k = anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Fermi multiplets Γ and Λj b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Superpotential W = Γ (det M − 1) + µΛj aM i jΦa i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This model has the same anomalies as our theory, and the superpotential is engineered in such a way that it flows to the SL(N, C) NLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is generally believed that LG models do not carry any non-perturbative physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus one could hope that the perturbative chiral algebra in this model could provide some useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It is captured by the βγ systems (V j i , M i j) (here V is the “beta” for M), (Ri a, Φa i ), and the bc systems (Γ, Γ) and (Λ a i , Λi a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The chiral algebra is defined in the cohomology of Q that acts according to: QΓ = det M − 1, QΛ = MΦ, QV j i = Γ∂ det M ∂M i j + µΛj aΦa i , QRi a = µΛj aM i j, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='46) and by zeros on the rest of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' All these βγ and bc systems are already globally defined, so one simply computes the cohomology of such Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The answer appears to be just Dk[SL(2, C)], which would be interesting to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' But more importantly, this, supposedly exact, answer in the LG model is the same as the perturbative VOA we find in the interval theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' This suggests that the exact non-perturbative physics in these models may depend on the UV completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Other open questions involve applications to the VOA[M4], which requires computing the interval reductions of more complicated gauge theories, and which we will study in the future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 5 Conclusion In this paper we considered the chiral algebra of a 3D N = 2 YM on R2 × [0, L] with the N = (0, 2) Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The algebra was computed both from the 3D and 2D perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We analyzed this protected sector using the holomorphic-topological twist of the 3D theory and, among other things, the holomorphic twist of the 2D theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' From the 3D perspective, the perturbative algebra was found to be an en- hancement of two affine vertex algebras living at the boundaries by their bimod- ules realized via the Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The boundary monopoles seem to modify the answer non-perturbatively, on which we proposed some conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The two-dimensional system after reduction in the right regime is an N = (0, 2) NLSM into GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The compactification algebra is the chiral algebra of this 2D model, and the beta-gamma system is the main tool to compute its perturbative approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The global sections corresponding to the left and right actions of the group on itself were explicitly found for G = SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In the abelian case, we find that the spectrally flowed modules of the βγ system are required to get the full result for the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Combining the latter perspective with the 3D analysis and with the known results on the sigma model into C∗, we obtain three different presentations of the chiral algebra (also called 34 the interval VOA) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' We also saw that the stress-energy tensor is de- composed in terms of the Sugawara stress tensors for the boundary symmetries, both in the abelian and the non-abelian cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The answers, when available, fully agree between the 2D and 3D calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Some puzzles and speculations are discussed towards the end and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Acknowledgements We benefited from the useful discussions and/or correspondence with: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Abanov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Creutzig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Dimofte, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Gaiotto, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Komargodski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Melnikov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Nekrasov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Niu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Roček.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A De Rham cohomology In this appendix we will show that H1 � SL (2, C) , Ω2,cl� → H3 dR (SL(2, C)) is in- jective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' H1 � SL (2, C) , Ω2,cl� is isomorphic to Zd � Ω3,0 ⊕ Ω2,1�/dΩ2,0 [Wit07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' There is an obvious map from Zd � Ω3,0 ⊕ Ω2,1�/dΩ2,0 to the third de Rham cohomology group H3 dR(SL(2, C)) given by [α] �→ [α] for any closed 2-form α ∈ Ω3,0 ⊕ Ω2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' For any [α] ∈ Zd � Ω3,0 ⊕ Ω2,1�/dΩ2,0 there exists β ∈ Zd � Ω3,0� such that [α] = [β], (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1) so there is an isomorphism: A ≡ Zd � Ω3,0 ⊕ Ω2,1� ⧸dΩ2,0 ∼= Zd � Ω3,0� ⧸dΩ2,0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2) where we have made use of a slight abuse of notation, and dΩ2,0 in the last quotient should be understood as dΩ2,0 ∩ Ω3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' A general form from A has the form α+β for some α ∈ Ω3,0 and β ∈ Ω2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The closeness conditions are ∂α + ∂β = 0, ∂β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3) The second condition says that β ∈ Z∂ � Ω2,1� , and using the fact that SL (2, C) is a Stein manifold with all positive degree Dolbeault cohomology groups vanishing H·,·≥1 ∂ = 0, one gets that the form β is in fact exact: β = ∂γ for some γ ∈ Ω2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' So, shifting α + β by −dγ, we get the desired representative in Ω3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ■ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' The map from A to H3 dR defined above is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Suppose we have a closed (3,0)-form ω that goes under the map to zero in H3 dR(SL(2, C)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ω = dα for some α ∈ Ω2,0 ⊕ Ω1,1 ⊕ Ω0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Let us represent α as α(2,0) + α(1,1) + α(0,2), where each α(p,q) ∈ Ωp,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Thus, ω = dα(2,0) + dα(1,1) + dα(0,2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='4) and as ω ∈ Ω3,0 we find that ∂α(0,2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Recalling that SL(2, C) has trivial non-zero Dolbeault cohomology groups as a Stein manifold, one obtains that 35 α(0,2) = ∂γ(0,1) for some γ(0,1) ∈ Ω0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' It means, however, that dα(0,2) ≡ ∂∂γ(0,1) + ∂ 2γ(0,1) = −∂∂γ(0,1) = ∂β(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Redefining α(1,1), ω = dα(2,0) + dα(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='5) Repeating the same argument with α(1,1), we obtain that ω = dα(2,0), meaning that it was trivial in A, which proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' ■ Let us show that the only generator of H3 dR(SL(2, C)) (which is Tr(g−1dg)3) after mapping to A indeed corresponds to the closed holomorphic (2,0)-form µ in H1 � SL (2, C) , Ω2,cl� from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='37): H ≡ Tr � g−1dg �3 = 2dγ1 ∧ dγ2 ∧ dγ3 γ1 = 2d�γ1 ∧ d�γ2 ∧ �dγ3 �γ2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='6) Following the general algorithm explicitly constructing the isomorphism between H1 � SL (2, C) , Ω2,cl� and A, we must find two (2,0)-forms T1 and T2 in each of the open sets U1 and U2, such that H = dT1 in U1 and H = dT2 in U2, and take their difference T12 ≡ T1 − T2 in U1 ∩ U2 to produce the element of A: dγ1 ∧ dγ2 ∧ dγ3 γ1 = d �γ3 γ1 dγ1 ∧ dγ2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='7) d�γ1 ∧ d�γ2 ∧ �dγ3 �γ2 = d ��γ3 �γ2 d�γ1 ∧ d�γ2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='8) γ3 γ1 dγ1 ∧ dγ2 − �γ3 �γ2 d�γ1 ∧ d�γ2 = −dγ1 ∧ dγ2 γ1γ2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='9) which indeed coincides with the Cech cohomology class defined by µ on U1∩U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' References [AHW82] Ian Affleck, Jeffrey A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' [BST19] Ilka Brunner, Jonathan Schulz, and Alexander Tabler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' “Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' In: JHEP 06 (2019), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1007/JHEP06(2019)046.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1007/JHEP10(2016)108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: 1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='08382 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 37 [BZ22] Mathew Bullimore and Daniel Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' “3d N = 4 Gauge Theories on an Elliptic Curve”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='10907 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' [CDT13] Oscar Chacaltana, Jacques Distler, and Yuji Tachikawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' “Nilpo- tent orbits and codimension-two defects of 6d N=(2,0) theories”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='nuclphysb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: hep-th/0403157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' [GGP16] 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: math/9906117 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='AG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' [GMS04] Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' “Gerbes of chiral differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' Vertex algebroids”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: 0801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='3955 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' [GJS15] Jirui Guo, Bei Jia, and Eric Sharpe.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' arXiv: math/0611517 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content='QA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} +page_content=' 45' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfP_bc/content/2301.00038v1.pdf'} diff --git a/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/2301.02389v1.pdf.txt b/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/2301.02389v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf074e358426a36048ba9db636034de87cbcda2e --- /dev/null +++ b/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/2301.02389v1.pdf.txt @@ -0,0 +1,2034 @@ +arXiv:2301.02389v1 [cs.LG] 6 Jan 2023 +Provable Reset-free Reinforcement Learning by No-Regret Reduction +Hoai-An Nguyen +Ching-An Cheng +Rutgers University +Microsoft Research +Abstract +Real-world reinforcement learning (RL) is of- +ten severely limited since typical RL algorithms +heavily rely on the reset mechanism to sample +proper initial states. In practice, the reset mech- +anism is expensive to implement due to the need +for human intervention or heavily engineered en- +vironments. To make learning more practical, +we propose a generic no-regret reduction to sys- +tematically design reset-free RL algorithms. Our +reduction turns reset-free RL into a two-player +game. We show that achieving sublinear regret +in this two player game would imply learning a +policy that has both sublinear performance regret +and sublinear total number of resets in the origi- +nal RL problem. This means that the agent even- +tually learns to perform optimally and avoid re- +sets. By this reduction, we design an instantia- +tion for linear Markov decision processes, which +is the first provably correct reset-free RL algo- +rithm to our knowledge. +1 +INTRODUCTION +Reinforcement learning (RL) enables an artificial agent to +learn problem-solving skills directly through interactions. +However, RL is notorious for its sample inefficiency, and +successful stories of RL so far are mostly limited to appli- +cations where an accurate simulator of the world is avail- +able (like in games). Real-world RL, such as robot learn- +ing, remains a challenging open question. +One key obstacle preventing the collection of a large num- +ber of samples in real-world RL is the need for reset- +ting the agent. +The ability to reset the agent to proper +initial states plays an important role in typical RL algo- +rithms, as it affects which region the agent can explore +and whether the agent can recover from its past mis- +takes (Kakade and Langford, 2002). In the absence of a +reset mechanism, agents may get stuck in absorbing states, +such as those where it has damaged itself or irreparably al- +tered the learning environment. Therefore, in most settings, +completely avoiding resets without prior knowledge of the +reset states or environment is infeasible. +For instance, a robot learning to walk would inevitably +fall before perfecting the skill, and timely intervention is +needed to prevent damaging the hardware and to return +the robot to a walkable configuration. Another example +we can consider is a robot manipulator learning to stack +three blocks on top of each other. Unrecoverable states that +would require intervention would include the robot knock- +ing a block off the table, or the robot smashing itself force- +fully into the table. Reset would then reconfigure the scene +to a meaningful initial state that is good for the robot to +learn from. +Resetting is a necessary part of the real-world learning pro- +cess if we want an agent to be able to adapt to any en- +vironment, but it is non-trivial. Unlike in simulation, we +cannot just set a real-world agent (e.g., a robot) to an ar- +bitrary initial state with a click of a button. Resetting in +the real world is usually quite expensive and requires con- +stant human monitoring and intervention. Consider again +the example of a robot learning to stack blocks. Normally, +a person would oversee the entire learning process. During +the process, they would manually reset the robot to a mean- +ingful starting state before it enters an unrecoverable state +where the problem can no longer be solved. Sometimes au- +tomatic resetting can be implemented by cleverly engineer- +ing the physical learning environment (Gupta et al., 2021), +but it is not always feasible. +An approach we can take to make real-world RL more +cost-efficient is through reset-free RL. The goal of reset- +free RL is to have an agent learn how to perform well +while minimizing the amount of external resets required. +Some examples of problems that have been approached in +reset-free RL include agents learning dexterity skills, such +as picking up an item or inserting a pipe, and learning +how to walk (Gupta et al., 2021; Ha et al., 2020). While +there has been numerous works proposing reset-free RL +algorithms using approaches such as multi-task learning +(Gupta et al., 2021; Ha et al., 2020), learning a reset pol- + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +icy (Eysenbach et al., 2018; Sharma et al., 2022), and skill- +space planning (Lu et al., 2020), to our knowledge, there +has not been any work with provable guarantees. +In this work, we take the first step to provide a provably +correct framework to design reset-free RL algorithms. Our +framework is based on the idea of a no-regret reduction. +First, we reduce the reset-free RL problem to a sequence of +safe RL problems with an adaptive initial state sequence, +where each safe RL problem is modeled as a constrained +Markov decision process (CMDP) with the states requiring +resets marked as unsafe. Then we derive our main no-regret +reduction, which further turns this sequence into a two- +player game between a primal player (updating the Marko- +vian policy) and a dual player (updating the Lagrange mul- +tiplier function of the constrained MDPs). Interestingly, +we show that such a reduction can be constructed without +using the typical Slater’s condition for strong duality and +despite the fact that CMDPs with different initial states in +general do not share a common Markovian optimal policy. +We show that if no regret is achieved in this game, then +the regret of the original RL problem and the total num- +ber of required resets are both provably sublinear. This +means that the agent eventually learns to perform optimally +and avoids resets. Using this reduction, we design a reset- +free RL algorithm instantiation under the linear MDP as- +sumption, using Ghosh et al. (2022) as the baseline algo- +rithm for the primal player and projected gradient descent +for the dual player. We prove that our algorithm achieves +˜O( +√ +d3H4K) regret and ˜O( +√ +d3H4K) resets with high +probability, where d is the feature dimension, H is the +length of an episode, and K is the total number of episodes. +2 +RELATED WORK +Reset-free RL is a relatively new concept in the literature, +and the work thus far, to our knowledge, has been limited to +non-provable approaches with empirical verification. One +such approach is by learning a reset policy in addition to the +main policy (Eysenbach et al., 2018; Sharma et al., 2022). +The idea is to learn a policy that will bring the agent back +to a safe initial state if they encounter a reset state concur- +rently with a policy that maximizes reward. A reset state is +a state in which human intervention normally would have +been required. This approach prevents the need for man- +ual resets; however, there is usually some required assump- +tions on knowledge of the reset policy reward function and +therefore knowledge of the reset states (Eysenbach et al., +2018). Sharma et al. (2022) avoid this assumption but as- +sume given demonstrations on how to accomplish the goal +and a fixed initial state distribution. +Another popular approach is using multi-task learning. +This is similar to learning a reset policy, but can be thought +of as a way to increase the amount of possible actions an +agent can take to perform a reset. The objective is to learn a +number of tasks so that a combination of them can achieve +the main goal, and in addition, some tasks can perform nat- +ural resets for other tasks. One problem that was tackled +by Gupta et al. (2021) was that of inserting a light bulb into +a lamp. The tasks their agent learns is recentering, insert- +ing, lifting, and flipping the bulb. Here, if the bulb starts +on the ground, the agent can recenter the bulb, lift it, flip it +over (if needed), and finally insert it. In addition, many of +the tasks perform resets for the others. For example, if the +agent drops the bulb while lifting it, it can recenter the bulb +and then try lifting it again. This approach breaks down the +reset process and (possibly) makes it easier to learn. How- +ever, this approach often requires the order in which tasks +should be learned to be provided manually (Gupta et al., +2021; Ha et al., 2020). +A related problem is infinite-horizon non-episodic RL +with provable guarantees (see Wei et al. (2020, 2019); +Dong et al. (2019) and the references within) as this prob- +lem is also motivated by not using resets. In this setting, +there is only one episode that goes on indefinitely. The ob- +jective is to maximize the cumulative reward, and progress +is usually measured in terms of regret with the compara- +tor being an optimal policy. However, compared with the +reset-free RL setting we study here, extra assumptions, +such as the absence or knowledge of absorbing states, are +usually required to achieve sublinear regret. In addition, +the objective does not necessarily lead to a minimization +of resets as the agent can leverage reset transitions to max- +imize rewards. Learning in infinite-horizon CMDPs has +been studied (Zheng and Ratliff, 2020; Jain et al., 2022), +but to our knowledge, all such works make strong assump- +tions such as a fixed initial state distribution or known dy- +namics. In this paper, we focus on an episodic setting of +reset-free RL (see Section 3); a non-episodic formulation +of reset-free RL could be an interesting one for further re- +search. +To our knowledge, we propose the first provable reset- +free RL technique in the literature. By borrowing ideas +from literature on the much more extensively studied area +of safe RL, we propose to associate states requiring re- +sets with the concept of unsafe states in safe RL. Safe +reinforcement learning involves solving the standard RL +problem while adhering to some safety constraints. There +has been a lot of work in safe RL, with approaches +such as utilizing a baseline safe (but not optimal) policy +(Huang et al., 2022; Garcia Polo and Fernandez Rebollo, +2011), pessimism (Amani and Yang, 2022), and shielding +(Alshiekh et al., 2018; Wagener et al., 2021). These works +have had promising empirical results but usually require +extra assumptions such as a given baseline policy or knowl- +edge of unsafe states. +There are also provable safe RL algorithms. +To our +knowledge, all involve framing safe RL as a CMDP. +Here, the safety constraints are modeled as a cost, and + +Hoai-An Nguyen, Ching-An Cheng +the overall goal is to maximize performance while keep- +ing the cost below a threshold. +The provable guaran- +tees are commonly either sublinear regret and constraint +violations, or sublinear regret with zero constraint vi- +olation (Wei et al., 2021; HasanzadeZonuzy et al., 2021; +Qiu et al., 2020; Wachi and Sui, 2020; Efroni et al., 2020; +Ghosh et al., 2022; Ding et al., 2021). +However, most +works (including all the aforementioned ones), consider the +episodic case where the initial state distribution of each +episode is fixed. This prevents a very natural extension +to reset-free learning as human intervention would be re- +quired to reset the environment at the end of each episode. +Works that allow for arbitrary initial state require fairly +strong assumptions, such as knowledge (and the existence) +of safe actions from each state (Amani et al., 2021). +In our work, we utilize techniques from provable safe RL +for reset-free RL, but weaken the typical assumptions to +allow for arbitrary initial states. This relaxation is neces- +sary for the reset-free RL problem and also allows for eas- +ier extensions to both lifelong and multi-task learning. We +achieve this relaxation with a key observation that identifies +a shared Markovian-policy saddle-point across CMDPs of +perfectly safe RL with different initial states (that is, the +constraint in the CMDP imposes perfect safety). This ob- +servation is new to our knowledge, and it is derived from +the particular structure of perfectly safe RL, which is a sub- +problem used in our reset-free RL reduction. We note that +general CMDPs with different initial states do not generally +admit shared Markovian-policy saddle-points. Therefore, +on the technical side, our algorithm can also be viewed as +the first safe RL algorithm that allows for arbitrary initial +state sequences without strong assumptions. +While we propose a generic reduction technique to de- +sign reset-free RL algorithms, our regret and constraint +violation bounds are still comparable to the above works +when specialized to their setting. Under the linear MDP +assumption, our algorithm achieves ˜O( +√ +d3H4K) regret +and violation (equivalently, the number of resets in reset- +free RL), which is asymptotically equivalent to Ghosh et al. +(2022) and comparable to the bounds of ˜O +√ +d2H6K from +Ding et al. (2021) for a fixed initial state. +3 +PRELIMINARY +We consider episodic reset-free RL: in each episode, the +agent aims to optimize for a fixed-horizon return starting +from the last state of the previous episode or some state +that the agent was reset to in the previous episode if reset +occurs (e.g., due to the robot falling over). +Problem Setup and Notation +Formally, we can de- +fine episodic reset-free RL as a Markov decision process +(MDP), (S, A, P, r, H), where S is the state space, A is +the action space, P = {Ph}H +h=1 is the transition dynamics, +r = {rh}H +h=1 is the reward function, and H is the task hori- +zon. We assume P and r are unknown. We allow S to be +large or continuous but assume A is relatively small so that +maxa∈A can be performed. We designate the set of reset +states as Sreset ⊆ S; we do not assume that the agent has +knowledge of Sreset. We also do not assume that there is a +reset-free action for each state, as opposed to (Amani et al., +2021). Therefore, the agent needs to plan for the long-term +to avoid resets. We assume rh : S × A → [0, 1], and +for simplicity, we assume rh is deterministic. However, we +note that it would be easy to extend this to the setting where +rewards are stochastic. +The agent interacts with the environment for K total +episodes. Following the convention of episodic problems, +we suppose the state space S is layered, and a state sτ ∈ S +at time τ is factored into two components sτ = (¯s, τ) +where ¯s denotes the time-invariant part. Reset happens at +time τ if ¯s ∈ Sreset (which we also write as sτ ∈ Sreset), +and the initial state of the next episode will be s1 = (¯s′, 1) +where ¯s′ is sampled from a fixed but unknown state distri- +bution. Otherwise, the initial state of the next episode is +the last state of the current episode, i.e., for episode k + 1, +sk+1 +1 += (¯s, 1) if sk +H = (¯s, H) in episode k.1 +We denote the set of Markovian policies as ∆, and a policy +π ∈ ∆ as π = {πh(ah|sh)}H +h=1. We define the state value +function and the state-action value function under π as2 +V π +r,h(s) := Eπ +� min(H,τ) +� +t=h +rt(st, at)|sh = s +� +(1) +Qπ +r,h(s, a) := rh(s, a) + E +� +V π +r,h+1(sh+1)|sh = s, ah = a +� +, +where h ≤ τ, and we recall τ is the time step when the +agent enters Sreset (if at all). +Objective +The overall goal is for the agent to learn a +Markovian policy to maximize its cumulative reward while +avoiding resets. Therefore, our performance measures are +1We can extend this setup to reset-free multi-task or lifelong +RL problems that are modeled as contextual MDPs since our algo- +rithm can work with any initial state sequence. In this case, we can +treat each state here as sτ = (¯s, c, τ), where c denotes the context +that stays constant within an episode. If no reset happens, the ini- +tial state of episode k + 1 can be written as sk+1 +1 += (¯s, ck+1, 1) +if sk +H = (¯s, ck, H) in episode k, where the new context ck+1 can +follow any distribution and may depend on the current context ck. +2This value function definition is the same as the H-step cu- +mulative reward in an MDP formulation where we place the agent +into a fictitious zero-reward absorbing state (i.e., a mega-state ab- +stracting Sreset) after the agent enters Sreset. We choose the cur- +rent formulation to make the definition of resets more transparent. + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +as follows (we seek to minimize both quantities): +Regret(K) = +max +π∈∆0(K) +K +� +k=1 +V π +r,1(sk +1) − V πk +r,1 (sk +1) +(2) +Resets(K) = +K +� +k=1 +Eπk + + +min(H,τ) +� +h=1 +1[sh ∈ Sreset] +���s1 = sk +1 + + +(3) +where ∆0(K) ⊆ ∆ is the set of Markovian policies that +avoid resets for all episodes, and πk is the policy used by +the agent in episode k. Note that by the reset mechanism +�min(H,τ) +h=1 +1[sh ∈ Sreset] ∈ {0, 1}. +Notice that the initial states in our regret and reset measures +are determined by the learner, not the optimal policy like in +some classical definitions of regret. Given the motivation +behind reset-free RL (see Section 1), we can expect that the +initial states here are by construction meaningful for perfor- +mance comparison; otherwise, a reset would have occurred +to set the learner to a meaningful state. A following impli- +cation is that all bad absorbing states are in Sreset; hence, +the agent cannot use the trivial solution of hiding in a bad +absorbing state to achieve small regret. +To make the problem feasible, we assume that achieving no +resets is possible. We state this formally in the assumption +below. +Assumption 1. For any sequence {sk +1}K +k=1, the set ∆0(K) +is not empty. That is, there is a Markovian policy π ∈ ∆ +such that Eπ[�H +h=1 1[sh ∈ Sreset]|s1 = sk +1] = 0. +This is a reasonable assumption in practice. If reset hap- +pens, the agent should be set to a state that the agent can +continue to operate in without reset; if the agent is at a state +where no such reset-free policy exists, reset should happen. +This assumption is similar to the assumption on the exis- +tence of a perfectly safe policy in safe RL literature, which +is a common and relatively weak assumption (Ghosh et al., +2022; Ding et al., 2021). If there were to be initial states +that inevitably lead to a reset, the problem would be infea- +sible. +4 +A NO-REGRET REDUCTION FOR +RESET-FREE RL +We present our main reduction of reset-free RL to regret +minimization in a two-player game. In the following, we +first show that reset-free RL can be framed as a sequence of +CMDPs of safe RL problems with an adaptive initial state +sequence. Then we design a two-player game based on a +primal-dual analysis of this sequence of CMDPs. Finally, +we show achieving sublinear regret in this two-player game +implies sublinear regret and resets in the original reset-free +RL problem in (2). The complete proofs for this section +can be found in Appendix A.1. +4.1 +Reset-free RL as a Sequence of CMDPs +The first step of our reduction is to cast the reset-free RL +problem in Section 3 to a sequence of CMDP problems +which share the same rewards, constraints, and dynamics, +but have different initial states. Each problem instance in +this sequence corresponds to an episode of the reset-free +RL problem, and its constraint describes the probability of +the agent entering a state that requires reset. +Specifically, we denote these constrained MDPs3 as +{(S, A, P, r, H, c, sk +1)}K +k=1: +in episode k, the CMDP problem is defined as +max +π∈∆ V π +r,1(sk +1), s.t. V π +c,1(sk +1) ≤ 0 +(4) +where we define the cost as +ch(s, a) := 1[s ∈ Sreset] +and V π +c,1, defined similarly to (1), is the state value func- +tion with respect to the cost c . We note that the initial +state, sk +1, depends on the past behaviors of the agent, and +Assumption 1 ensures each CMDP in (4) is a feasible prob- +lem (i.e., there is a Markovian policy satisfying the con- +straint). We can interpret each CMDP in (4) as a safe RL +problem by treating Sreset as the unsafe states that a safe +RL agent should avoid. From this perspective, the con- +straint in (4) can be viewed as the probability of a trajectory +entering an unsafe state. +Since CMDPs are typically defined without early episode +termination unlike (1), with abuse of notation, we extend +the definitions of P, S, r, c as follows so that the CMDP +definition above is consistent with the common literature. +We introduce a fictitious absorbing state denoted as s† in +S, where rh(s†, a) = 0 and ch(s†, a) = 0; once the agent +enters s†, it stays there until the end of the episode. We +extend the definition P such that, after the agent is in a state +s ∈ Sreset, any action it takes brings it to s† in the next +time step. In this way, we can write the value function, +e.g. for reward, as V π +r,h(s) = Eπ +� �H +t=h rt(st, at)|sh = s +� +in terms of this extended dynamics. We note that these +two formulations are mathematically the same for the pur- +pose of learning; when the agent enters s†, it means that the +agent is reset in the episode. +By the construction above, we can write +Resets(K) = +K +� +k=1 +V πk +c,1 (sk +1) +which is the same as the number of total constraint vio- +lations across the CMDPs. Because we do not make any +3In general, the solution (i.e., optimal policy) to a CMDP +depends on its initial state, unlike in MDPs (see remark 2.2 in +Altman (1999)). + +Hoai-An Nguyen, Ching-An Cheng +assumptions about the agent’s knowledge of the constraint +function (e.g., the agent does not know states ∈ Sreset), we +allow the agent to reset during learning while minimizing +the total number of resets over all K episodes. +4.2 +Reduction to Two-Player Game +From the problem formulation above, we see that the ma- +jor difficulty of reset-free RL is the coupling between an +adaptive initial state sequence and the constraint on reset +probability. If we were to remove either of them, we can +use standard algorithms, since the problem will become a +single CMDP problem (Altman, 1999) or an episodic RL +problem with varying initial states (Jin et al., 2019). +We propose a reduction to systematically design algorithms +for this sequence of CMDPs and therefore for reset-free +RL. The main idea is to approximately solve the saddle +point problem of each CMDP in (4), i.e., +max +π∈∆ min +λ≥0 V π +r,1(sk +1) − λV π +c,1(sk +1) +(5) +where λ denotes the dual variable (i.e. +the La- +grange multiplier). +Each CMDP can be framed as +a linear program (Hern´andez-Lerma and Lasserre, 2002) +whose primal and dual optimal values match (see +section 8.1 in Hazan et al. (2016)). +Therefore, for +each CMDP, maxπ∈∆ minλ≥0 V π +r,1(sk +1) − λV π +c,1(sk +1) = +minλ≥0 maxπ∈∆ V π +r,1(sk +1) − λV π +c,1(sk +1). +While using a primal-dual algorithm to solve for the sad- +dle point of a single CMDP is straightforward and known, +using this approach for a sequence of CMDPs is not obvi- +ous. Each CMDP’s optimal policy and Lagrange multiplier +can be a function of the initial state (Altman, 1999), and +in general, a common saddle point of Markovian polices +and Lagrange multipliers does not necessarily exist for a +sequence of CMDPs with varying initial states.4 As a re- +sult, it is unclear if there exists a primal-dual algorithm to +solve this sequence, especially given that the initial states +here are adaptively chosen. +Existence of a Shared Saddle-Point +Fortunately, there +is a shared saddle-point with a Markovian policy across all +the CMDPs considered here due to the special structure of +reset-free RL. It is a proof that does not use Slater’s condi- +tion for strong duality, unlike similar literature, but attains +the desired property. Instead we use Assumption 1 and the +fact that the cost c is non-negative. We formalize this be- +low. +Theorem 1. There exist a function ˆλ(·) where for each s, +ˆλ(s) ∈ arg min +y≥0 +� +max +π∈∆ V π +r,1(s) − yV π +c,1(s) +� +, +4A shared saddle-point with a non-Markovian policy always +exists on the other hand. +and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a +saddle-point to the CMDPs +max +π∈∆ V π +r,1(s1), s.t. V π +c,1(s1) ≤ 0 +for all initial states s1 ∈ S such that the CMDP is feasible. +That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S, +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) ≥ V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1) +≥ V π +r,1(s1) − ˆλ(s1)V π +c,1(s1). +(6) +Corollary 1. +For π∗ +in +Theorem 1, +it holds that +Regret(K) = �K +k=1 V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1). +We prove for ease of construction that the pair (π∗, λ∗) +where λ∗(·) = ˆλ(·) + 1 is also a saddle-point. +Corollary 2. For any saddle-point to the CMDPs +max +π∈∆ V π +r,1(s1), s.t. V π +c,1(s1) ≤ 0 +of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also +a saddle-point as defined in eq (6). +Therefore, the pair (π∗, λ∗) in Corollary 2 is a saddle-point +to all the CMDPs the agent faces. This makes potentially +designing a two-player game reduction possible. In the +next section, we give the details of our construction. +Two-Player Game +Our two-player game proceeds itera- +tively in the following manner: in episode k, a dual player +determines a state value function λk : S → R, and a +primal player determines a policy πk which can depend +on λk. +Then the primal and dual player receive losses +Lk(πk, λ) and −Lk(π, λk), respectively, where Lk(π, λ) +is a Lagrangian function defined as +Lk(π, λ) := V π +r,1(sk +1) − λ(sk +1)V π +c,1(sk +1). +(7) +The regret of these two players are defined as follows. +Definition 1. Let πc and λc be comparators. The regret of +the primal and the dual players are +Rp({πk}K +k=1, πc) := +K +� +k=1 +Lk(πc, λk) − Lk(πk, λk) (8) +Rd({λk}K +k=1, λc) := +K +� +k=1 +Lk(πk, λk) − Lk(πk, λc). (9) +We present our main reduction theorem for reset-free RL +below. By Theorem 2, if both players have sublinear re- +gret in the two-player game, then the resulting policy se- +quence will have sublinear performance regret and a sub- +linear number of resets in the original RL problem. Since +there are many standard techniques (Hazan et al., 2016) + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +from online learning to solve such a two-player game, we +can leverage them to systematically design reset-free RL +algorithms. In the next section, we will give an example +algorithm of this reduction for linear MDPs. +Theorem 2. Under Assumption 1, for any sequences +{πk}K +k=1 and {λk}K +k=1 , it holds that +Regret(K) ≤ Rp({πk}K +k=1, π∗) + Rd({λk}K +k=1, 0) +Resets(K) ≤ Rp({πk}K +k=1, π∗) + Rd({λk}K +k=1, λ∗) +where (π∗, λ∗) is the saddle-point defined in Corollary 2. +Proof +Sketch +of +Theorem 1 +Let +Q∗ +c(s, a) += +minπ∈∆ Qπ +c (s, a) and V ∗ +c (s) += +minπ∈∆ V π +c (s). +We +define π∗ in Theorem 1 as the optimal policy to the +following MDP: (S, A, P, r, H), +where we define a +state-dependent action space A as +As = {a ∈ A : Q∗ +c(s, a) ≤ V ∗ +c (s)}. +By definition, As is non-empty for all s. +We also define a shorthand notation: we write π ∈ A(s) if +Eπ[�H +t=1 1{at /∈ Ast}|s1 = s] = 0. We have the follow- +ing lemma, which is an application of the performance dif- +ference lemma (see Lemma 6.1 in (Kakade and Langford, +2002) and Lemma A.1 in (Cheng et al., 2021)). +Lemma 1. For any s1 ∈ S such that V ∗ +c (s1) = 0 and any +π ∈ ∆, it is true that π ∈ A(s1) if and only if V π +c (s1) = 0. +We prove our main claim, (6), below. Because V π∗ +c,1 (s1) = +0, the first inequality is trivial: V π∗ +r,1 (s1)−λ(s1)V π∗ +c,1 (s1) = +V π∗ +r,1 (s1) = V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1). +To prove the second inequality, we use Lemma 1: +V π +r,1(s1) − ˆλ(s1)V π +c,1(s1) +≤ max +π∈∆ V π +r,1(s1) − ˆλ(s1)V π +c,1(s1) += min +y≥0 max +π∈∆ V π +r,1(s1) − yV π +c,1(s1) += +max +π∈Ac(s1) +V π +r,1(s1) +(By Lemma 1 ) +=V π∗ +r,1 (s1) = V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1). +Proof Sketch of Theorem 2 +We first establish the fol- +lowing intermediate result that will help us with our de- +composition. +Lemma 2. For any primal-dual sequence {πk, λk}K +k=1, +�K +k=1(Lk(π∗, λ′) − Lk(πk, λk)) +≤ +Rp({π}K +k=1, π∗), +where (π∗, λ′) is the saddle-point defined in either +Theorem 1 or Corollary 2. +Then we upper bound Regret(K) and Resets(K) by +Rp({πk}K +k=1, πc) and Rd({λk}K +k=1, λc) for suitable com- +parators. This decomposition is inspired by the techniques +used in Ho-Nguyen and Kılınc¸-Karzan (2018). +We first bound Resets(K). +Lemma 3. For any primal-dual sequence {πk, λk}K +k=1, +�K +k=1 V πk +c,1 (sk +1) ≤ Rp({π}K +k=1, π∗) + Rd({λ}K +k=1, λ∗), +where (π∗, λ∗) is the saddle-point defined in Corollary 2. +Proof. Notice �K +k=1 V πk +c,1 (sk +1) += +�K +k=1 Lk(πk, ˆλ) − +Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined +in Theorem 1. +By (6), and adding and subtracting +�K +k=1 Lk(πk, λk), we can bound this difference by +K +� +k=1 +Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗). +Using Lemma 2 and Definition 1 to upper bound the above, +we get the desired result. +Lastly, we bound Regret(K) with the lemma below and +Corollary 1. +Lemma 4. For any primal-dual sequence {πk, λk}K +k=1, +�K +k=1(V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1)) +≤ +Rp({π}K +k=1, π∗) + +Rd({λ}K +k=1, 0), where (π∗, λ∗) is the saddle-point defined +in Corollary 2. +Proof. Note that L(π∗, λ∗) = L(π∗, 0) since V π∗ +c,1 = 0 for +all k ∈ [K] = {1, ..., K}. Since by definition, for any π, +Lk(π, 0) = V π +r,1(sk +1), we have the following: +K +� +k=1 +V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1) = +K +� +k=1 +Lk(π∗, λ∗) − Lk(πk, 0) += +K +� +k=1 +Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0) +≤Rp({π}K +k=1, π∗) + Rd({λ}K +k=1, 0) +where the last inequality follows from Lemma 2 and +Definition 1. +5 +RESET-FREE LEARNING FOR +LINEAR MDP +To demonstrate the utility of our reduction, we design a +provably correct algorithm instantiation for reset-free RL. +We consider a linear MDP setting, which is common in the +RL theory literature (Jin et al., 2019). +Assumption 2. We assume (S, A, P, r, c, H) is linear with +a known feature map φ : S × A → Rd: for any h ∈ [H], +there exists d unknown signed measures µh = {µ1 +h, ..., µd +h} +over S such that for any (s, a, s′) ∈ S × A × S, we have +Ph(s′|a) = ⟨φ(s, a), µh(s′)⟩, + +Hoai-An Nguyen, Ching-An Cheng +and there exists unknown vectors ωr,h, ωc,h ∈ Rd such that +for any (s, a) ∈ S × A, +rh(s, a) = ⟨φ(s, a), ωr,h⟩ +ch(s, a) = ⟨φ(s, a), ωc,h⟩. +We assume, for all (s, a, h) ∈ S×A×[H], ||φ(s, a)||2 ≤ 1, +and max{||µh(s)||2, ||ωr,h||2, ||ωc,h||2} ≤ +√ +d. +In addition, we make a linearity assumption on the function +λ∗ defined in Theorem 1. +Assumption 3. We assume the knowledge of a feature ξ : +S → Rd such that ∀s ∈ S, ||ξ(s)||2 ≤ 1 and λ∗(s) = +⟨ξ(s), θ∗⟩, for some unknown vector θ∗ ∈ Rd. In addition, +we assume the knowledge of a convex set5 U ⊆ Rd such +that θ∗, 0 ∈ U and ∀θ ∈ U, ||θ||2 ≤ B and ⟨ξ(s), θ⟩ ≥ 0. 6 +5.1 +Algorithm +The basis of our algorithm lies between the interaction be- +tween the primal and dual players. We let the dual player +perform projected gradient descent and the primal player +update policies based on upper confidence bound with the +knowledge of the decision of the dual player. This sequen- +tial strategy resembles the optimistic style update in online +learning (Mertikopoulos et al., 2018). +Specifically, in each episode, upon receiving the initial +state, we execute actions according to the policy based on +a softmax (lines 5-8). Then, we perform the dual update +through projected gradient descent. The dual player plays +for the next round, k + 1, after observing its loss after the +primal player plays for the current round, k. The projec- +tion is to a l2 ball containing λ∗(·) (lines 9-11). Finally, we +perform the update of the primal player by computing the +Q-functions for both the reward and cost with a bonus to +encourage exploration (lines 12-20). +This algorithm builds upon Ghosh et al. (2022). However, +notably, we extend it to handle the adaptive initial state se- +quence seen in reset-free RL by Theorems 1 and 2. +5.2 +Analysis +We show below that our algorithm achieves regret and +number of resets that are sublinear in the total number of +time steps, KH, using Theorem 2. This result is asymptot- +ically equivalent to Ghosh et al. (2022) and comparable to +the bounds of ˜O +√ +d2H6K from Ding et al. (2021). +5Such a set can be constructed by upper bounding the values +using scaling and ensuring non-negativity using a sum of squares +approach. +6From the previous section, we can see that the optimal func- +tion for the dual player is not necessarily unique. +So, we as- +sume bounds on at least one optimal function that we designate +as λ∗(s). +Theorem 3. Under Assumptions 1, 2, and 3, with high +probability, Regret(K) +≤ +˜O((B + 1) +√ +d3H4K) and +Resets(K) ≤ ˜O((B + 1) +√ +d3H4K). +Proof Sketch of Theorem 3 +We provide a proof sketch +here and defer the complete proof to Appendix A.2. We +first bound the regret of {πk}K +k=1 and {λk}K +k=1, and then +use this to prove the bounds on our algorithm’s regret and +number of resets with Theorem 2. +We first bound the regret of {λk}K +k=1. +Lemma 5. Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈ +U. Then it holds that Rd({λk}K +k=1, λc) ≤ 1.5B +√ +K + +�K +k=1(λk(sk +1) − λc(sk +1))(V k +c,1(sk +1) − V πk +c,1 (sk +1)). +Proof. We notice first an equality. +Rd({λk}K +k=1, λc) = +K +� +k=1 +Lk(πk, λk) − Lk(πk, λc) += +K +� +k=1 +λc(sk +1)V πk +c,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1) += +K +� +k=1 +(λk(sk +1) − λc(sk +1))(−V k +c,1(sk +1)) ++ +K +� +k=1 +(λk(sk +1) − λc(sk +1))(V k +c,1(sk +1) − V πk +c,1 (sk +1)). +We observe that the first term is an online linear problem +for θk (the parameter of λk(·)). +In episode k ∈ [K], +λk is played, and then the loss is revealed. +Since the +space of θk is convex, we use standard results (Lemma +3.1 (Hazan et al., 2016)) to show that updating θk through +projected gradient descent results in an upper bound for +�K +k=1(λk(sk +1) − λc(sk +1))(−V k +c,1(sk +1)). +We now bound the regret of {π}K +k=1. +Lemma 6. Consider any πc. +With high probability, +Rp({π}K +k=1, πc) ≤ 2H(1 + B + H) + �K +k=1 V k +r,1(sk +1) − +V πk +r,1 (sk +1) + λk(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)). +Proof. First we expand the regret into two terms. +Rp({π}K +k=1, πc) = +K +� +k=1 +Lk(πc, λk) − Lk(πk, λk) += +K +� +k=1 +V πc +r,1 (sk +1) − λk(sk +1)V πc +c,1(sk +1) − [V πk +r,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1)] += +K +� +k=1 +V πc +r,1 (sk +1) − λk(sk +1)V πc +c,1(sk +1) − [V k +r,1(sk +1) − λk(sk +1)V k +c,1(sk +1)] ++ +K +� +k=1 +V k +r,1(sk +1) − V πk +r,1 (sk +1) + λk(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)). + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +Algorithm 1 Primal-Dual Reset-Free RL Algorithm for Linear MDP with Adaptive Initial States +1: Input: Feature maps φ and ξ. Failure probability p. Some universal constant c. +2: Initialization: θ1 = 0, wr,h = 0, wc,h = 0, α = +log(|A|)K +2(1 + B + H), β = cdH +� +log(4 log |A|dKH/p) +3: for episodes k = 1, ...K do +4: +Observe the initial state sk +1. +5: +for step h = 1, ..., H do +6: +Compute πh,k(a|·) ← +exp(α(Qk +r,h(·, a) − λk(sk +1)Qk +c,h(·, a))) +� +a exp(α(Qk +r,h(·, a) − λk(sk +1)Qk +c,h(·, a))). +7: +Take action ak +h ∼ πh,k(·|sk +h) and observe sk +h+1. +8: +end for +9: +ηk ← B +√ +k +10: +Update θk+1 ← ProjU(θk + ηk · ξ(sk +1)V k +c,1(sk +1)) +11: +λk+1(·) ← ⟨θk+1, ξ(·)⟩ +12: +for step h = H, ..., 1 do +13: +Λk+1 +h +← +k� +i=1 +φ(si +h, ai +h)φ(si +h, ai +h)T + λI. +14: +wk+1 +r,h ← (Λk+1 +h +)−1[ +k� +i=1 +φ(si +h, ai +h)[rh(si +h, ai +h) + V k+1 +r,h+1(si +h+1)]] +15: +wk+1 +c,h ← (Λk+1 +h +)−1[ +k� +i=1 +φ(si +h, ai +h)[ch(si +h, ai +h) + V k+1 +c,h+1(si +h+1)]] +16: +Qk+1 +r,h (·, ·) ← max{min{⟨wk+1 +r,h , φ(·, ·)⟩ + β(φ(·, ·)T (Λk+1 +h +)−1φ(·, ·))1/2, H − h + 1}, 0} +17: +Qk+1 +c,h (·, ·) ← max{min{⟨wk+1 +c,h , φ(·, ·)⟩ − β(φ(·, ·)T (Λk+1 +h +)−1φ(·, ·))1/2, 1}, 0} +18: +V k+1 +r,h (·) = � +a πh,k(a|·)Qk+1 +r,h (·, a) +19: +V k+1 +c,h (·) = � +a πh,k(a|·)Qk+1 +c,h (·, a) +20: +end for +21: end for +To bound the first term, we use Lemma 3 from Ghosh et al. +(2022), which characterizes the property of upper confi- +dence bound. +Lastly, +we derive a bound on Rd({λk}K +k=1, λc) + +Rp({πk}K +k=1, πc), which directly implies our final up- +per bound on Regret(K) and Resets(K) in Theorem 3 by +Theorem 2. Combining the upper bounds in Lemma 5 and +Lemma 6, we have a high-probability upper bound +Rd({λk}K +k=1, λc) + Rp({πk}K +k=1, πc) +≤ 1.5B +√ +K + 2H(1 + B + H)+ ++ +K +� +k=1 +V k +r,1(sk +1) − V πk +r,1 (sk +1) + λc(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)) +where the last term is the overestimation error due to opti- +mism. Note that for all k ∈ [K], V k +r,1(sk +1) and V k +c,1(sk +1) are +as defined in Algorithm 1 and are optimistic estimates of +V π∗ +r,1 (sk +1) and V π∗ +c,1 (sk +1). To bound this term, we use Lemma +4 from (Ghosh et al., 2022). +6 +CONCLUSION +We propose a generic no-regret reduction for designing +provable reset-free RL algorithms. +Our reduction casts +reset-free RL into the regret minimization problem of a +two-player game, for which many existing no-regret al- +gorithms are available. As a result, we can reuse these +techniques to systematically build new reset-free RL algo- +rithms. 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PMLR. + +Hoai-An Nguyen, Ching-An Cheng +A +Appendix +A.1 +Missing Proofs for Section 4 +A.1.1 +Proof of Theorem 1 +Theorem 1. There exist a function ˆλ(·) where for each s, +ˆλ(s) ∈ arg min +y≥0 +� +max +π∈∆ V π +r,1(s) − yV π +c,1(s) +� +, +and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a saddle-point to the CMDPs +max +π∈∆ V π +r,1(s1), s.t. V π +c,1(s1) ≤ 0 +for all initial states s1 ∈ S such that the CMDP is feasible. That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S, +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) ≥ V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1) +≥ V π +r,1(s1) − ˆλ(s1)V π +c,1(s1). +(6) +For policy π∗, we define it by the following construction (we ignore writing out the time dependency for simplicity): first, +we define a cost-based MDP Mc = (S, A, P, c, H). Let Q∗ +c(s, a) = minπ∈∆ Qπ +c (s, a) and V ∗ +c (s) = minπ∈∆ V π +c (s) be +the optimal values, where we recall V π +c and Qπ +c are the state and state-action values under policy π with respect to the cost. +Now we construct another reward-based MDP M = (S, A, P, r, H), where we define the state-dependent action space A +as +As = {a ∈ A : Q∗ +c(s, a) ≤ V ∗ +c (s)}. +By definition, As is non-empty for all s. We define a shorthand notation: we write π ∈ A(s) if Eπ[�H +t=1 1{at /∈ +Ast}|s1 = s] = 0. Then we have the following lemma, which is a straightforward application of the performance +difference lemma. +Lemma 1. For any s1 ∈ S such that V ∗ +c (s1) = 0 and any π ∈ ∆, it is true that π ∈ A(s1) if and only if V π +c (s1) = 0. +Proof. By performance difference lemma (Kakade and Langford, 2002), we can write +V π +c (s1) − V ∗ +c (s1) = Eπ +� H +� +t=1 +Q∗ +c(st, at) − V ∗ +c (st)|s1 = s1 +� +. +If for some s1 ∈ S, π ∈ A(s1), then Eπ +��H +t=1 Q∗ +c(st, at) − V ∗ +c (st) +� +≤ 0, which implies V π +c (s1) ≤ V ∗ +c (s1). But since V ∗ +c +is optimal, V π +c (s1) = V ∗ +c (s1). On the other hand, suppose V π +c (s1) = 0. It implies Eπ +��H +t=1 Q∗ +c(st, at) − V ∗ +c (st) +� += 0 +since V ∗ +c (s1) = 0. Because by definition of optimality Q∗ +c(st, at) − V ∗ +c (st) ≥ 0, this implies π ∈ A(s1). +We set our candidate policy π∗ as the optimal policy of this M. By Lemma 1, we have V π∗ +c +(s) = V ∗ +c (s), so it is also an +optimal policy to Mc. We prove our main claim of Theorem 1 below: +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) ≥ V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1) ≥ V π +r,1(s1) − ˆλ(s1)V π +c,1(s1). +Proof. Because V π∗ +c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial: +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) = V π∗ +r,1 (s1) = V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1). + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +For the second inequality, we use Lemma 1: +V π +r,1(s1) − ˆλ(s1)V π +c,1(s1) ≤ max +π∈∆ V π +r,1(s1) − ˆλ(s1)V π +c,1(s1) += min +y≥0 max +π∈∆ V π +r,1(s1) − yV π +c,1(s1) += +max +π∈Ac(s1) +V π +r,1(s1) +(By Lemma 1 ) += V π∗ +r,1 (s1) += V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1). +A.1.2 +Proof of Corollary 1 +Corollary 1. For π∗ in Theorem 1, it holds that Regret(K) = �K +k=1 V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1). +Proof. To +prove +Regret(K) += +�K +k=1 V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1), +it +suffices +to +prove +�K +k=1 V π∗ +r,1 (sk +1) += +maxπ∈∆0(K) +�K +k=1 V π +r,1(sk +1). +By Lemma 1 and under Assumption 1, we notice that maxπ∈∆0(K) +�K +k=1 V π +r,1(sk +1) = +maxπ∈A(sk +1 ),∀k∈[K] +�K +k=1 V π +r,1(sk +1). This is equal to �K +k=1 V π∗ +r,1 (sk +1) by the definition of π∗ in the proof of Theorem 1. +A.1.3 +Proof of Corollary 2 +Corollary 2. For any saddle-point to the CMDPs +max +π∈∆ V π +r,1(s1), s.t. V π +c,1(s1) ≤ 0 +of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also a saddle-point as defined in eq (6). +Proof. We prove that eq (6) holds for (π∗, λ∗), that is +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) ≥ V π∗ +r,1 (s1) − λ∗(s1)V π∗ +c,1 (s1) ≥ V π +r,1(s1) − λ∗(s1)V π +c,1(s1). +Because V π∗ +c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial: +V π∗ +r,1 (s1) − λ(s1)V π∗ +c,1 (s1) = V π∗ +r,1 (s1) = V π∗ +r,1 (s1) − λ∗(s1)V π∗ +c,1 (s1). +For the second inequality, we use Theorem 1: +V π +r,1(s1) − λ∗(s1)V π +c,1(s1) ≤V π +r,1(s1) − ˆλ(s1)V π +c,1(s1) +≤V π∗ +r,1 (s1) − ˆλ(s1)V π∗ +c,1 (s1) +=V π∗ +r,1 (s1) − λ∗(s1)V π∗ +c,1 (s1) +where the first step is because V π +c,1(s1) by definition is in [0, 1] and λ∗ = ˆλ + 1, and the second step is by Theorem 1. +A.1.4 +Proof of Theorem 2 +Theorem 2. Under Assumption 1, for any sequences {πk}K +k=1 and {λk}K +k=1 , it holds that +Regret(K) ≤ Rp({πk}K +k=1, π∗) + Rd({λk}K +k=1, 0) +Resets(K) ≤ Rp({πk}K +k=1, π∗) + Rd({λk}K +k=1, λ∗) +where (π∗, λ∗) is the saddle-point defined in Corollary 2. +We first establish the following intermediate result that will help us with our decomposition. + +Hoai-An Nguyen, Ching-An Cheng +Lemma 2. For any primal-dual sequence {πk, λk}K +k=1, �K +k=1(Lk(π∗, λ′) − Lk(πk, λk)) ≤ Rp({π}K +k=1, π∗), where +(π∗, λ′) is the saddle-point defined in either Theorem 1 or Corollary 2. +Proof. We derive this lemma by Theorem 1 and Corollary 2. First notice by Theorem 1 and Corollary 2 that for λ′ = λ∗, ˆλ, +K +� +k=1 +Lk(π∗, λ′) = +K +� +k=1 +V π∗ +r,1 (sk +1) − λ′(sk +1)V π∗ +c,1 (sk +1) +≤ +K +� +k=1 +V π∗ +r,1 (sk +1) − λk(sk +1)V π∗ +c,1 (sk +1) = +K +� +k=1 +Lk(π∗, λk). +Then we can derive +K +� +k=1 +(Lk(π∗, λ′) − Lk(πk, λk)) = +K +� +k=1 +Lk(π∗, λ′) − Lk(π∗, λk) + Lk(π∗, λk) − Lk(πk, λk) +≤ +K +� +k=1 +Lk(π∗, λk) − Lk(πk, λk) = Rp({π}K +k=1, π∗) +which finishes the proof. +Then we upper bound Regret(K) and Resets(K) by Rp({πk}K +k=1, πc) and Rd({λk}K +k=1, λc) for suitable comparators. This +decomposition is inspired by the techniques used in Ho-Nguyen and Kılınc¸-Karzan (2018). +We first bound Resets(K). +Lemma 3. For any primal-dual sequence {πk, λk}K +k=1, �K +k=1 V πk +c,1 (sk +1) ≤ Rp({π}K +k=1, π∗) + Rd({λ}K +k=1, λ∗), where +(π∗, λ∗) is the saddle-point defined in Corollary 2. +Proof. Notice �K +k=1 V πk +c,1 (sk +1) = �K +k=1 Lk(πk, ˆλ) − Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined in Theorem 1. +This is because, as defined, λ∗ = ˆλ + 1. Therefore, we bound the RHS. We have +K +� +k=1 +Lk(πk, ˆλ) − Lk(πk, λ∗) = +K +� +k=1 +Lk(πk, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗) +≤ +K +� +k=1 +Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗) +≤Rp({π}K +k=1, π∗) + Rd({λ}K +k=1, λ∗) +where second inequality is because �K +k=1 Lk(π∗, ˆλ) ≥ �K +k=1 Lk(πk, ˆλ) by Theorem 1, and the first inequality follows +from Lemma 2 and Definition 1. +Lastly, we bound Regret(K) with the lemma below and Corollary 1. +Lemma 4. For any primal-dual sequence {πk, λk}K +k=1, �K +k=1(V π∗ +r,1 (sk +1)−V πk +r,1 (sk +1)) ≤ Rp({π}K +k=1, π∗)+Rd({λ}K +k=1, 0), +where (π∗, λ∗) is the saddle-point defined in Corollary 2. +Proof. Note that L(π∗, λ∗) = L(π∗, 0) since V π∗ +c,1 (sk +1) = 0 for all k ∈ [K]. Since by definition, for any π, Lk(π, 0) = +V π +r,1(sk +1), we have the following: +K +� +k=1 +V π∗ +r,1 (sk +1) − V πk +r,1 (sk +1) = +K +� +k=1 +Lk(π∗, λ∗) − Lk(πk, 0) += +K +� +k=1 +Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0) +≤Rp({π}K +k=1, π∗) + Rd({λ}K +k=1, 0) +where the last inequality follows from Lemma 2 and Definition 1. + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +A.2 +Missing Proofs for Section 5 +A.2.1 +Proof of Theorem 3 +Theorem 3. Under Assumptions 1, 2, and 3, with high probability, Regret(K) ≤ ˜O((B + 1) +√ +d3H4K) and Resets(K) ≤ +˜O((B + 1) +√ +d3H4K). +We first bound the regret of {πk}K +k=1 and {λk}K +k=1, and then use this to prove the bounds on our algorithm’s regret and +number of resets with Theorem 2. +We first bound the regret of {λk}K +k=1. +Lemma 5. Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈ U. +Then it holds that Rd({λk}K +k=1, λc) ≤ 1.5B +√ +K + +�K +k=1(λk(sk +1) − λc(sk +1))(V k +c,1(sk +1) − V πk +c,1 (sk +1)). +Proof. We notice first an equality. +Rd({λk}K +k=1, λc) = +K +� +k=1 +Lk(πk, λk) − Lk(πk, λc) += +K +� +k=1 +λc(sk +1)V πk +c,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1) += +K +� +k=1 +λc(sk +1)V πk +c,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1) ++ +K +� +k=1 +λc(sk +1)V k +c,1(sk +1) − λc(sk +1)V k +c,1(sk +1) + λk(sk +1)V k +c,1(sk +1) − λk(sk +1)V k +c,1(sk +1) += +K +� +k=1 +(λk(sk +1) − λc(sk +1))(−V k +c,1(sk +1)) + +K +� +k=1 +(λk(sk +1) − λc(sk +1))(V k +c,1(sk +1) − V πk +c,1 (sk +1)). +We observe that the first term is an online linear problem for θk (the parameter of λk(·)). +In episode k ∈ [K], +λk is played, and then the loss is revealed. +Since the space of θk is convex, we use standard results (Lemma +3.1 (Hazan et al., 2016)) to show that updating θk through projected gradient descent results in an upper bound for +�K +k=1(λk(sk +1) − λc(sk +1))(−V k +c,1(sk +1)). We restate the lemma here. +Lemma 7 (Lemma 3.1 (Hazan et al., 2016)). Let S ⊆ Rd be a bounded convex and closed set in Euclidean space. +Denote D as an upper bound on the diameter of S, and G as an upper bound on the norm of the subgradients of convex +cost functions fk over S. Using online projected gradient descent to generate sequence {xk}K +k=1 with step sizes {ηk = +D +G +√ +k, k ∈ [K]} guarantees, for all K ≥ 1: +RegretK = max +x∗∈K +K +� +k=1 +fk(xk) − fk(x∗) ≤ 1.5GD +√ +K. +Let us bound D. By Assumption 3, λ∗ = ⟨ξ(s), θ∗⟩ and ||θ∗||2 ≤ B. Since the comparator we use is λ∗, we can set D to +be B. To bound G, we observe that the subgradient of our loss function is ξ(s)V k +c,1(sk +1) for each k ∈ [K]. Therefore, since +V k +c,1(sk +1) ∈ [0, 1] and ||ξ(s)||2 ≤ 1 by Assumption 3, we can set G to be 1. +We now bound the regret of {π}K +k=1. +Lemma 6. Consider any πc. With high probability, Rp({π}K +k=1, πc) ≤ 2H(1 + B + H) + �K +k=1 V k +r,1(sk +1) − V πk +r,1 (sk +1) + +λk(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)). + +Hoai-An Nguyen, Ching-An Cheng +Proof. First we expand the regret into two terms. +Rp({π}K +k=1, πc) = +K +� +k=1 +Lk(πc, λk) − Lk(πk, λk) += +K +� +k=1 +V πc +r,1(sk +1) − λk(sk +1)V πc +c,1(sk +1) − [V πk +r,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1)] += +K +� +k=1 +V πc +r,1(sk +1) − λk(sk +1)V πc +c,1(sk +1) − [V πk +r,1 (sk +1) − λk(sk +1)V πk +c,1 (sk +1)] ++ +K +� +k=1 +[V k +r,1(sk +1) − λk(sk +1)V k +c,1(sk +1)] − [V k +r,1(sk +1) − λk(sk +1)V k +c,1(sk +1)] += +K +� +k=1 +V πc +r,1(sk +1) − λk(sk +1)V πc +c,1(sk +1) − [V k +r,1(sk +1) − λk(sk +1)V k +c,1(sk +1)] ++ +K +� +k=1 +V k +r,1(sk +1) − V πk +r,1 (sk +1) + λk(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)). +To bound the first term, we use Lemma 3 from Ghosh et al. (2022), which characterize the property of upper confidence +bound. For completeness, we re-write the lemma here. 7 +Lemma 8 (Lemma 3 (Ghosh et al., 2022)). With probability 1−p/2, it holds that T1 = �K +k=1 +� +V πc +r,1(sk +1)−λkV πc +c,1(sk +1) +� +− +� +V k +r,1(sk +1) − λkV k +c,1(sk +1) +� +≤ KH log(|A|)/α. Hence, for α = +log(|A|)K +2(1 + C + H), T1 ≤ 2H(1 + C + H), where C is such +that λk ≤ C. +In our problem setting, we can set C = B in the lemma above. Therefore, the first term is bounded by 2H(1+B+H). +Lastly, we derive a bound on Rd({λk}K +k=1, λc) + Rp({πk}K +k=1, πc), which directly implies our final upper bound on +Regret(K) and Resets(K) in Theorem 3 by Theorem 2. +Lemma 9. For any πc and λc(s) = ⟨ξ(s), θc⟩ such that ∥θc∥ ≤ B, we have with probability 1 − p, Rd({λk}K +k=1, λc) + +Rp({πk}K +k=1, πc) ≤ 1.5B +√ +K + 2H(1 + B + H) + O((B + 1) +√ +d3H4Kι2) where ι = log[log(|A|)4dKH/p]. +Proof. Combining the upper bounds in Lemma 5 and Lemma 6, we have an upper bound of +Rd({λk}K +k=1, λc) + Rp({πk}K +k=1, πc) =1.5B +√ +K + +K +� +k=1 +(λk(sk +1) − λc(sk +1))(V k +c,1(sk +1) − V πk +c,1 (sk +1)) ++ 2H(1 + B + H) + +K +� +k=1 +V k +r,1(sk +1) − V πk +r,1 (sk +1) + λk(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)) +=1.5B +√ +K + 2H(1 + B + H)+ ++ +K +� +k=1 +V k +r,1(sk +1) − V πk +r,1 (sk +1) + λc(sk +1)(V πk +c,1 (sk +1) − V k +c,1(sk +1)) +where the last term is the overestimation error due to optimism. To bound this term, we use Lemma 4 from Ghosh et al. +(2022). We re-write the lemma here. +Lemma 10 (Lemma 4 (Ghosh et al., 2022)). WIth probability at least 1 − p/2, for any λ ∈ [0, C], �K +k=1 +� +V k +r,1(sk +1) − +V πk +r,1 (sk +1) +� ++ λ �K +k=1 +� +V πk +c,1 (sk +1) − V k +c,1(sk +1) +� +≤ O((λ + 1) +√ +d3H4Kι2) where ι = log[log(|A|)4dKH/p]. +7Note that Ghosh et al. (2022) use a utility function rather than a cost function to denote the constraint on the MDP (cost is just −1× +utility). Also note that their Lemma 3 is proved for an arbitrary initial state sequence and for any comparator (which includes π∗). + +Provable Reset-free Reinforcement Learning by No-Regret Reduction +Since we have a bound on all λc(sk +1) of B for all k ∈ [K], we have a bound of O((B + 1) +√ +d3H4Kι2). + diff --git a/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/load_file.txt b/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75b152b53cf4dde9b8d2e79ddf666ef27ddfb2ea --- /dev/null +++ b/4dE0T4oBgHgl3EQfeQDL/content/tmp_files/load_file.txt @@ -0,0 +1,824 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf,len=823 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='02389v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='LG] 6 Jan 2023 Provable Reset-free Reinforcement Learning by No-Regret Reduction Hoai-An Nguyen Ching-An Cheng Rutgers University Microsoft Research Abstract Real-world reinforcement learning (RL) is of- ten severely limited since typical RL algorithms heavily rely on the reset mechanism to sample proper initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In practice, the reset mech- anism is expensive to implement due to the need for human intervention or heavily engineered en- vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To make learning more practical, we propose a generic no-regret reduction to sys- tematically design reset-free RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Our reduction turns reset-free RL into a two-player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We show that achieving sublinear regret in this two player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the origi- nal RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This means that the agent even- tually learns to perform optimally and avoid re- sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By this reduction, we design an instantia- tion for linear Markov decision processes, which is the first provably correct reset-free RL algo- rithm to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 1 INTRODUCTION Reinforcement learning (RL) enables an artificial agent to learn problem-solving skills directly through interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' However, RL is notorious for its sample inefficiency, and successful stories of RL so far are mostly limited to appli- cations where an accurate simulator of the world is avail- able (like in games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Real-world RL, such as robot learn- ing, remains a challenging open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' One key obstacle preventing the collection of a large num- ber of samples in real-world RL is the need for reset- ting the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The ability to reset the agent to proper initial states plays an important role in typical RL algo- rithms, as it affects which region the agent can explore and whether the agent can recover from its past mis- takes (Kakade and Langford, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In the absence of a reset mechanism, agents may get stuck in absorbing states, such as those where it has damaged itself or irreparably al- tered the learning environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, in most settings, completely avoiding resets without prior knowledge of the reset states or environment is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For instance, a robot learning to walk would inevitably fall before perfecting the skill, and timely intervention is needed to prevent damaging the hardware and to return the robot to a walkable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Another example we can consider is a robot manipulator learning to stack three blocks on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Unrecoverable states that would require intervention would include the robot knock- ing a block off the table, or the robot smashing itself force- fully into the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Reset would then reconfigure the scene to a meaningful initial state that is good for the robot to learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Resetting is a necessary part of the real-world learning pro- cess if we want an agent to be able to adapt to any en- vironment, but it is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Unlike in simulation, we cannot just set a real-world agent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', a robot) to an ar- bitrary initial state with a click of a button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Resetting in the real world is usually quite expensive and requires con- stant human monitoring and intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Consider again the example of a robot learning to stack blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Normally, a person would oversee the entire learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' During the process, they would manually reset the robot to a mean- ingful starting state before it enters an unrecoverable state where the problem can no longer be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Sometimes au- tomatic resetting can be implemented by cleverly engineer- ing the physical learning environment (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021), but it is not always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' An approach we can take to make real-world RL more cost-efficient is through reset-free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The goal of reset- free RL is to have an agent learn how to perform well while minimizing the amount of external resets required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Some examples of problems that have been approached in reset-free RL include agents learning dexterity skills, such as picking up an item or inserting a pipe, and learning how to walk (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' While there has been numerous works proposing reset-free RL algorithms using approaches such as multi-task learning (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020), learning a reset pol- Provable Reset-free Reinforcement Learning by No-Regret Reduction icy (Eysenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022), and skill- space planning (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020), to our knowledge, there has not been any work with provable guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In this work, we take the first step to provide a provably correct framework to design reset-free RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Our framework is based on the idea of a no-regret reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' First, we reduce the reset-free RL problem to a sequence of safe RL problems with an adaptive initial state sequence, where each safe RL problem is modeled as a constrained Markov decision process (CMDP) with the states requiring resets marked as unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we derive our main no-regret reduction, which further turns this sequence into a two- player game between a primal player (updating the Marko- vian policy) and a dual player (updating the Lagrange mul- tiplier function of the constrained MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Interestingly, we show that such a reduction can be constructed without using the typical Slater’s condition for strong duality and despite the fact that CMDPs with different initial states in general do not share a common Markovian optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We show that if no regret is achieved in this game, then the regret of the original RL problem and the total num- ber of required resets are both provably sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This means that the agent eventually learns to perform optimally and avoids resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Using this reduction, we design a reset- free RL algorithm instantiation under the linear MDP as- sumption, using Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022) as the baseline algo- rithm for the primal player and projected gradient descent for the dual player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We prove that our algorithm achieves ˜O( √ d3H4K) regret and ˜O( √ d3H4K) resets with high probability, where d is the feature dimension, H is the length of an episode, and K is the total number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 2 RELATED WORK Reset-free RL is a relatively new concept in the literature, and the work thus far, to our knowledge, has been limited to non-provable approaches with empirical verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' One such approach is by learning a reset policy in addition to the main policy (Eysenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The idea is to learn a policy that will bring the agent back to a safe initial state if they encounter a reset state concur- rently with a policy that maximizes reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A reset state is a state in which human intervention normally would have been required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This approach prevents the need for man- ual resets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' however, there is usually some required assump- tions on knowledge of the reset policy reward function and therefore knowledge of the reset states (Eysenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022) avoid this assumption but as- sume given demonstrations on how to accomplish the goal and a fixed initial state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Another popular approach is using multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This is similar to learning a reset policy, but can be thought of as a way to increase the amount of possible actions an agent can take to perform a reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The objective is to learn a number of tasks so that a combination of them can achieve the main goal, and in addition, some tasks can perform nat- ural resets for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' One problem that was tackled by Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2021) was that of inserting a light bulb into a lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The tasks their agent learns is recentering, insert- ing, lifting, and flipping the bulb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Here, if the bulb starts on the ground, the agent can recenter the bulb, lift it, flip it over (if needed), and finally insert it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In addition, many of the tasks perform resets for the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For example, if the agent drops the bulb while lifting it, it can recenter the bulb and then try lifting it again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This approach breaks down the reset process and (possibly) makes it easier to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' How- ever, this approach often requires the order in which tasks should be learned to be provided manually (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A related problem is infinite-horizon non-episodic RL with provable guarantees (see Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2020, 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2019) and the references within) as this prob- lem is also motivated by not using resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In this setting, there is only one episode that goes on indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The ob- jective is to maximize the cumulative reward, and progress is usually measured in terms of regret with the compara- tor being an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' However, compared with the reset-free RL setting we study here, extra assumptions, such as the absence or knowledge of absorbing states, are usually required to achieve sublinear regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In addition, the objective does not necessarily lead to a minimization of resets as the agent can leverage reset transitions to max- imize rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Learning in infinite-horizon CMDPs has been studied (Zheng and Ratliff, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022), but to our knowledge, all such works make strong assump- tions such as a fixed initial state distribution or known dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In this paper, we focus on an episodic setting of reset-free RL (see Section 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' a non-episodic formulation of reset-free RL could be an interesting one for further re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To our knowledge, we propose the first provable reset- free RL technique in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By borrowing ideas from literature on the much more extensively studied area of safe RL, we propose to associate states requiring re- sets with the concept of unsafe states in safe RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Safe reinforcement learning involves solving the standard RL problem while adhering to some safety constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' There has been a lot of work in safe RL, with approaches such as utilizing a baseline safe (but not optimal) policy (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Garcia Polo and Fernandez Rebollo, 2011), pessimism (Amani and Yang, 2022), and shielding (Alshiekh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Wagener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' These works have had promising empirical results but usually require extra assumptions such as a given baseline policy or knowl- edge of unsafe states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' There are also provable safe RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To our knowledge, all involve framing safe RL as a CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Here, the safety constraints are modeled as a cost, and Hoai-An Nguyen, Ching-An Cheng the overall goal is to maximize performance while keep- ing the cost below a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The provable guaran- tees are commonly either sublinear regret and constraint violations, or sublinear regret with zero constraint vi- olation (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' HasanzadeZonuzy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Wachi and Sui, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Efroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' However, most works (including all the aforementioned ones), consider the episodic case where the initial state distribution of each episode is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This prevents a very natural extension to reset-free learning as human intervention would be re- quired to reset the environment at the end of each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Works that allow for arbitrary initial state require fairly strong assumptions, such as knowledge (and the existence) of safe actions from each state (Amani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In our work, we utilize techniques from provable safe RL for reset-free RL, but weaken the typical assumptions to allow for arbitrary initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This relaxation is neces- sary for the reset-free RL problem and also allows for eas- ier extensions to both lifelong and multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We achieve this relaxation with a key observation that identifies a shared Markovian-policy saddle-point across CMDPs of perfectly safe RL with different initial states (that is, the constraint in the CMDP imposes perfect safety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This ob- servation is new to our knowledge, and it is derived from the particular structure of perfectly safe RL, which is a sub- problem used in our reset-free RL reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We note that general CMDPs with different initial states do not generally admit shared Markovian-policy saddle-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, on the technical side, our algorithm can also be viewed as the first safe RL algorithm that allows for arbitrary initial state sequences without strong assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' While we propose a generic reduction technique to de- sign reset-free RL algorithms, our regret and constraint violation bounds are still comparable to the above works when specialized to their setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Under the linear MDP assumption, our algorithm achieves ˜O( √ d3H4K) regret and violation (equivalently, the number of resets in reset- free RL), which is asymptotically equivalent to Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022) and comparable to the bounds of ˜O √ d2H6K from Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2021) for a fixed initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 3 PRELIMINARY We consider episodic reset-free RL: in each episode, the agent aims to optimize for a fixed-horizon return starting from the last state of the previous episode or some state that the agent was reset to in the previous episode if reset occurs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', due to the robot falling over).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Problem Setup and Notation Formally, we can de- fine episodic reset-free RL as a Markov decision process (MDP), (S, A, P, r, H), where S is the state space, A is the action space, P = {Ph}H h=1 is the transition dynamics, r = {rh}H h=1 is the reward function, and H is the task hori- zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We assume P and r are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We allow S to be large or continuous but assume A is relatively small so that maxa∈A can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We designate the set of reset states as Sreset ⊆ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' we do not assume that the agent has knowledge of Sreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We also do not assume that there is a reset-free action for each state, as opposed to (Amani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, the agent needs to plan for the long-term to avoid resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We assume rh : S × A → [0, 1], and for simplicity, we assume rh is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' However, we note that it would be easy to extend this to the setting where rewards are stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The agent interacts with the environment for K total episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Following the convention of episodic problems, we suppose the state space S is layered, and a state sτ ∈ S at time τ is factored into two components sτ = (¯s, τ) where ¯s denotes the time-invariant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Reset happens at time τ if ¯s ∈ Sreset (which we also write as sτ ∈ Sreset), and the initial state of the next episode will be s1 = (¯s′, 1) where ¯s′ is sampled from a fixed but unknown state distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Otherwise, the initial state of the next episode is the last state of the current episode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', for episode k + 1, sk+1 1 = (¯s, 1) if sk H = (¯s, H) in episode k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 We denote the set of Markovian policies as ∆, and a policy π ∈ ∆ as π = {πh(ah|sh)}H h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We define the state value function and the state-action value function under π as2 V π r,h(s) := Eπ � min(H,τ) � t=h rt(st, at)|sh = s � (1) Qπ r,h(s, a) := rh(s, a) + E � V π r,h+1(sh+1)|sh = s, ah = a � , where h ≤ τ, and we recall τ is the time step when the agent enters Sreset (if at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Objective The overall goal is for the agent to learn a Markovian policy to maximize its cumulative reward while avoiding resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, our performance measures are 1We can extend this setup to reset-free multi-task or lifelong RL problems that are modeled as contextual MDPs since our algo- rithm can work with any initial state sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In this case, we can treat each state here as sτ = (¯s, c, τ), where c denotes the context that stays constant within an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' If no reset happens, the ini- tial state of episode k + 1 can be written as sk+1 1 = (¯s, ck+1, 1) if sk H = (¯s, ck, H) in episode k, where the new context ck+1 can follow any distribution and may depend on the current context ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 2This value function definition is the same as the H-step cu- mulative reward in an MDP formulation where we place the agent into a fictitious zero-reward absorbing state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', a mega-state ab- stracting Sreset) after the agent enters Sreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We choose the cur- rent formulation to make the definition of resets more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Provable Reset-free Reinforcement Learning by No-Regret Reduction as follows (we seek to minimize both quantities): Regret(K) = max π∈∆0(K) K � k=1 V π r,1(sk 1) − V πk r,1 (sk 1) (2) Resets(K) = K � k=1 Eπk \uf8ee \uf8f0 min(H,τ) � h=1 1[sh ∈ Sreset] ���s1 = sk 1 \uf8f9 \uf8fb (3) where ∆0(K) ⊆ ∆ is the set of Markovian policies that avoid resets for all episodes, and πk is the policy used by the agent in episode k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Note that by the reset mechanism �min(H,τ) h=1 1[sh ∈ Sreset] ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Notice that the initial states in our regret and reset measures are determined by the learner, not the optimal policy like in some classical definitions of regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Given the motivation behind reset-free RL (see Section 1), we can expect that the initial states here are by construction meaningful for perfor- mance comparison;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' otherwise, a reset would have occurred to set the learner to a meaningful state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A following impli- cation is that all bad absorbing states are in Sreset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' hence, the agent cannot use the trivial solution of hiding in a bad absorbing state to achieve small regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To make the problem feasible, we assume that achieving no resets is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We state this formally in the assumption below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any sequence {sk 1}K k=1, the set ∆0(K) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' That is, there is a Markovian policy π ∈ ∆ such that Eπ[�H h=1 1[sh ∈ Sreset]|s1 = sk 1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This is a reasonable assumption in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' If reset hap- pens, the agent should be set to a state that the agent can continue to operate in without reset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' if the agent is at a state where no such reset-free policy exists, reset should happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This assumption is similar to the assumption on the exis- tence of a perfectly safe policy in safe RL literature, which is a common and relatively weak assumption (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' If there were to be initial states that inevitably lead to a reset, the problem would be infea- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 4 A NO-REGRET REDUCTION FOR RESET-FREE RL We present our main reduction of reset-free RL to regret minimization in a two-player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In the following, we first show that reset-free RL can be framed as a sequence of CMDPs of safe RL problems with an adaptive initial state sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we design a two-player game based on a primal-dual analysis of this sequence of CMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Finally, we show achieving sublinear regret in this two-player game implies sublinear regret and resets in the original reset-free RL problem in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The complete proofs for this section can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 Reset-free RL as a Sequence of CMDPs The first step of our reduction is to cast the reset-free RL problem in Section 3 to a sequence of CMDP problems which share the same rewards, constraints, and dynamics, but have different initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Each problem instance in this sequence corresponds to an episode of the reset-free RL problem, and its constraint describes the probability of the agent entering a state that requires reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Specifically, we denote these constrained MDPs3 as {(S, A, P, r, H, c, sk 1)}K k=1: in episode k, the CMDP problem is defined as max π∈∆ V π r,1(sk 1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' V π c,1(sk 1) ≤ 0 (4) where we define the cost as ch(s, a) := 1[s ∈ Sreset] and V π c,1, defined similarly to (1), is the state value func- tion with respect to the cost c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We note that the initial state, sk 1, depends on the past behaviors of the agent, and Assumption 1 ensures each CMDP in (4) is a feasible prob- lem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', there is a Markovian policy satisfying the con- straint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We can interpret each CMDP in (4) as a safe RL problem by treating Sreset as the unsafe states that a safe RL agent should avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' From this perspective, the con- straint in (4) can be viewed as the probability of a trajectory entering an unsafe state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since CMDPs are typically defined without early episode termination unlike (1), with abuse of notation, we extend the definitions of P, S, r, c as follows so that the CMDP definition above is consistent with the common literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We introduce a fictitious absorbing state denoted as s† in S, where rh(s†, a) = 0 and ch(s†, a) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' once the agent enters s†, it stays there until the end of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We extend the definition P such that, after the agent is in a state s ∈ Sreset, any action it takes brings it to s† in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In this way, we can write the value function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' for reward, as V π r,h(s) = Eπ � �H t=h rt(st, at)|sh = s � in terms of this extended dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We note that these two formulations are mathematically the same for the pur- pose of learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' when the agent enters s†, it means that the agent is reset in the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By the construction above, we can write Resets(K) = K � k=1 V πk c,1 (sk 1) which is the same as the number of total constraint vio- lations across the CMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Because we do not make any 3In general, the solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', optimal policy) to a CMDP depends on its initial state, unlike in MDPs (see remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2 in Altman (1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Hoai-An Nguyen, Ching-An Cheng assumptions about the agent’s knowledge of the constraint function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', the agent does not know states ∈ Sreset), we allow the agent to reset during learning while minimizing the total number of resets over all K episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2 Reduction to Two-Player Game From the problem formulation above, we see that the ma- jor difficulty of reset-free RL is the coupling between an adaptive initial state sequence and the constraint on reset probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' If we were to remove either of them, we can use standard algorithms, since the problem will become a single CMDP problem (Altman, 1999) or an episodic RL problem with varying initial states (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We propose a reduction to systematically design algorithms for this sequence of CMDPs and therefore for reset-free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The main idea is to approximately solve the saddle point problem of each CMDP in (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', max π∈∆ min λ≥0 V π r,1(sk 1) − λV π c,1(sk 1) (5) where λ denotes the dual variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' the La- grange multiplier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Each CMDP can be framed as a linear program (Hern´andez-Lerma and Lasserre, 2002) whose primal and dual optimal values match (see section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 in Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, for each CMDP, maxπ∈∆ minλ≥0 V π r,1(sk 1) − λV π c,1(sk 1) = minλ≥0 maxπ∈∆ V π r,1(sk 1) − λV π c,1(sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' While using a primal-dual algorithm to solve for the sad- dle point of a single CMDP is straightforward and known, using this approach for a sequence of CMDPs is not obvi- ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Each CMDP’s optimal policy and Lagrange multiplier can be a function of the initial state (Altman, 1999), and in general, a common saddle point of Markovian polices and Lagrange multipliers does not necessarily exist for a sequence of CMDPs with varying initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='4 As a re- sult, it is unclear if there exists a primal-dual algorithm to solve this sequence, especially given that the initial states here are adaptively chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Existence of a Shared Saddle-Point Fortunately, there is a shared saddle-point with a Markovian policy across all the CMDPs considered here due to the special structure of reset-free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' It is a proof that does not use Slater’s condi- tion for strong duality, unlike similar literature, but attains the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Instead we use Assumption 1 and the fact that the cost c is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We formalize this be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' There exist a function ˆλ(·) where for each s, ˆλ(s) ∈ arg min y≥0 � max π∈∆ V π r,1(s) − yV π c,1(s) � , 4A shared saddle-point with a non-Markovian policy always exists on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a saddle-point to the CMDPs max π∈∆ V π r,1(s1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' V π c,1(s1) ≤ 0 for all initial states s1 ∈ S such that the CMDP is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S, V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) ≥ V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1) ≥ V π r,1(s1) − ˆλ(s1)V π c,1(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (6) Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For π∗ in Theorem 1, it holds that Regret(K) = �K k=1 V π∗ r,1 (sk 1) − V πk r,1 (sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We prove for ease of construction that the pair (π∗, λ∗) where λ∗(·) = ˆλ(·) + 1 is also a saddle-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any saddle-point to the CMDPs max π∈∆ V π r,1(s1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' V π c,1(s1) ≤ 0 of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also a saddle-point as defined in eq (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, the pair (π∗, λ∗) in Corollary 2 is a saddle-point to all the CMDPs the agent faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This makes potentially designing a two-player game reduction possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In the next section, we give the details of our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Two-Player Game Our two-player game proceeds itera- tively in the following manner: in episode k, a dual player determines a state value function λk : S → R, and a primal player determines a policy πk which can depend on λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then the primal and dual player receive losses Lk(πk, λ) and −Lk(π, λk), respectively, where Lk(π, λ) is a Lagrangian function defined as Lk(π, λ) := V π r,1(sk 1) − λ(sk 1)V π c,1(sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (7) The regret of these two players are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Let πc and λc be comparators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The regret of the primal and the dual players are Rp({πk}K k=1, πc) := K � k=1 Lk(πc, λk) − Lk(πk, λk) (8) Rd({λk}K k=1, λc) := K � k=1 Lk(πk, λk) − Lk(πk, λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (9) We present our main reduction theorem for reset-free RL below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By Theorem 2, if both players have sublinear re- gret in the two-player game, then the resulting policy se- quence will have sublinear performance regret and a sub- linear number of resets in the original RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since there are many standard techniques (Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2016) Provable Reset-free Reinforcement Learning by No-Regret Reduction from online learning to solve such a two-player game, we can leverage them to systematically design reset-free RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In the next section, we will give an example algorithm of this reduction for linear MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Under Assumption 1, for any sequences {πk}K k=1 and {λk}K k=1 , it holds that Regret(K) ≤ Rp({πk}K k=1, π∗) + Rd({λk}K k=1, 0) Resets(K) ≤ Rp({πk}K k=1, π∗) + Rd({λk}K k=1, λ∗) where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof Sketch of Theorem 1 Let Q∗ c(s, a) = minπ∈∆ Qπ c (s, a) and V ∗ c (s) = minπ∈∆ V π c (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We define π∗ in Theorem 1 as the optimal policy to the following MDP: (S, A, P, r, H), where we define a state-dependent action space A as As = {a ∈ A : Q∗ c(s, a) ≤ V ∗ c (s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By definition, As is non-empty for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We also define a shorthand notation: we write π ∈ A(s) if Eπ[�H t=1 1{at /∈ Ast}|s1 = s] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We have the follow- ing lemma, which is an application of the performance dif- ference lemma (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 in (Kakade and Langford, 2002) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 in (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any s1 ∈ S such that V ∗ c (s1) = 0 and any π ∈ ∆, it is true that π ∈ A(s1) if and only if V π c (s1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We prove our main claim, (6), below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Because V π∗ c,1 (s1) = 0, the first inequality is trivial: V π∗ r,1 (s1)−λ(s1)V π∗ c,1 (s1) = V π∗ r,1 (s1) = V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To prove the second inequality, we use Lemma 1: V π r,1(s1) − ˆλ(s1)V π c,1(s1) ≤ max π∈∆ V π r,1(s1) − ˆλ(s1)V π c,1(s1) = min y≥0 max π∈∆ V π r,1(s1) − yV π c,1(s1) = max π∈Ac(s1) V π r,1(s1) (By Lemma 1 ) =V π∗ r,1 (s1) = V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof Sketch of Theorem 2 We first establish the fol- lowing intermediate result that will help us with our de- composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1(Lk(π∗, λ′) − Lk(πk, λk)) ≤ Rp({π}K k=1, π∗), where (π∗, λ′) is the saddle-point defined in either Theorem 1 or Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we upper bound Regret(K) and Resets(K) by Rp({πk}K k=1, πc) and Rd({λk}K k=1, λc) for suitable com- parators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This decomposition is inspired by the techniques used in Ho-Nguyen and Kılınc¸-Karzan (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound Resets(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1 V πk c,1 (sk 1) ≤ Rp({π}K k=1, π∗) + Rd({λ}K k=1, λ∗), where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Notice �K k=1 V πk c,1 (sk 1) = �K k=1 Lk(πk, ˆλ) − Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By (6), and adding and subtracting �K k=1 Lk(πk, λk), we can bound this difference by K � k=1 Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Using Lemma 2 and Definition 1 to upper bound the above, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lastly, we bound Regret(K) with the lemma below and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1(V π∗ r,1 (sk 1) − V πk r,1 (sk 1)) ≤ Rp({π}K k=1, π∗) + Rd({λ}K k=1, 0), where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Note that L(π∗, λ∗) = L(π∗, 0) since V π∗ c,1 = 0 for all k ∈ [K] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since by definition, for any π, Lk(π, 0) = V π r,1(sk 1), we have the following: K � k=1 V π∗ r,1 (sk 1) − V πk r,1 (sk 1) = K � k=1 Lk(π∗, λ∗) − Lk(πk, 0) = K � k=1 Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0) ≤Rp({π}K k=1, π∗) + Rd({λ}K k=1, 0) where the last inequality follows from Lemma 2 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 5 RESET-FREE LEARNING FOR LINEAR MDP To demonstrate the utility of our reduction, we design a provably correct algorithm instantiation for reset-free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We consider a linear MDP setting, which is common in the RL theory literature (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We assume (S, A, P, r, c, H) is linear with a known feature map φ : S × A → Rd: for any h ∈ [H], there exists d unknown signed measures µh = {µ1 h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', µd h} over S such that for any (s, a, s′) ∈ S × A × S, we have Ph(s′|a) = ⟨φ(s, a), µh(s′)⟩, Hoai-An Nguyen, Ching-An Cheng and there exists unknown vectors ωr,h, ωc,h ∈ Rd such that for any (s, a) ∈ S × A, rh(s, a) = ⟨φ(s, a), ωr,h⟩ ch(s, a) = ⟨φ(s, a), ωc,h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We assume, for all (s, a, h) ∈ S×A×[H], ||φ(s, a)||2 ≤ 1, and max{||µh(s)||2, ||ωr,h||2, ||ωc,h||2} ≤ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In addition, we make a linearity assumption on the function λ∗ defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We assume the knowledge of a feature ξ : S → Rd such that ∀s ∈ S, ||ξ(s)||2 ≤ 1 and λ∗(s) = ⟨ξ(s), θ∗⟩, for some unknown vector θ∗ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In addition, we assume the knowledge of a convex set5 U ⊆ Rd such that θ∗, 0 ∈ U and ∀θ ∈ U, ||θ||2 ≤ B and ⟨ξ(s), θ⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 Algorithm The basis of our algorithm lies between the interaction be- tween the primal and dual players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We let the dual player perform projected gradient descent and the primal player update policies based on upper confidence bound with the knowledge of the decision of the dual player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This sequen- tial strategy resembles the optimistic style update in online learning (Mertikopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Specifically, in each episode, upon receiving the initial state, we execute actions according to the policy based on a softmax (lines 5-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then, we perform the dual update through projected gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The dual player plays for the next round, k + 1, after observing its loss after the primal player plays for the current round, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' The projec- tion is to a l2 ball containing λ∗(·) (lines 9-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Finally, we perform the update of the primal player by computing the Q-functions for both the reward and cost with a bonus to encourage exploration (lines 12-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This algorithm builds upon Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' However, notably, we extend it to handle the adaptive initial state se- quence seen in reset-free RL by Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2 Analysis We show below that our algorithm achieves regret and number of resets that are sublinear in the total number of time steps, KH, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This result is asymptot- ically equivalent to Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022) and comparable to the bounds of ˜O √ d2H6K from Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 5Such a set can be constructed by upper bounding the values using scaling and ensuring non-negativity using a sum of squares approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 6From the previous section, we can see that the optimal func- tion for the dual player is not necessarily unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' So, we as- sume bounds on at least one optimal function that we designate as λ∗(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Under Assumptions 1, 2, and 3, with high probability, Regret(K) ≤ ˜O((B + 1) √ d3H4K) and Resets(K) ≤ ˜O((B + 1) √ d3H4K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof Sketch of Theorem 3 We provide a proof sketch here and defer the complete proof to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound the regret of {πk}K k=1 and {λk}K k=1, and then use this to prove the bounds on our algorithm’s regret and number of resets with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound the regret of {λk}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then it holds that Rd({λk}K k=1, λc) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + �K k=1(λk(sk 1) − λc(sk 1))(V k c,1(sk 1) − V πk c,1 (sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We notice first an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Rd({λk}K k=1, λc) = K � k=1 Lk(πk, λk) − Lk(πk, λc) = K � k=1 λc(sk 1)V πk c,1 (sk 1) − λk(sk 1)V πk c,1 (sk 1) = K � k=1 (λk(sk 1) − λc(sk 1))(−V k c,1(sk 1)) + K � k=1 (λk(sk 1) − λc(sk 1))(V k c,1(sk 1) − V πk c,1 (sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We observe that the first term is an online linear problem for θk (the parameter of λk(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In episode k ∈ [K], λk is played, and then the loss is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since the space of θk is convex, we use standard results (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2016)) to show that updating θk through projected gradient descent results in an upper bound for �K k=1(λk(sk 1) − λc(sk 1))(−V k c,1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We now bound the regret of {π}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Consider any πc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' With high probability, Rp({π}K k=1, πc) ≤ 2H(1 + B + H) + �K k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λk(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' First we expand the regret into two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Rp({π}K k=1, πc) = K � k=1 Lk(πc, λk) − Lk(πk, λk) = K � k=1 V πc r,1 (sk 1) − λk(sk 1)V πc c,1(sk 1) − [V πk r,1 (sk 1) − λk(sk 1)V πk c,1 (sk 1)] = K � k=1 V πc r,1 (sk 1) − λk(sk 1)V πc c,1(sk 1) − [V k r,1(sk 1) − λk(sk 1)V k c,1(sk 1)] + K � k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λk(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Provable Reset-free Reinforcement Learning by No-Regret Reduction Algorithm 1 Primal-Dual Reset-Free RL Algorithm for Linear MDP with Adaptive Initial States 1: Input: Feature maps φ and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Failure probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Some universal constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 2: Initialization: θ1 = 0, wr,h = 0, wc,h = 0, α = log(|A|)K 2(1 + B + H), β = cdH � log(4 log |A|dKH/p) 3: for episodes k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='K do 4: Observe the initial state sk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 5: for step h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', H do 6: Compute πh,k(a|·) ← exp(α(Qk r,h(·, a) − λk(sk 1)Qk c,h(·, a))) � a exp(α(Qk r,h(·, a) − λk(sk 1)Qk c,h(·, a))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 7: Take action ak h ∼ πh,k(·|sk h) and observe sk h+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 8: end for 9: ηk ← B √ k 10: Update θk+1 ← ProjU(θk + ηk · ξ(sk 1)V k c,1(sk 1)) 11: λk+1(·) ← ⟨θk+1, ξ(·)⟩ 12: for step h = H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 1 do 13: Λk+1 h ← k� i=1 φ(si h, ai h)φ(si h, ai h)T + λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 14: wk+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h ← (Λk+1 h )−1[ k� i=1 φ(si h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ai h)[rh(si h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ai h) + V k+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h+1(si h+1)]] 15: wk+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h ← (Λk+1 h )−1[ k� i=1 φ(si h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ai h)[ch(si h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ai h) + V k+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h+1(si h+1)]] 16: Qk+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·) ← max{min{⟨wk+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·)⟩ + β(φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·)T (Λk+1 h )−1φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·))1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' H − h + 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 0} 17: Qk+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·) ← max{min{⟨wk+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·)⟩ − β(φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·)T (Λk+1 h )−1φ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' ·))1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 0} 18: V k+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·) = � a πh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='k(a|·)Qk+1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' a) 19: V k+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·) = � a πh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='k(a|·)Qk+1 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='h (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' a) 20: end for 21: end for To bound the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' we use Lemma 3 from Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022), which characterizes the property of upper confi- dence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lastly, we derive a bound on Rd({λk}K k=1, λc) + Rp({πk}K k=1, πc), which directly implies our final up- per bound on Regret(K) and Resets(K) in Theorem 3 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Combining the upper bounds in Lemma 5 and Lemma 6, we have a high-probability upper bound Rd({λk}K k=1, λc) + Rp({πk}K k=1, πc) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + 2H(1 + B + H)+ + K � k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λc(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)) where the last term is the overestimation error due to opti- mism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Note that for all k ∈ [K], V k r,1(sk 1) and V k c,1(sk 1) are as defined in Algorithm 1 and are optimistic estimates of V π∗ r,1 (sk 1) and V π∗ c,1 (sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To bound this term, we use Lemma 4 from (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 6 CONCLUSION We propose a generic no-regret reduction for designing provable reset-free RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Our reduction casts reset-free RL into the regret minimization problem of a two-player game, for which many existing no-regret al- gorithms are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' As a result, we can reuse these techniques to systematically build new reset-free RL algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In particular, we design a reset-free RL algorithm for linear MDPs using our new reduction techniques, taking the first step towards designing provable reset-free RL al- gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Extending these techniques to nonlinear function approximators and verifying their effectiveness empirically are important future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Acknowledgements Part 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', and Zeilinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', editors, Proceedings of the 2nd Conference on Learning for Dynamics and Control, vol- ume 120 of Proceedings of Machine Learning Research, pages 620–629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Hoai-An Nguyen, Ching-An Cheng A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 Missing Proofs for Section 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 Proof of Theorem 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' There exist a function ˆλ(·) where for each s, ˆλ(s) ∈ arg min y≥0 � max π∈∆ V π r,1(s) − yV π c,1(s) � , and a Markovian policy π∗ ∈ ∆, such that (π∗, ˆλ) is a saddle-point to the CMDPs max π∈∆ V π r,1(s1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' V π c,1(s1) ≤ 0 for all initial states s1 ∈ S such that the CMDP is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' That is, for all π ∈ ∆, λ : S → R, and s1 ∈ S, V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) ≥ V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1) ≥ V π r,1(s1) − ˆλ(s1)V π c,1(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (6) For policy π∗, we define it by the following construction (we ignore writing out the time dependency for simplicity): first, we define a cost-based MDP Mc = (S, A, P, c, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Let Q∗ c(s, a) = minπ∈∆ Qπ c (s, a) and V ∗ c (s) = minπ∈∆ V π c (s) be the optimal values, where we recall V π c and Qπ c are the state and state-action values under policy π with respect to the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Now we construct another reward-based MDP M = (S, A, P, r, H), where we define the state-dependent action space A as As = {a ∈ A : Q∗ c(s, a) ≤ V ∗ c (s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By definition, As is non-empty for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We define a shorthand notation: we write π ∈ A(s) if Eπ[�H t=1 1{at /∈ Ast}|s1 = s] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we have the following lemma, which is a straightforward application of the performance difference lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any s1 ∈ S such that V ∗ c (s1) = 0 and any π ∈ ∆, it is true that π ∈ A(s1) if and only if V π c (s1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By performance difference lemma (Kakade and Langford, 2002), we can write V π c (s1) − V ∗ c (s1) = Eπ � H � t=1 Q∗ c(st, at) − V ∗ c (st)|s1 = s1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' If for some s1 ∈ S, π ∈ A(s1), then Eπ ��H t=1 Q∗ c(st, at) − V ∗ c (st) � ≤ 0, which implies V π c (s1) ≤ V ∗ c (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' But since V ∗ c is optimal, V π c (s1) = V ∗ c (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' On the other hand, suppose V π c (s1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' It implies Eπ ��H t=1 Q∗ c(st, at) − V ∗ c (st) � = 0 since V ∗ c (s1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Because by definition of optimality Q∗ c(st, at) − V ∗ c (st) ≥ 0, this implies π ∈ A(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We set our candidate policy π∗ as the optimal policy of this M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By Lemma 1, we have V π∗ c (s) = V ∗ c (s), so it is also an optimal policy to Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We prove our main claim of Theorem 1 below: V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) ≥ V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1) ≥ V π r,1(s1) − ˆλ(s1)V π c,1(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Because V π∗ c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial: V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) = V π∗ r,1 (s1) = V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Provable Reset-free Reinforcement Learning by No-Regret Reduction For the second inequality, we use Lemma 1: V π r,1(s1) − ˆλ(s1)V π c,1(s1) ≤ max π∈∆ V π r,1(s1) − ˆλ(s1)V π c,1(s1) = min y≥0 max π∈∆ V π r,1(s1) − yV π c,1(s1) = max π∈Ac(s1) V π r,1(s1) (By Lemma 1 ) = V π∗ r,1 (s1) = V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2 Proof of Corollary 1 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For π∗ in Theorem 1, it holds that Regret(K) = �K k=1 V π∗ r,1 (sk 1) − V πk r,1 (sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To prove Regret(K) = �K k=1 V π∗ r,1 (sk 1) − V πk r,1 (sk 1), it suffices to prove �K k=1 V π∗ r,1 (sk 1) = maxπ∈∆0(K) �K k=1 V π r,1(sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By Lemma 1 and under Assumption 1, we notice that maxπ∈∆0(K) �K k=1 V π r,1(sk 1) = maxπ∈A(sk 1 ),∀k∈[K] �K k=1 V π r,1(sk 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This is equal to �K k=1 V π∗ r,1 (sk 1) by the definition of π∗ in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='3 Proof of Corollary 2 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any saddle-point to the CMDPs max π∈∆ V π r,1(s1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' V π c,1(s1) ≤ 0 of (π∗, ˆλ) from Theorem 1, (π∗, ˆλ + 1) =: (π∗, λ∗) is also a saddle-point as defined in eq (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We prove that eq (6) holds for (π∗, λ∗), that is V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) ≥ V π∗ r,1 (s1) − λ∗(s1)V π∗ c,1 (s1) ≥ V π r,1(s1) − λ∗(s1)V π c,1(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Because V π∗ c,1 (s1) = 0 (for an initial state s1 such that the CMDP is feasible), the first inequality is trivial: V π∗ r,1 (s1) − λ(s1)V π∗ c,1 (s1) = V π∗ r,1 (s1) = V π∗ r,1 (s1) − λ∗(s1)V π∗ c,1 (s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For the second inequality, we use Theorem 1: V π r,1(s1) − λ∗(s1)V π c,1(s1) ≤V π r,1(s1) − ˆλ(s1)V π c,1(s1) ≤V π∗ r,1 (s1) − ˆλ(s1)V π∗ c,1 (s1) =V π∗ r,1 (s1) − λ∗(s1)V π∗ c,1 (s1) where the first step is because V π c,1(s1) by definition is in [0, 1] and λ∗ = ˆλ + 1, and the second step is by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='4 Proof of Theorem 2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Under Assumption 1, for any sequences {πk}K k=1 and {λk}K k=1 , it holds that Regret(K) ≤ Rp({πk}K k=1, π∗) + Rd({λk}K k=1, 0) Resets(K) ≤ Rp({πk}K k=1, π∗) + Rd({λk}K k=1, λ∗) where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first establish the following intermediate result that will help us with our decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Hoai-An Nguyen, Ching-An Cheng Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1(Lk(π∗, λ′) − Lk(πk, λk)) ≤ Rp({π}K k=1, π∗), where (π∗, λ′) is the saddle-point defined in either Theorem 1 or Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We derive this lemma by Theorem 1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' First notice by Theorem 1 and Corollary 2 that for λ′ = λ∗, ˆλ, K � k=1 Lk(π∗, λ′) = K � k=1 V π∗ r,1 (sk 1) − λ′(sk 1)V π∗ c,1 (sk 1) ≤ K � k=1 V π∗ r,1 (sk 1) − λk(sk 1)V π∗ c,1 (sk 1) = K � k=1 Lk(π∗, λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we can derive K � k=1 (Lk(π∗, λ′) − Lk(πk, λk)) = K � k=1 Lk(π∗, λ′) − Lk(π∗, λk) + Lk(π∗, λk) − Lk(πk, λk) ≤ K � k=1 Lk(π∗, λk) − Lk(πk, λk) = Rp({π}K k=1, π∗) which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then we upper bound Regret(K) and Resets(K) by Rp({πk}K k=1, πc) and Rd({λk}K k=1, λc) for suitable comparators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This decomposition is inspired by the techniques used in Ho-Nguyen and Kılınc¸-Karzan (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound Resets(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1 V πk c,1 (sk 1) ≤ Rp({π}K k=1, π∗) + Rd({λ}K k=1, λ∗), where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Notice �K k=1 V πk c,1 (sk 1) = �K k=1 Lk(πk, ˆλ) − Lk(πk, λ∗) where (π∗, ˆλ) is the saddle-point defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' This is because, as defined, λ∗ = ˆλ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, we bound the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We have K � k=1 Lk(πk, ˆλ) − Lk(πk, λ∗) = K � k=1 Lk(πk, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗) ≤ K � k=1 Lk(π∗, ˆλ) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, λ∗) ≤Rp({π}K k=1, π∗) + Rd({λ}K k=1, λ∗) where second inequality is because �K k=1 Lk(π∗, ˆλ) ≥ �K k=1 Lk(πk, ˆλ) by Theorem 1, and the first inequality follows from Lemma 2 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lastly, we bound Regret(K) with the lemma below and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any primal-dual sequence {πk, λk}K k=1, �K k=1(V π∗ r,1 (sk 1)−V πk r,1 (sk 1)) ≤ Rp({π}K k=1, π∗)+Rd({λ}K k=1, 0), where (π∗, λ∗) is the saddle-point defined in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Note that L(π∗, λ∗) = L(π∗, 0) since V π∗ c,1 (sk 1) = 0 for all k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since by definition, for any π, Lk(π, 0) = V π r,1(sk 1), we have the following: K � k=1 V π∗ r,1 (sk 1) − V πk r,1 (sk 1) = K � k=1 Lk(π∗, λ∗) − Lk(πk, 0) = K � k=1 Lk(π∗, λ∗) − Lk(πk, λk) + Lk(πk, λk) − Lk(πk, 0) ≤Rp({π}K k=1, π∗) + Rd({λ}K k=1, 0) where the last inequality follows from Lemma 2 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Provable Reset-free Reinforcement Learning by No-Regret Reduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2 Missing Proofs for Section 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 Proof of Theorem 3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Under Assumptions 1, 2, and 3, with high probability, Regret(K) ≤ ˜O((B + 1) √ d3H4K) and Resets(K) ≤ ˜O((B + 1) √ d3H4K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound the regret of {πk}K k=1 and {λk}K k=1, and then use this to prove the bounds on our algorithm’s regret and number of resets with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We first bound the regret of {λk}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Consider λc(s) = ⟨ξ(s), θc⟩ for some θc ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Then it holds that Rd({λk}K k=1, λc) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + �K k=1(λk(sk 1) − λc(sk 1))(V k c,1(sk 1) − V πk c,1 (sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We notice first an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Rd({λk}K k=1, λc) = K � k=1 Lk(πk, λk) − Lk(πk, λc) = K � k=1 λc(sk 1)V πk c,1 (sk 1) − λk(sk 1)V πk c,1 (sk 1) = K � k=1 λc(sk 1)V πk c,1 (sk 1) − λk(sk 1)V πk c,1 (sk 1) + K � k=1 λc(sk 1)V k c,1(sk 1) − λc(sk 1)V k c,1(sk 1) + λk(sk 1)V k c,1(sk 1) − λk(sk 1)V k c,1(sk 1) = K � k=1 (λk(sk 1) − λc(sk 1))(−V k c,1(sk 1)) + K � k=1 (λk(sk 1) − λc(sk 1))(V k c,1(sk 1) − V πk c,1 (sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We observe that the first term is an online linear problem for θk (the parameter of λk(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In episode k ∈ [K], λk is played, and then the loss is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since the space of θk is convex, we use standard results (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2016)) to show that updating θk through projected gradient descent results in an upper bound for �K k=1(λk(sk 1) − λc(sk 1))(−V k c,1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We restate the lemma here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 7 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (Hazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Let S ⊆ Rd be a bounded convex and closed set in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Denote D as an upper bound on the diameter of S, and G as an upper bound on the norm of the subgradients of convex cost functions fk over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Using online projected gradient descent to generate sequence {xk}K k=1 with step sizes {ηk = D G √ k, k ∈ [K]} guarantees, for all K ≥ 1: RegretK = max x∗∈K K � k=1 fk(xk) − fk(x∗) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5GD √ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Let us bound D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' By Assumption 3, λ∗ = ⟨ξ(s), θ∗⟩ and ||θ∗||2 ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Since the comparator we use is λ∗, we can set D to be B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To bound G, we observe that the subgradient of our loss function is ξ(s)V k c,1(sk 1) for each k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, since V k c,1(sk 1) ∈ [0, 1] and ||ξ(s)||2 ≤ 1 by Assumption 3, we can set G to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We now bound the regret of {π}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Consider any πc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' With high probability, Rp({π}K k=1, πc) ≤ 2H(1 + B + H) + �K k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λk(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Hoai-An Nguyen, Ching-An Cheng Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' First we expand the regret into two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Rp({π}K k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' πc) = K � k=1 Lk(πc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' λk) − Lk(πk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' λk) = K � k=1 V πc r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V πc c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − [V πk r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1) − λk(sk 1)V πk c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1)] = K � k=1 V πc r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V πc c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − [V πk r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1) − λk(sk 1)V πk c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1)] + K � k=1 [V k r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V k c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1)] − [V k r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V k c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1)] = K � k=1 V πc r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V πc c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − [V k r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − λk(sk 1)V k c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1)] + K � k=1 V k r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1) − V πk r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1) + λk(sk 1)(V πk c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1 (sk 1) − V k c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='1(sk 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To bound the first term, we use Lemma 3 from Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022), which characterize the property of upper confidence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For completeness, we re-write the lemma here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 7 Lemma 8 (Lemma 3 (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' With probability 1−p/2, it holds that T1 = �K k=1 � V πc r,1(sk 1)−λkV πc c,1(sk 1) � − � V k r,1(sk 1) − λkV k c,1(sk 1) � ≤ KH log(|A|)/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Hence, for α = log(|A|)K 2(1 + C + H), T1 ≤ 2H(1 + C + H), where C is such that λk ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' In our problem setting, we can set C = B in the lemma above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Therefore, the first term is bounded by 2H(1+B+H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lastly, we derive a bound on Rd({λk}K k=1, λc) + Rp({πk}K k=1, πc), which directly implies our final upper bound on Regret(K) and Resets(K) in Theorem 3 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' For any πc and λc(s) = ⟨ξ(s), θc⟩ such that ∥θc∥ ≤ B, we have with probability 1 − p, Rd({λk}K k=1, λc) + Rp({πk}K k=1, πc) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + 2H(1 + B + H) + O((B + 1) √ d3H4Kι2) where ι = log[log(|A|)4dKH/p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Combining the upper bounds in Lemma 5 and Lemma 6, we have an upper bound of Rd({λk}K k=1, λc) + Rp({πk}K k=1, πc) =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + K � k=1 (λk(sk 1) − λc(sk 1))(V k c,1(sk 1) − V πk c,1 (sk 1)) + 2H(1 + B + H) + K � k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λk(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)) =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content='5B √ K + 2H(1 + B + H)+ + K � k=1 V k r,1(sk 1) − V πk r,1 (sk 1) + λc(sk 1)(V πk c,1 (sk 1) − V k c,1(sk 1)) where the last term is the overestimation error due to optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' To bound this term, we use Lemma 4 from Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' We re-write the lemma here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Lemma 10 (Lemma 4 (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' WIth probability at least 1 − p/2, for any λ ∈ [0, C], �K k=1 � V k r,1(sk 1) − V πk r,1 (sk 1) � + λ �K k=1 � V πk c,1 (sk 1) − V k c,1(sk 1) � ≤ O((λ + 1) √ d3H4Kι2) where ι = log[log(|A|)4dKH/p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' 7Note that Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' (2022) use a utility function rather than a cost function to denote the constraint on the MDP (cost is just −1× utility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Also note that their Lemma 3 is proved for an arbitrary initial state sequence and for any comparator (which includes π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} +page_content=' Provable Reset-free Reinforcement Learning by No-Regret Reduction Since we have a bound on all λc(sk 1) of B for all k ∈ [K], we have a bound of O((B + 1) √ d3H4Kι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE0T4oBgHgl3EQfeQDL/content/2301.02389v1.pdf'} diff --git a/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf b/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..35e226aacc71ea183f4aa2804270589f3448f1d3 --- /dev/null +++ b/59AzT4oBgHgl3EQfEfpg/content/2301.00994v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bc0427e5723046b37a54a97870c91e9e0bf241b08146eee52bab8da386b7d9c7 +size 3192602 diff 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sha256:113748cf17819c0287bda1ca2616302abc0a1ade62b27e017dc53e50d0f3feb9 +size 166770 diff --git a/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/2301.05512v1.pdf.txt b/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/2301.05512v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e15ab2be6879e85df64a5ddd90f474817b151eda --- /dev/null +++ b/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/2301.05512v1.pdf.txt @@ -0,0 +1,1183 @@ +arXiv:2301.05512v1 [math.PR] 13 Jan 2023 +Almost sure invariance principle for the Kantorovich distance +between the empirical and the marginal distributions of strong +mixing sequences +J´erˆome Dedecker∗, Florence Merlev`ede † +January 16, 2023 +Abstract +We prove a strong invariance principle for the Kantorovich distance between the empirical +distribution and the marginal distribution of stationary α-mixing sequences. +Running head. ASIP for the empirical W1 distance. +Keywords. Empirical process, Wasserstein distance, Almost sure invariance principle, Compact +law of the iterated logarithm, Bounded law of the iterated logarithm, Conditional Value at Risk +Mathematics Subject Classification (2010). 60F15, 60G10, 60B12 +1 +Introduction and notations +Let (Xi)i∈Z be a strictly stationary sequence of real-valued random variables. Define the two +σ-algebras F0 = σ(Xi, i ≤ 0) and Gk = σ(Xi, i ≥ k), and recall that the strong mixing coefficients +(α(k))k≥0 of Rosenblatt [13] are defined by +α(k) = +sup +A∈F0,B∈Gk +|P(A ∩ B) − P(A)P(B)| . +(1.1) +Let µ be the common distribution of the Xi’s, and let +µn = 1 +n +n +� +k=1 +δXk +be the empirical measure based on X1, . . . , Xn. +In this paper, we prove a strong invariance +principle for the Kantorovich distance W1(µn, µ) between µn and µ under a condition on the +mixing coefficients α(k). Recall that the Kantorovich distance (also called Wasserstein distance +of order 1) between two probability measures µ and ν is defined by +W1(µ, ν) = +inf +π∈M(µ,ν) +� +|x − y|π(dx, dy) , +where M(µ, ν) is the set of probability measures on R2 with marginals µ and ν. We shall use the +following well known representation for probabilities on the real line: +W1(µ, ν) = +� +|Fµ(x) − Fν(x)|dx , +(1.2) +∗J´erˆome Dedecker, Universit´e de Paris, CNRS, MAP5, UMR 8145, 45 rue des Saints-P`eres, F-75006 Paris, France. +†Florence Merlev`ede, Universit´e Gustave Eiffel, LAMA, UMR 8050 CNRS, F-77454 Marne-La-Vall´ee, France. +1 + +where Fµ is the cumulative distribution function of µ. +Let H : t → P([X0| > t) be the tail function of |X0|. In the case where (Xi)i∈Z is a sequence +of independent and identically distributed (i.i.d.) random variables, del Barrio et al. [2] used the +representation (1.2) and a general result of Jain [7] for Banach-valued random variables to prove a +central limit theorem for √nW1(µn, µ). More precisely, they showed that √nW1(µn, µ) converges +in distribution to the L1(dt) norm of an L1(dt)-valued Gaussian random variable, provided that +� ∞ +0 +� +H(t) dt < ∞ . +(1.3) +They also proved that √nW1(µn, µ) is stochastically bounded iff (1.3) holds, proving that this +condition is necessary and sufficient for the weak convergence of √nW1(µn, µ). +Still in the i.i.d. case, we easily deduce from Chapters 8 and 10 in Ledoux and Talagrand [8] +that: if (1.3) holds, then the sequence +√n +√2 log log nW1(µn, µ) +(1.4) +satisfies a compact law of the iterated logarithm. +For strongly mixing sequences in the sense of Rosenblatt [13], we proved in [6] the central limit +theorem for √nW1(µn, µ) under the condition +� ∞ +0 +� +� +� +� +∞ +� +k=0 +(α(k) ∧ H(t)) dt < ∞ +(1.5) +(where a ∧ b means the minimum between two reals a and b), and we give sufficient conditions +for (1.5) to hold. Note that, in [6], we used a weaker version of the α-mixing coefficients, that +enables to deal with a large class of non-mixing processes in the sense of Rosenblatt [13]. +In Section 2 of this paper, we prove a strong invariance principle for W1(µn, µ) under the +condition (1.5). The compact law of the iterated logarithm for (1.4) easily follows from this strong +invariance principle. In Section 3, we apply our general result to derive the almost sure rate of +convergence of the empirical estimator of the Conditional Value at Risk (CV aR) for stationary +α-mixing sequences. +In the rest of the paper, we shall use the following notation: for two sequences (an)n≥1 and +(bn)n≥1 of positive reals, an ≪ bn means there exists a positive constant C not depending on n +such that an ≤ Cbn for any n ≥ 1. +2 +Main result +Our main result is the following strong invariance principle for W1(µn, µ). +Theorem 2.1. Assume that (1.5) is satisfied. Then, enlarging the probability space if necessary, +there exists a sequence of i.i.d. L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with +covariance function defined as follows: for any f, g ∈ L∞(dt), +Γ(f, g) = Cov +�� +f(t)Z1(t) dt, +� +g(t)Z1(t) dt +� += +� +k∈Z +�� +f(t)g(s)Cov(1X0≤t, 1Xk≤s) ds dt , +(2.1) +and such that +nW1(µn, µ) − +� ����� +n +� +k=1 +Zk(t) +����� dt = o( +� +n log log n) +almost surely. +2 + +Remark 2.2. In [4], Cuny proved a strong invariance principle for W1(µn, µ). under the condition +∞ +� +k=0 +1 +√ +k + 1 +� ∞ +0 +� +α(k) ∧ H(t) dt < ∞ +(2.2) +(in fact, he proved the result for a weaker version of the α-mixing coefficient, the same as that +used in [6] for the central limit theorem). It follows from Section 5 of [6], that the condition (1.5) +is always less restrictive than (2.2). +As a consequence of Theorem 2.1, we get the compact law of the iterated logarithm. Let K +be the unit ball of the reproducing kernel Hilbert space (RKHS) associated with Γ, and C be the +image of K by the L1(dt) norm. The following corollary holds: +Corollary 2.1. Assume that (1.5) is satisfied. Then the sequence +√n +√2 log log nW1(µn, µ) +is almost surely relatively compact, with limit set C. +The proof of Theorem 2.1 is based on two ingredients: a martingale approximation in L1(dt), +as in [6], and the following version of the bounded law of the iterated logarithm, which has an +interest in itself. +Proposition 2.1. Assume that (1.5) holds, and let +V = +� ∞ +0 +� +� +� +� +∞ +� +k=0 +(α(k) ∧ H(t)) dt . +(2.3) +Then, there exists a universal constant η such that for any ε > 0, +� +n≥2 +1 +nP +� +max +1≤k≤n kW1(µk, µ) > (ηV + ε) +� +n log log n +� +< ∞ . +(2.4) +Remark 2.3. (The bivariate case). Let (Xi, Yi)i∈Z be a stationary sequence of R2-valued random +variables, and define the coefficients α(k) as in (1.1), with the two σ-algebras F0 = σ(Xi, Yi, i ≤ 0) +and Gk = σ(Xi, Yi, i ≥ k). Let µX (resp. µY ) be the common distribution of the Xi’s (resp. the +Yi’s), and let +µn,X = 1 +n +n +� +k=1 +δXk +and +µn,Y = 1 +n +n +� +k=1 +δYk . +Combining the arguments in [3] and the proof of Theorem 2.1, one can prove the following strong +invariance principle for n (W1(µn,X, µn,Y ) − W1(µX, µY )). +Let ϕ be the continuous function from L1(dt) to R defined by +ϕ(x) = +� � +sign{FX(t) − FY (t)} x(t)1FX(t)̸=FY (t) + |x(t)|1FX(t)=FY (t) +� +dt , +where FX (resp. FY ) is the cumulative distribution function of µX (resp. µY ). Assume that +� ∞ +0 +� +� +� +� +∞ +� +k=0 +(α(k) ∧ HX(t)) dt < ∞ +and +� ∞ +0 +� +� +� +� +∞ +� +k=0 +(α(k) ∧ HY (t)) dt < ∞ . +Then, enlarging the probability space if necessary, there exists a sequence of i.i.d. L1(dt)-valued +centered Gaussian random variables (Zi)i≥1 with covariance function given by: for any f, g ∈ +L∞(dt), +�Γ(f, g) = Cov +�� +f(t)Z1(t) dt, +� +g(t)Z1(t) dt +� += +� +k∈Z +�� +f(t)g(s)Cov(1X0≤t − 1Y0≤t, 1Xk≤s − 1Yk≤s) ds dt , +3 + +and such that +n (W1(µn,X, µn,Y ) − W1(µX, µY )) − ϕ +� n +� +k=1 +Zk +� += o( +� +n log log n) +almost surely. +3 +Rates of convergence of the empirical estimator of +the Conditional Value at Risk +The Conditional Value at Risk at level u ∈ (0, 1] of a real-valued integrable random variable X +(CV aRu(X)) is a “risk measure” (according to the definition of Acerbi and Tasche [1]), which is +widely used in mathematical finance. It is sometimes called Expected Shortfall of Average Value +at Risk. We refer to the paper [1] for a clear definition of that indicator, and for its relation with +other well known measures, such as the Value at Risk, the Worst Conditional Expectation, the +Tail Conditional Expectation... According to Acerbi and Tasche [1], CV aRu(X) can be expressed +as +CV aRu(X) = − 1 +u +� u +0 +F −1 +X (x)dx , +where FX is the cumulative distribution function of the variable X, and F −1 +X +is its usual cadlag +inverse: F −1 +X (u) = inf{x ∈ R : FX(x) ≥ u}. +Concerning the difference between the Conditional Value at Risk of two random variables X +and Y , the following elementary inequality holds (see for instance [12]): +|CV aRu(X) − CV aRu(Y )| ≤ 1 +u +� 1 +0 +|F −1 +X (x) − F −1 +Y (x)|dx = 1 +uW1(µX, µY ) , +(3.1) +where µX (resp. µY ) is the distribution of X (resp. Y ). +Consider now the problem of estimating CV aRu(X) from the random variables X1, ..., Xn, +where (Xi)i∈Z is a stationary sequence of α-mixing random variables with common distribution +µ = µX. A natural estimator is then +� +CV aRu,n = − 1 +u +� u +0 +F −1 +n (x)dx , +where Fn is the empirical distribution function based on X1, . . . , Xn. From (3.1), we get the upper +bound +���CV aRu(X) − � +CV aRu,n +��� ≤ 1 +u +� 1 +0 +|F −1 +X (x) − F −1 +n (x)|dx = 1 +uW1(µn, µ) , +From Corollary 2.1, we obtain the almost sure rate of convergence of � +CV aRu,n: if (1.5) holds, +then +lim sup +n→∞ +√n +√2 log log n +���CV aRu(X) − � +CV aRu,n +��� ≤ κ(Γ) +u +almost surely, +where κ(Γ) is the largest value of the compact set C of Corollary 2.1 (recall that the covariance +function Γ is defined in (2.1)). It is well known (see for instance Section 8 in [8]) that the constant +κ(Γ) can be expressed as +κ(Γ) = +sup +f:∥f∥∞≤1 +� +Var +�� +f(t)Z(t)dt +��1/2 +≤ +���� +� +|Z(t)|dt +���� +2 +, +where Z is an L1(dt)-valued centered random variable with covariance function Γ. +4 + +4 +Proofs +4.1 +Proof of Theorem 2.1 +Let (Ω, A, P) be the underlying probability space. +By a standard argument, one may assume +that Xi = X0 ◦ T , where T : Ω �→ Ω is a bijective, bi-measurable transformation, preserving the +probability P. Let also Fi = σ(Xk, k ≤ i). +Let Y0(t) = 1X0≤t − F(t), and Yk(t) = Y0(t) ◦ T k = 1Xk≤t − F(t). With these notations and +the representation (1.2) one has that +nW1(µn, µ) = +� ����� +n +� +k=1 +Yk(t) +����� dt . +(4.1) +From Section 4 in [6], we know that, if (1.5) holds, then +Y0(t) = D0(t) + A(t) − A(t) ◦ T, +(4.2) +where D0 is such that E(D1(t)|F−1) = 0 almost surely and +� +∥D0(t)∥2 dt < ∞, and A is such that +� +∥A(t)∥1 dt < ∞. Moreover, the covariance operator of D0 is exactly Γ: for any f, g ∈ L∞(dt), +Γ(f, g) = Cov +�� +f(t)D0(t) dt, +� +g(t)D0(t) dt +� +. +(4.3) +Let Dk(t) = D0 ◦ T k. From (4.2), it follows that +n +� +k=1 +Yk = +n +� +k=1 +Dk + A ◦ T − A ◦ T n . +(4.4) +From [4, Proposition 3.3], we know that, enlarging the probability space if necessary, there exists +a sequence of i.i.d. L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance +function Γ such that +� ����� +n +� +k=1 +Dk(t) − +n +� +k=1 +Zk(t) +����� dt = o +�� +n log log n +� +almost surely. +(4.5) +Hence, the result will follow from (4.1), (4.4) and (4.5) if we can prove that +lim +n→∞ +1 +√n log log n +� +|A(t) ◦ T n| dt = 0 +almost surely. +(4.6) +To prove (4.6), we start by considering the integral over [−M, M]c, for M > 0. Applying again +[4, Proposition 3.3], we infer that +lim sup +n→∞ +1 +√2n log log n +� +[−M,M]c +����� +n +� +k=1 +Dk(t) +����� dt ≤ +� +[−M,M]c ∥D0(t)∥2 dt +almost surely. +(4.7) +Now, as will be clear from the proof, Proposition 2.1 also holds on the space L1([−M, M]c, dt), +and implies that there exists a universal constant η such that, for any positive ε, +lim sup +n→∞ +1 +√n log log n +� +[−M,M]c +����� +n +� +k=1 +Yk(t) +����� dt ≤ ε+η +� ∞ +M +� +� +� +� +∞ +� +k=0 +min {α(k), H(t)} dt +almost surely. +(4.8) +From (4.7) and (4.8), we infer that +lim +M→∞ lim sup +n→∞ +1 +√n log log n +� +[−M,M]c |A(t) ◦ T n| dt = 0 +almost surely. +5 + +Hence the proof of (4.6) will be complete if we prove that, for any M > 0, +lim sup +n→∞ +1 +√n log log n +� M +−M +|A(t) ◦ T n| dt = 0 +almost surely. +(4.9) +To prove (4.9), we work in the space H = L2([−M, M], dt), and we denote by ∥ · ∥H and ⟨·, ·⟩ +the usual norm and scalar product on H. Since E(∥D0∥2 +H) < ∞, we know from [4] that �n +k=1 Dk +satisfies the compact law of the iterated logarithm in H. Since � +k≥0 α(k) < ∞ and Y0 is bounded +in H, we infer from [5] that �n +k=1 Yk satisfies also the compact law of the iterated logarithm in H. +Now, arguing exactly as in the end of the proof of [5, Theorem 4], one has: for any f in H +lim +n→∞ +⟨f, A ◦ T n⟩ +√n log log n = 0 +almost surely. +(4.10) +Let (ei)i≥1 be a complete orthonormal basis of H and PN(f) = �N +k=1⟨f, ek⟩ek be the projection +of f on the space spanned by the first N elements of the basis. From (4.10), we get that +lim +n→∞ +PN(A ◦ T n) +√n log log n = 0 +almost surely. +(4.11) +On another hand, applying again [4, Proposition 3.3] (as done in (4.7)), we get +lim +N→∞ lim sup +n→∞ +1 +√n log log n +�����(I − PN) +� n +� +k=1 +Dk +������ +H += 0 +almost surely, +(4.12) +and applying [5, Theorem 4], +lim +N→∞ lim sup +n→∞ +1 +√n log log n +�����(I − PN) +� n +� +k=1 +Yk +������ +H += 0 +almost surely. +(4.13) +From (4.4), (4.12) and (4.13), we infer that +lim +N→∞ lim sup +n→∞ +∥(I − PN)A ◦ T n∥H +√n log log n += 0 +almost surely, +which, together with (4.11), implies (4.9). The proof of Theorem 2.1 is complete. ⋄ +4.2 +Proof of Proposition 2.1 +For any n ∈ N, let us introduce the following notations: +R(u) = min{q ∈ N∗ : α(q) ≤ u}Q(u) +and +R−1(x) = inf{u ∈ [0, 1] : R(u) ≤ x} . +For a positive real a that will be specified later, let +mn = a +� +n +log log n , +vn = R−1(mn) , +Mn = Q(vn) . +(4.14) +For any M > 0, let gM(y) = (y ∧ M) ∨ (−M). For any integer i, define +X′ +i = gMn(Xi) and X′′ +i = Xi − X′ +i . +(4.15) +We first recall that, by the dual expression of W1(µn, µ), +nW1(µn, µ) = sup +f∈Λ1 +n +� +i=1 +(f(Xi) − E(f(Xi))) . +6 + +where Λ1 is the set of Lipschitz functions such that |f(x) − f(y)| ≤ |x − y|. Hence, +nW1(µn, µ) ≤ sup +f∈Λ1 +n +� +i=1 +� +f(X′ +i) − E(f(X′ +i)) +� ++ sup +f∈Λ1 +n +� +i=1 +� +f(Xi) − f(X′ +i) − E(f(Xi) − f(X′ +i)) +� +. +Therefore, setting, +F ′ +n(t) = 1 +n +n +� +k=1 +1{X′ +k≤t} +and +F ′(t) = P(X′ +1 ≤ t) , +and noticing that +k∥F ′ +k − F ′∥1 = sup +f∈Λ1 +k +� +i=1 +� +f(X′ +i) − E(f(X′ +i) +� +, +we get +max +1≤k≤n kW1(µk, µ) ≤ max +1≤k≤n k∥F ′ +k − F ′∥1 + +n +� +i=1 +(|X′′ +i | + E(|X′′ +i |) . +(4.16) +Now, note that +� +n≥2 +1 +√n log log nE(|X′′ +n|) ≤ +� +n≥2 +1 +√n log log n +� +∞ +0 +P +� +|X0|1|X0|>Q(vn) > t +� +dt +≤ +� +n≥2 +1 +√n log log n +� +∞ +Q(vn) +H(t)dt ≤ +� +n≥2 +1 +√n log log n +� vn +0 +Q(u)du +≤ +� +n≥2 +1 +√n log log n +� 1 +0 +Q(u)1mn≤R(u)du ≪ +� 1 +0 +R(u)Q(u)du. +But, according to Propositions 5.1 and 5.2 in [6], condition (1.5) implies that +� 1 +0 +R(u)Q(u)du < ∞ . +(4.17) +Hence, to prove (2.4) it suffices to show that there exists an universal constant η such that for +any ε > 0, +� +n≥2 +1 +nP +� +max +1≤k≤n k∥F ′ +k − F ′∥1 > ηV +� +n log log n +� +< ∞ . +(4.18) +For this purpose, let +qn = min{k ∈ N∗ : α(k) ≤ vn} ∧ n . +(4.19) +Since R is right continuous, we have R(R−1(w)) ≤ w for any w, hence +qnMn = R(vn) = R(R−1(mn)) ≤ mn . +(4.20) +Assume first that qn = n. Bounding f(X′ +i) − E(f(X′ +i)) by 2Mn, we obtain +max +1≤k≤n k∥F ′ +k − F ′∥1 ≤ 2nMn = 2qnMn ≤ 2mn . +(4.21) +Taking into account the definition of mn, it follows that there exists n0 depending on a, V and η, +such that for any n ≥ n0, 8mn ≤ κV √n log log n. This proves the proposition in the case where +qn = n. +From now on, we assume that qn < n. Therefore qn = min{k ∈ N∗ : α(k) ≤ vn} and then +α(qn) ≤ vn. For any integer i, define +Ui(t) = +iqn +� +k=(i−1)qn+1 +� +1X′ +k≤t − E +� +1X′ +k≤t +�� +. +7 + +and notice that +max +1≤k≤n k∥F ′ +k − F ′∥1 ≤ 2qnMn + +� Mn +−Mn +max +1≤j≤[n/qn] +����� +j +� +i=1 +Ui(t) +����� dt . +Let kn = [n/qn]. For any t, applying Rio’s coupling lemma (see [11, Lemma 5.2]) recursively, +we can construct random variables (U ∗ +i (t))1≤i≤kn such that +• U ∗ +i (t) has the same distribution as U ′ +i for all 1 ≤ i ≤ kn, +• the random variables (U ∗ +2i(t))2≤2i≤kn are independent, as well as the random variables +(U ∗ +2i−1(t))1≤2i−1≤kn, +• we can suitably control ∥Ui(t) − U ∗ +i (t)∥1 as follows: for any i ≥ 1, +∥Ui(t) − U ∗ +i (t)∥1 ≤ 4qnα(qn) . +(4.22) +Substituting U ∗ +i (t) to Ui(t), we obtain +max +1≤k≤n k∥F ′ +k − F ′∥1 ≤ 2qnMn + +max +2≤2j≤[n/qn] +����� +j +� +i=1 +U ∗ +2i(t) +����� ++ +max +1≤2j−1≤[n/qn] +����� +j +� +i=1 +U ∗ +2i−1(t) +�����+ +[n/qn] +� +i=1 +|Ui(t) − U ∗ +i (t)| . +(4.23) +Therefore, setting κ = η/4, for n ≥ n0, +P +� +max +1≤k≤n k∥F ′ +k − F ′∥1 ≥ 4V κ +� +n log log n +� +≤ I1(n) + I2(n) + I3(n) , +(4.24) +where +I1(n) = P + + +� Mn +−Mn +[n/qn] +� +i=1 +|Ui(t) − U ∗ +i (t)| dt ≥ V κ +� +n log log n + + +I2(n) = P +�� Mn +−Mn +max +2≤2j≤[n/qn] +����� +j +� +i=1 +U ∗ +2i(t) +����� dt ≥ V κ +� +n log log n +� +I3(n) = P +�� Mn +−Mn +max +1≤2j−1≤[n/qn] +����� +j +� +i=1 +U ∗ +2i−1(t) +����� dt ≥ V κ +� +n log log n +� +. +Using Markov’s inequality and (4.22), we get +I1(n) ≪ +n +√n log log nMnα(qn) ≪ +n +√n log log nvnQ(vn) ≪ +n +√n log log n +� R−1(mn) +0 +Q(u)du . +Hence, by (4.17), +� +n≥2 +1 +nI1(n) ≪ +� +n≥2 +1 +√n log log n +� R−1(mn) +0 +Q(u)du ≪ +� 1 +0 +R(u)Q(u)du < ∞ . +To handle now the term I2(n) (as well as I3(n)) in the decomposition (4.24), we shall use +again Markov’s inequality but this time at the order p ≥ 2. Hence for p ≥ 2, taking into account +the stationarity, we get +I2(n) ≤ +1 +(V κ)p(n log log n)p/2 + + +� Q(vn) +−Q(vn) +����� +max +2≤2j≤[n/qn] +����� +j +� +i=1 +˜U2i(t) +����� +����� +p +dt + + +p +. +8 + +Applying Rosenthal’s inequality (see for instance [9, Theorem 4.1]) and taking into account the +stationarity, there exist two positive universal constants c1 and c2 not depending on p such that +����� +max +2≤2j≤[n/qn] +����� +j +� +i=1 +U ∗ +2i(t) +����� +����� +p +p +≤ cp +1pp/2(n/qn)p/2∥U2(t)∥p +2 + cp +2pp(n/qn)∥U2(t)∥p +p := J1(t) + J2(t) . +(4.25) +Using similar arguments as to handle the quantity I2(n) in the proof of [6, Proposition 3.4], we +have +� Q(vn) +−Q(vn) +∥U2(t)∥2dt = +� Q(vn) +−Q(vn) +� +Var +� qn +� +i=1 +1{X′ +i≤t} +��1/2 +dt +≤ 2 +√ +2√qn +� Q(vn) +0 +�qn−1 +� +k=0 +α(k) ∧ H(t) +�1/2 +dt ≤ 2V +� +2qn . +(4.26) +Hence +� +n≥2 +1 +n(V κ)p(n log log n)p/2 +�� Q(vn) +−Q(vn) +J1(t)1/pdt +�p +≤ +� +n≥2 +(2 +√ +2c1√p)p +nκp(log log n)p/2 . +Let now +p = pn = max{c log log n, 2}, +where c will be specified later. Set n1 = min{n ≥ 2 : c log log n ≥ 2}. It follows that +� +n≥n1 +1 +n(V κ)p(n log log n)p/2 +�� Q(vn) +−Q(vn) +J1(t)1/pdt +�p +≤ +� +n≥n1 +1 +n +�2c1 +√ +2c +κ +�c log log n +, +which is finite provided we take κ such that 2c1 +√ +2c +κ += α−1 with α > 1 and c > (log α)−1. +On another hand, proceeding as in (4.26), we deduce that, for any t > 0, +∥U2(t)∥p +p = +����� +qn +� +i=1 +� +1{X′ +i≤t} − P(X′ +i ≤ t) +������ +p +p +≤ qp−2 +n +����� +qn +� +i=1 +� +1{X′ +i≤t} − P(X′ +i ≤ t) +������ +2 +2 +≤ 2qp−1 +n +qn−1 +� +k=0 +(α(k) ∧ H(t)). +In addition +� Q(vn) +0 +�qn−1 +� +k=0 +α(k) ∧ H(t) +�1/p +dt = +� Q(vn) +0 +�� H(t) +0 +(α−1(u) ∧ qn)du +�1/p +dt +≤ +� Q(vn) +0 +� +vnqn + +� H(t) +vn +(α−1(u) ∧ qn)du +�1/p +dt . +Note that u < H(t) ⇐⇒ t < Q(u). Consequently u < H(t) implies that Q−2(u) < t−2. Hence +� Q(vn) +0 +�qn−1 +� +k=0 +α(k) ∧ H(t) +�1/p +dt +≤ (vnqn)1/pQ(vn) + +� Q(vn) +0 +� +t−2 +� H(t) +vn +(α−1(u) ∧ qn)Q2(u)du +�1/p +≤ (vnqn)1/pQ(vn) + +�� 1 +vn +R(u)Q(u)du +�1/p � Q(vn) +0 +t−2/pdt +≤ (vnqn)1/pQ(vn) + +�� 1 +0 +R(u)Q(u)du +�1/p +p(p − 2)−1Q(vn)1−2/p . +9 + +Set n2 = min{n ≥ 2 : c log log n ≥ 4}. It follows that +� +n≥n2 +1 +n(V κ)p(n log log n)p/2 +�� Q(vn) +−Q(vn) +J2(t)1/pdt +�p +≤ 2 +� +n≥n2 +(4c2p)p +(κV )p(n log log n)p/2 qp−2 +n +� +vnqnQp(vn) + 2pQ(vn)p−2 +� 1 +0 +R(u)Q(u)du +� +. +Note that +vnqnQ2(vn) = vnα−1(vn)Q2(vn) ≤ +� 1 +0 +R(u)Q(u)du . +Hence, since qnMn ≤ mn, we get +� +n≥n2 +1 +n(V κ)p(n log log n)p/2 +�� Q(vn) +−Q(vn) +J2(t)1/pdt +�p +≤ 4 +� 1 +0 +R(u)Q(u)du +� +n≥n2 +(8c2p)p +(κV )p(n log log n)p/2 mp−2 +n +≤ 4a−2 +� 1 +0 +R(u)Q(u)du +� +n≥n2 +�8ac2c +κV +�p log log n +n +, +which is finite by taking into account (4.17), and if we choose a = (c1κV )/(2c2 +√ +2c). Indeed, in +this case, +8ac2c +κV += 2c1 +√ +2c +κ +× 2ac2 +√ +2c +c1κV += α−1 . +This ends the proof of the proposition. ⋄ +References +[1] C. Acerbi and D. Tasche (2002), On the coherence of Expected Shortfall. Journal of Banking +and Finance 26 1487-1503. +[2] E. del Barrio, E. Gin´e and C. Matr´an (1999), Central limit theorems for the Wasserstein +distance between the empirical and the true distributions. Ann. Probab. 27 1009-1071. +[3] P. Berthet, J. Dedecker, and F. Merlev`ede, Central limit theorem and almost sure results for +bivariate empirical W1 distances. (2020) https://hal.archives-ouvertes.fr/hal-02881842 +[4] C. Cuny (2017), Invariance principles under the Maxwell-Woodroofe condition in Banach +spaces. Ann. Probab. 45 1578–1611. +[5] J. Dedecker and F. Merlev`ede (2010), On the almost sure invariance principle for stationary +sequences of Hilbert-valued random variables. Dependence in probability, analysis and number +theory, 157–175, Kendrick Press, Heber City, UT. +[6] J. Dedecker and F. Merlev`ede (2017), Behavior of the Wasserstein distance between the empir- +ical and the marginal distributions of stationary α-dependent sequences. Bernoulli 23 2083– +2127. +[7] N. C. Jain (1977), Central limit theorems and related questions in Banach space. Proceedings +of Symposium in Pure and Applied Mathematics 31 55-65. Amer. Math. Soc. Providence, RI. +[8] M. Ledoux and M. Talagrand (1991), Probability in Banach spaces. Isoperimetry and pro- +cesses. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23 Springer-Verlag, Berlin, +xii+ 480 pp. +[9] I. Pinelis (1994), Optimum bounds for the distributions of martingales in Banach spaces. Ann. +Probab. 22 1679–1706. +10 + +[10] E. Rio (1995), The functional law of the iterated logarithm for stationary α-mixing sequences. +Ann. Probab. 23 1188-1203. +[11] E. Rio (2000), Th´eorie asymptotique des processus al´eatoires faiblement d´ependants. Math. +Appl. 31 Berlin. +[12] E. Rio (2017), About the conditional value at risk of partial sums. C. R. Math. Acad. Sci. +Paris 355 1190-1195. +[13] M. Rosenblatt (1956), A central limit theorem and a strong mixing condition, Proc. Nat. +Acad. Sci. U.S.A. 42 43-47. +11 + diff --git a/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/load_file.txt b/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2d586eb93e3ec6ea3f81567da244d1add696cef --- /dev/null +++ b/79E5T4oBgHgl3EQfQQ7U/content/tmp_files/load_file.txt @@ -0,0 +1,339 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf,len=338 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='05512v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='PR] 13 Jan 2023 Almost sure invariance principle for the Kantorovich distance between the empirical and the marginal distributions of strong mixing sequences J´erˆome Dedecker∗, Florence Merlev`ede † January 16, 2023 Abstract We prove a strong invariance principle for the Kantorovich distance between the empirical distribution and the marginal distribution of stationary α-mixing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Running head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' ASIP for the empirical W1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Empirical process, Wasserstein distance, Almost sure invariance principle, Compact law of the iterated logarithm, Bounded law of the iterated logarithm, Conditional Value at Risk Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 60F15, 60G10, 60B12 1 Introduction and notations Let (Xi)i∈Z be a strictly stationary sequence of real-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Define the two σ-algebras F0 = σ(Xi, i ≤ 0) and Gk = σ(Xi, i ≥ k), and recall that the strong mixing coefficients (α(k))k≥0 of Rosenblatt [13] are defined by α(k) = sup A∈F0,B∈Gk |P(A ∩ B) − P(A)P(B)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1) Let µ be the common distribution of the Xi’s, and let µn = 1 n n � k=1 δXk be the empirical measure based on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In this paper, we prove a strong invariance principle for the Kantorovich distance W1(µn, µ) between µn and µ under a condition on the mixing coefficients α(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Recall that the Kantorovich distance (also called Wasserstein distance of order 1) between two probability measures µ and ν is defined by W1(µ, ν) = inf π∈M(µ,ν) � |x − y|π(dx, dy) , where M(µ, ν) is the set of probability measures on R2 with marginals µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' We shall use the following well known representation for probabilities on the real line: W1(µ, ν) = � |Fµ(x) − Fν(x)|dx , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2) ∗J´erˆome Dedecker, Universit´e de Paris, CNRS, MAP5, UMR 8145, 45 rue des Saints-P`eres, F-75006 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' †Florence Merlev`ede, Universit´e Gustave Eiffel, LAMA, UMR 8050 CNRS, F-77454 Marne-La-Vall´ee, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 1 where Fµ is the cumulative distribution function of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let H : t → P([X0| > t) be the tail function of |X0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In the case where (Xi)i∈Z is a sequence of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=') random variables, del Barrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' [2] used the representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2) and a general result of Jain [7] for Banach-valued random variables to prove a central limit theorem for √nW1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' More precisely, they showed that √nW1(µn, µ) converges in distribution to the L1(dt) norm of an L1(dt)-valued Gaussian random variable, provided that � ∞ 0 � H(t) dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3) They also proved that √nW1(µn, µ) is stochastically bounded iff (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3) holds, proving that this condition is necessary and sufficient for the weak convergence of √nW1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Still in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' case, we easily deduce from Chapters 8 and 10 in Ledoux and Talagrand [8] that: if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3) holds, then the sequence √n √2 log log nW1(µn, µ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) satisfies a compact law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' For strongly mixing sequences in the sense of Rosenblatt [13], we proved in [6] the central limit theorem for √nW1(µn, µ) under the condition � ∞ 0 � � � � ∞ � k=0 (α(k) ∧ H(t)) dt < ∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) (where a ∧ b means the minimum between two reals a and b), and we give sufficient conditions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Note that, in [6], we used a weaker version of the α-mixing coefficients, that enables to deal with a large class of non-mixing processes in the sense of Rosenblatt [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In Section 2 of this paper, we prove a strong invariance principle for W1(µn, µ) under the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' The compact law of the iterated logarithm for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) easily follows from this strong invariance principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In Section 3, we apply our general result to derive the almost sure rate of convergence of the empirical estimator of the Conditional Value at Risk (CV aR) for stationary α-mixing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In the rest of the paper, we shall use the following notation: for two sequences (an)n≥1 and (bn)n≥1 of positive reals, an ≪ bn means there exists a positive constant C not depending on n such that an ≤ Cbn for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 2 Main result Our main result is the following strong invariance principle for W1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Then, enlarging the probability space if necessary, there exists a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance function defined as follows: for any f, g ∈ L∞(dt), Γ(f, g) = Cov �� f(t)Z1(t) dt, � g(t)Z1(t) dt � = � k∈Z �� f(t)g(s)Cov(1X0≤t, 1Xk≤s) ds dt , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1) and such that nW1(µn, µ) − � ����� n � k=1 Zk(t) ����� dt = o( � n log log n) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 2 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In [4], Cuny proved a strong invariance principle for W1(µn, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' under the condition ∞ � k=0 1 √ k + 1 � ∞ 0 � α(k) ∧ H(t) dt < ∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2) (in fact, he proved the result for a weaker version of the α-mixing coefficient, the same as that used in [6] for the central limit theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' It follows from Section 5 of [6], that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) is always less restrictive than (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' As a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1, we get the compact law of the iterated logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let K be the unit ball of the reproducing kernel Hilbert space (RKHS) associated with Γ, and C be the image of K by the L1(dt) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' The following corollary holds: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Then the sequence √n √2 log log nW1(µn, µ) is almost surely relatively compact, with limit set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 is based on two ingredients: a martingale approximation in L1(dt), as in [6], and the following version of the bounded law of the iterated logarithm, which has an interest in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Assume that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) holds, and let V = � ∞ 0 � � � � ∞ � k=0 (α(k) ∧ H(t)) dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3) Then, there exists a universal constant η such that for any ε > 0, � n≥2 1 nP � max 1≤k≤n kW1(µk, µ) > (ηV + ε) � n log log n � < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (The bivariate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let (Xi, Yi)i∈Z be a stationary sequence of R2-valued random variables, and define the coefficients α(k) as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1), with the two σ-algebras F0 = σ(Xi, Yi, i ≤ 0) and Gk = σ(Xi, Yi, i ≥ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' µY ) be the common distribution of the Xi’s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' the Yi’s), and let µn,X = 1 n n � k=1 δXk and µn,Y = 1 n n � k=1 δYk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Combining the arguments in [3] and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1, one can prove the following strong invariance principle for n (W1(µn,X, µn,Y ) − W1(µX, µY )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let ϕ be the continuous function from L1(dt) to R defined by ϕ(x) = � � sign{FX(t) − FY (t)} x(t)1FX(t)̸=FY (t) + |x(t)|1FX(t)=FY (t) � dt , where FX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' FY ) is the cumulative distribution function of µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' µY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Assume that � ∞ 0 � � � � ∞ � k=0 (α(k) ∧ HX(t)) dt < ∞ and � ∞ 0 � � � � ∞ � k=0 (α(k) ∧ HY (t)) dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Then, enlarging the probability space if necessary, there exists a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance function given by: for any f, g ∈ L∞(dt), �Γ(f, g) = Cov �� f(t)Z1(t) dt, � g(t)Z1(t) dt � = � k∈Z �� f(t)g(s)Cov(1X0≤t − 1Y0≤t, 1Xk≤s − 1Yk≤s) ds dt , 3 and such that n (W1(µn,X, µn,Y ) − W1(µX, µY )) − ϕ � n � k=1 Zk � = o( � n log log n) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 3 Rates of convergence of the empirical estimator of the Conditional Value at Risk The Conditional Value at Risk at level u ∈ (0, 1] of a real-valued integrable random variable X (CV aRu(X)) is a “risk measure” (according to the definition of Acerbi and Tasche [1]), which is widely used in mathematical finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' It is sometimes called Expected Shortfall of Average Value at Risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' We refer to the paper [1] for a clear definition of that indicator, and for its relation with other well known measures, such as the Value at Risk, the Worst Conditional Expectation, the Tail Conditional Expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' According to Acerbi and Tasche [1], CV aRu(X) can be expressed as CV aRu(X) = − 1 u � u 0 F −1 X (x)dx , where FX is the cumulative distribution function of the variable X, and F −1 X is its usual cadlag inverse: F −1 X (u) = inf{x ∈ R : FX(x) ≥ u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Concerning the difference between the Conditional Value at Risk of two random variables X and Y , the following elementary inequality holds (see for instance [12]): |CV aRu(X) − CV aRu(Y )| ≤ 1 u � 1 0 |F −1 X (x) − F −1 Y (x)|dx = 1 uW1(µX, µY ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1) where µX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' µY ) is the distribution of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Consider now the problem of estimating CV aRu(X) from the random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=', Xn, where (Xi)i∈Z is a stationary sequence of α-mixing random variables with common distribution µ = µX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' A natural estimator is then � CV aRu,n = − 1 u � u 0 F −1 n (x)dx , where Fn is the empirical distribution function based on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1), we get the upper bound ���CV aRu(X) − � CV aRu,n ��� ≤ 1 u � 1 0 |F −1 X (x) − F −1 n (x)|dx = 1 uW1(µn, µ) , From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1, we obtain the almost sure rate of convergence of � CV aRu,n: if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) holds, then lim sup n→∞ √n √2 log log n ���CV aRu(X) − � CV aRu,n ��� ≤ κ(Γ) u almost surely, where κ(Γ) is the largest value of the compact set C of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 (recall that the covariance function Γ is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' It is well known (see for instance Section 8 in [8]) that the constant κ(Γ) can be expressed as κ(Γ) = sup f:∥f∥∞≤1 � Var �� f(t)Z(t)dt ��1/2 ≤ ���� � |Z(t)|dt ���� 2 , where Z is an L1(dt)-valued centered random variable with covariance function Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 4 4 Proofs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 Let (Ω, A, P) be the underlying probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' By a standard argument, one may assume that Xi = X0 ◦ T , where T : Ω �→ Ω is a bijective, bi-measurable transformation, preserving the probability P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let also Fi = σ(Xk, k ≤ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let Y0(t) = 1X0≤t − F(t), and Yk(t) = Y0(t) ◦ T k = 1Xk≤t − F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' With these notations and the representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2) one has that nW1(µn, µ) = � ����� n � k=1 Yk(t) ����� dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1) From Section 4 in [6], we know that, if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) holds, then Y0(t) = D0(t) + A(t) − A(t) ◦ T, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2) where D0 is such that E(D1(t)|F−1) = 0 almost surely and � ∥D0(t)∥2 dt < ∞, and A is such that � ∥A(t)∥1 dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Moreover, the covariance operator of D0 is exactly Γ: for any f, g ∈ L∞(dt), Γ(f, g) = Cov �� f(t)D0(t) dt, � g(t)D0(t) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3) Let Dk(t) = D0 ◦ T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2), it follows that n � k=1 Yk = n � k=1 Dk + A ◦ T − A ◦ T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) From [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3], we know that, enlarging the probability space if necessary, there exists a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' L1(dt)-valued centered Gaussian random variables (Zi)i≥1 with covariance function Γ such that � ����� n � k=1 Dk(t) − n � k=1 Zk(t) ����� dt = o �� n log log n � almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) Hence, the result will follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) if we can prove that lim n→∞ 1 √n log log n � |A(t) ◦ T n| dt = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='6) To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='6), we start by considering the integral over [−M, M]c, for M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Applying again [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3], we infer that lim sup n→∞ 1 √2n log log n � [−M,M]c ����� n � k=1 Dk(t) ����� dt ≤ � [−M,M]c ∥D0(t)∥2 dt almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='7) Now, as will be clear from the proof, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 also holds on the space L1([−M, M]c, dt), and implies that there exists a universal constant η such that, for any positive ε, lim sup n→∞ 1 √n log log n � [−M,M]c ����� n � k=1 Yk(t) ����� dt ≤ ε+η � ∞ M � � � � ∞ � k=0 min {α(k), H(t)} dt almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='8) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='8), we infer that lim M→∞ lim sup n→∞ 1 √n log log n � [−M,M]c |A(t) ◦ T n| dt = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 5 Hence the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='6) will be complete if we prove that, for any M > 0, lim sup n→∞ 1 √n log log n � M −M |A(t) ◦ T n| dt = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='9) To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='9), we work in the space H = L2([−M, M], dt), and we denote by ∥ · ∥H and ⟨·, ·⟩ the usual norm and scalar product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Since E(∥D0∥2 H) < ∞, we know from [4] that �n k=1 Dk satisfies the compact law of the iterated logarithm in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Since � k≥0 α(k) < ∞ and Y0 is bounded in H, we infer from [5] that �n k=1 Yk satisfies also the compact law of the iterated logarithm in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Now, arguing exactly as in the end of the proof of [5, Theorem 4], one has: for any f in H lim n→∞ ⟨f, A ◦ T n⟩ √n log log n = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='10) Let (ei)i≥1 be a complete orthonormal basis of H and PN(f) = �N k=1⟨f, ek⟩ek be the projection of f on the space spanned by the first N elements of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='10), we get that lim n→∞ PN(A ◦ T n) √n log log n = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='11) On another hand, applying again [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='3] (as done in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='7)), we get lim N→∞ lim sup n→∞ 1 √n log log n �����(I − PN) � n � k=1 Dk ������ H = 0 almost surely, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='12) and applying [5, Theorem 4], lim N→∞ lim sup n→∞ 1 √n log log n �����(I − PN) � n � k=1 Yk ������ H = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='13) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='13), we infer that lim N→∞ lim sup n→∞ ∥(I − PN)A ◦ T n∥H √n log log n = 0 almost surely, which, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='11), implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' ⋄ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 For any n ∈ N, let us introduce the following notations: R(u) = min{q ∈ N∗ : α(q) ≤ u}Q(u) and R−1(x) = inf{u ∈ [0, 1] : R(u) ≤ x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' For a positive real a that will be specified later, let mn = a � n log log n , vn = R−1(mn) , Mn = Q(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='14) For any M > 0, let gM(y) = (y ∧ M) ∨ (−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' For any integer i, define X′ i = gMn(Xi) and X′′ i = Xi − X′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='15) We first recall that, by the dual expression of W1(µn, µ), nW1(µn, µ) = sup f∈Λ1 n � i=1 (f(Xi) − E(f(Xi))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 6 where Λ1 is the set of Lipschitz functions such that |f(x) − f(y)| ≤ |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Hence, nW1(µn, µ) ≤ sup f∈Λ1 n � i=1 � f(X′ i) − E(f(X′ i)) � + sup f∈Λ1 n � i=1 � f(Xi) − f(X′ i) − E(f(Xi) − f(X′ i)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Therefore, setting, F ′ n(t) = 1 n n � k=1 1{X′ k≤t} and F ′(t) = P(X′ 1 ≤ t) , and noticing that k∥F ′ k − F ′∥1 = sup f∈Λ1 k � i=1 � f(X′ i) − E(f(X′ i) � , we get max 1≤k≤n kW1(µk, µ) ≤ max 1≤k≤n k∥F ′ k − F ′∥1 + n � i=1 (|X′′ i | + E(|X′′ i |) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='16) Now, note that � n≥2 1 √n log log nE(|X′′ n|) ≤ � n≥2 1 √n log log n � +∞ 0 P � |X0|1|X0|>Q(vn) > t � dt ≤ � n≥2 1 √n log log n � +∞ Q(vn) H(t)dt ≤ � n≥2 1 √n log log n � vn 0 Q(u)du ≤ � n≥2 1 √n log log n � 1 0 Q(u)1mn≤R(u)du ≪ � 1 0 R(u)Q(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' But, according to Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2 in [6], condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='5) implies that � 1 0 R(u)Q(u)du < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='17) Hence, to prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4) it suffices to show that there exists an universal constant η such that for any ε > 0, � n≥2 1 nP � max 1≤k≤n k∥F ′ k − F ′∥1 > ηV � n log log n � < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='18) For this purpose, let qn = min{k ∈ N∗ : α(k) ≤ vn} ∧ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='19) Since R is right continuous, we have R(R−1(w)) ≤ w for any w, hence qnMn = R(vn) = R(R−1(mn)) ≤ mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='20) Assume first that qn = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Bounding f(X′ i) − E(f(X′ i)) by 2Mn, we obtain max 1≤k≤n k∥F ′ k − F ′∥1 ≤ 2nMn = 2qnMn ≤ 2mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='21) Taking into account the definition of mn, it follows that there exists n0 depending on a, V and η, such that for any n ≥ n0, 8mn ≤ κV √n log log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' This proves the proposition in the case where qn = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' From now on, we assume that qn < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Therefore qn = min{k ∈ N∗ : α(k) ≤ vn} and then α(qn) ≤ vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' For any integer i, define Ui(t) = iqn � k=(i−1)qn+1 � 1X′ k≤t − E � 1X′ k≤t �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 7 and notice that max 1≤k≤n k∥F ′ k − F ′∥1 ≤ 2qnMn + � Mn −Mn max 1≤j≤[n/qn] ����� j � i=1 Ui(t) ����� dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let kn = [n/qn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' For any t, applying Rio’s coupling lemma (see [11, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='2]) recursively, we can construct random variables (U ∗ i (t))1≤i≤kn such that U ∗ i (t) has the same distribution as U ′ i for all 1 ≤ i ≤ kn, the random variables (U ∗ 2i(t))2≤2i≤kn are independent, as well as the random variables (U ∗ 2i−1(t))1≤2i−1≤kn, we can suitably control ∥Ui(t) − U ∗ i (t)∥1 as follows: for any i ≥ 1, ∥Ui(t) − U ∗ i (t)∥1 ≤ 4qnα(qn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='22) Substituting U ∗ i (t) to Ui(t), we obtain max 1≤k≤n k∥F ′ k − F ′∥1 ≤ 2qnMn + max 2≤2j≤[n/qn] ����� j � i=1 U ∗ 2i(t) ����� + max 1≤2j−1≤[n/qn] ����� j � i=1 U ∗ 2i−1(t) �����+ [n/qn] � i=1 |Ui(t) − U ∗ i (t)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='23) Therefore, setting κ = η/4, for n ≥ n0, P � max 1≤k≤n k∥F ′ k − F ′∥1 ≥ 4V κ � n log log n � ≤ I1(n) + I2(n) + I3(n) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='24) where I1(n) = P \uf8eb \uf8ed � Mn −Mn [n/qn] � i=1 |Ui(t) − U ∗ i (t)| dt ≥ V κ � n log log n \uf8f6 \uf8f8 I2(n) = P �� Mn −Mn max 2≤2j≤[n/qn] ����� j � i=1 U ∗ 2i(t) ����� dt ≥ V κ � n log log n � I3(n) = P �� Mn −Mn max 1≤2j−1≤[n/qn] ����� j � i=1 U ∗ 2i−1(t) ����� dt ≥ V κ � n log log n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Using Markov’s inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='22), we get I1(n) ≪ n √n log log nMnα(qn) ≪ n √n log log nvnQ(vn) ≪ n √n log log n � R−1(mn) 0 Q(u)du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='17), � n≥2 1 nI1(n) ≪ � n≥2 1 √n log log n � R−1(mn) 0 Q(u)du ≪ � 1 0 R(u)Q(u)du < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' To handle now the term I2(n) (as well as I3(n)) in the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='24), we shall use again Markov’s inequality but this time at the order p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Hence for p ≥ 2, taking into account the stationarity, we get I2(n) ≤ 1 (V κ)p(n log log n)p/2 \uf8eb \uf8ed � Q(vn) −Q(vn) ����� max 2≤2j≤[n/qn] ����� j � i=1 ˜U2i(t) ����� ����� p dt \uf8f6 \uf8f8 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 8 Applying Rosenthal’s inequality (see for instance [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='1]) and taking into account the stationarity, there exist two positive universal constants c1 and c2 not depending on p such that ����� max 2≤2j≤[n/qn] ����� j � i=1 U ∗ 2i(t) ����� ����� p p ≤ cp 1pp/2(n/qn)p/2∥U2(t)∥p 2 + cp 2pp(n/qn)∥U2(t)∥p p := J1(t) + J2(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='25) Using similar arguments as to handle the quantity I2(n) in the proof of [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='4], we have � Q(vn) −Q(vn) ∥U2(t)∥2dt = � Q(vn) −Q(vn) � Var � qn � i=1 1{X′ i≤t} ��1/2 dt ≤ 2 √ 2√qn � Q(vn) 0 �qn−1 � k=0 α(k) ∧ H(t) �1/2 dt ≤ 2V � 2qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='26) Hence � n≥2 1 n(V κ)p(n log log n)p/2 �� Q(vn) −Q(vn) J1(t)1/pdt �p ≤ � n≥2 (2 √ 2c1√p)p nκp(log log n)p/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Let now p = pn = max{c log log n, 2}, where c will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Set n1 = min{n ≥ 2 : c log log n ≥ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' It follows that � n≥n1 1 n(V κ)p(n log log n)p/2 �� Q(vn) −Q(vn) J1(t)1/pdt �p ≤ � n≥n1 1 n �2c1 √ 2c κ �c log log n , which is finite provided we take κ such that 2c1 √ 2c κ = α−1 with α > 1 and c > (log α)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' On another hand, proceeding as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='26), we deduce that, for any t > 0, ∥U2(t)∥p p = ����� qn � i=1 � 1{X′ i≤t} − P(X′ i ≤ t) ������ p p ≤ qp−2 n ����� qn � i=1 � 1{X′ i≤t} − P(X′ i ≤ t) ������ 2 2 ≤ 2qp−1 n qn−1 � k=0 (α(k) ∧ H(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' In addition � Q(vn) 0 �qn−1 � k=0 α(k) ∧ H(t) �1/p dt = � Q(vn) 0 �� H(t) 0 (α−1(u) ∧ qn)du �1/p dt ≤ � Q(vn) 0 � vnqn + � H(t) vn (α−1(u) ∧ qn)du �1/p dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Note that u < H(t) ⇐⇒ t < Q(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Consequently u < H(t) implies that Q−2(u) < t−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Hence � Q(vn) 0 �qn−1 � k=0 α(k) ∧ H(t) �1/p dt ≤ (vnqn)1/pQ(vn) + � Q(vn) 0 � t−2 � H(t) vn (α−1(u) ∧ qn)Q2(u)du �1/p ≤ (vnqn)1/pQ(vn) + �� 1 vn R(u)Q(u)du �1/p � Q(vn) 0 t−2/pdt ≤ (vnqn)1/pQ(vn) + �� 1 0 R(u)Q(u)du �1/p p(p − 2)−1Q(vn)1−2/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' 9 Set n2 = min{n ≥ 2 : c log log n ≥ 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' It follows that � n≥n2 1 n(V κ)p(n log log n)p/2 �� Q(vn) −Q(vn) J2(t)1/pdt �p ≤ 2 � n≥n2 (4c2p)p (κV )p(n log log n)p/2 qp−2 n � vnqnQp(vn) + 2pQ(vn)p−2 � 1 0 R(u)Q(u)du � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Note that vnqnQ2(vn) = vnα−1(vn)Q2(vn) ≤ � 1 0 R(u)Q(u)du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Hence, since qnMn ≤ mn, we get � n≥n2 1 n(V κ)p(n log log n)p/2 �� Q(vn) −Q(vn) J2(t)1/pdt �p ≤ 4 � 1 0 R(u)Q(u)du � n≥n2 (8c2p)p (κV )p(n log log n)p/2 mp−2 n ≤ 4a−2 � 1 0 R(u)Q(u)du � n≥n2 �8ac2c κV �p log log n n , which is finite by taking into account (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content='17), and if we choose a = (c1κV )/(2c2 √ 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' Indeed, in this case, 8ac2c κV = 2c1 √ 2c κ × 2ac2 √ 2c c1κV = α−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' This ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E5T4oBgHgl3EQfQQ7U/content/2301.05512v1.pdf'} +page_content=' ⋄ References [1] C.' metadata={'source': 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Tanabe1 and Ke Li2 +1Faculty of Environment and Information Sciences, Yokohama National University, Yokohama, Japan +2Department of Computer Science, University of Exeter, EX4 4QF, Exeter, UK +∗Email: rt.ryoji.tanabe@gmail.com, k.li@exeter.ac.uk +Abstract: +Some quality indicators have been proposed for benchmarking preference-based evolu- +tionary multi-objective optimization algorithms using a reference point. Although a systematic review +and analysis of the quality indicators are helpful for both benchmarking and practical decision-making, +neither has been conducted. In this context, first, this paper reviews existing regions of interest and +quality indicators for preference-based evolutionary multi-objective optimization using the reference +point. We point out that each quality indicator was designed for a different region of interest. Then, +this paper investigates the properties of the quality indicators. We demonstrate that an achievement +scalarizing function value is not always consistent with the distance from a solution to the reference +point in the objective space. We observe that the regions of interest can be significantly different +depending on the position of the reference point and the shape of the Pareto front. We identify un- +desirable properties of some quality indicators. We also show that the ranking of preference-based +evolutionary multi-objective optimization algorithms significantly depends on the choice of quality +indicators. +Keywords: +Preference-based evolutionary multi-objective optimization, quality indicators, +benchmarking +1 +Introduction +The ultimate goal of multi-objective optimization is to facilitate multi-criterion decision-making +(MCDM) that finds the Pareto-optimal solution(s) satisfying the decision maker’s aspirations [1]. +Partially due to the population-based property, evolutionary algorithms (EAs) have been widely rec- +ognized as an effective approach for multi-objective optimization, as known as evolutionary multi- +objective optimization (EMO). Conventional EMO algorithms, such as NSGA-II [2], IBEA [3], and +MOEA/D [4], are designed to search for a set of trade-off alternatives that approximate the Pareto- +optimal front (PF) without considering any preference information [5]. Thereafter, this solution set +is handed over to the decision maker (DM) for an a posteriori MCDM to choose the solution(s) of +interest (SOI). On the other hand, if the DM’s preference information is available a priori, it can be +used to navigate an EMO algorithm, also known as preference-based EMO (PBEMO) algorithm [6–8], +to search for a set of “preferred” trade-off solutions lying in a region of interest (ROI), i.e., a subregion +of the PF specified according to the DM’s preference information [9]. From the perspective of EMO, +approximating an ROI can be relatively easier than approximating the complete PF, especially when +having many objectives. From the perspective of MCDM, using the preference information can reduce +the DM’s workload since she/he is only asked to investigate her/his potentially preferred solutions. +Reference point, also known as an aspiration level vector [10], which consists of desirable objective +values specified by the DM, is one of the most popular approaches for expressing the preference in- +formation in the EMO literature [11,12]. Comparing to the other preference formats [7,8], specifying +∗This manuscript is submitted for potential publication. Reviewers can use this version in peer review. +1 +arXiv:2301.12148v1 [cs.NE] 28 Jan 2023 + +a reference point is relatively more intuitive and easier for the DM to elicit her/his preference infor- +mation. Many conventional EMO algorithms have been extended to PBEMO using a reference point, +such as R-NSGA-II [13], PBEA [14], and MOEA/D-NUMS [15]. +Besides algorithm development, in the EMO literature, quality indicators play a vital role in +quantitatively benchmarking EMO algorithms for approximating the whole PF [16–18]. Representative +quality indicators are the hypervolume (HV) [19], the additive ϵ-indicator (Iϵ+) [17], the generational +distance (GD) [20], and the inverted GD (IGD) [21]. It is worth noting that none of these quality +indicators take any preference information into account in quality assessment. Thus, they are not +suitable for evaluating the performance of PBEMO algorithms for approximating the ROI(s). +In +fact, quality assessment on PBEMO algorithms have not received significant attention in the EMO +community until [22]. Early studies mainly relied on visual comparisons which are neither reliable nor +scalable to many objectives [13,23]. On the other hand, some studies around 2010 (e.g., [24,25]) directly +applied conventional quality indicators thus are likely to lead to some misleading conclusions [26]. To +the best of our knowledge, the first quality indicator for PBEMO was proposed in [22]. Although it +has several technical flaws, this quality indicator had a significant impact on the quality assessment +for PBEMO as discussed in [26] and [27]. +Motivation for a review. Although there have been a number of preference-based quality indicators +proposed since [22] in 2010, there is no systematic survey along this line of research. Some survey +papers on quality indicators [16–18] are available, but they are hardly about preference-based ones. +Afsar et al. [12] conducted a survey on how to evaluate the performance of interactive preference- +based multi-objective optimizers, but they focused on experimental conditions rather than quality +indicators. Bechikh et al. [8] presented an exhaustive review of PBEMO algorithms yet on quality +indicators. In addition, some previous studies implicitly proposed quality indicators. For example, +Ruiz et al. [9] proposed WASF-GA. In [9], they also designed a new quality indicator called HVz to +evaluate the performance of WASF-GA. However, they did not clearly state that the design of HVz +was their contribution. For this reason, most previous studies on preference-based quality indicators +(e.g., [26,28,29]) overlooked HVz. +Motivation for analysis. +The properties of quality indicators are not obvious, including which +point set a quality indicator prefers and which quality indicators are consistent/inconsistent with +each other. Thus, it is likely to incorrectly evaluate the performance of EMO algorithms when using a +particular quality indicator. To address this issue, some previous studies analyzed quality indicators in +various ways [30]. Nevertheless, little is known about the properties of quality indicators for PBEMO. +Although some previous studies (e.g., [29,31]) analyzed a few quality indicators for PBEMO, the scale +of their experiments is relatively small. +Apart from this issue of quality indicators, Li et. al. [11] reported the pathological behavior of +some PBEMO algorithms when setting the reference point far from the PF. They showed that R- +NSGA-II [13], r-NSGA-II [24], and R-MEAD2 [32] unexpectedly obtain points on the edge of the +PF, which are far from the reference point. They also showed that only MOEA/D-NUMS [15] works +expectedly even in this case. Since the DM does not know any information about the PF in real-world +applications, these undesirable behavior can be observed in practice. However, the previous study [11] +could not determine what caused these undesirable behavior. +Contributions. +Motivated by the above discussion, first, we review ROIs and preference-based +quality indicators proposed in the literature. We clarify the quality indicators based on their target +ROIs. Then, we analyze the quality indicators. Through an analysis, we address the following four +research questions: +RQ1: Does a Pareto-optimal point with the minimum achievement scalarizing function (ASF) value +always minimize the distance from the reference point? +RQ2: What are the differences of the definitions of ROIs considered in previous studies? How do these +differences influence the behavior of EMO algorithms? +RQ3: What are the properties of existing quality indicators for PBEMO? +RQ4: How does the choice of quality indicator affect the ranking of PBEMO algorithms? +2 + +Outline. The rest of this paper is organized as follows. Section 2 provides some preliminary knowledge +pertinent to this paper. Section 3 reviews and analyzes three ROIs considered in previous studies. +Section 4 reviews 14 preference-based quality indicators developed in the literature. Our experimental +settings are provided in Section 5 while the results are analyzed in Section 6. Section 7 concludes this +paper. +Supplementary file. This paper has a supplementary file. Figure S.∗ and Table S.∗ indicate a figure +and a table in the supplementary file, respectively. +Code availability. The Python implementation of all preference-based quality indicators investigated +in this work is available at https://github.com/ryojitanabe/prefqi. +2 +Preliminaries +2.1 +Multi-objective optimization +The multi-objective optimization problem (MOP) considered in this paper is formulated as: +minimize +F(x) = (f1(x), . . . , fm(x))⊤ +subject to +x ∈ Ω +, +(1) +where x = (x1, . . . , xn)⊤ is an n-dimensional decision vector, and F(x) is an m-dimensional objective +vector. Ω is the feasible set in the decision space Rn and F : Ω → Rm is the corresponding attainable set +in the objective space Rm. A solution x1 is said to Pareto dominate x2 if and only if fi(x1) ≤ fi(x2) for +all i ∈ {1, . . . , m} and fi(x1) < fi(x2) for at least one index i. We denote x1 ≺ x2 when x1 dominates +x2. In addition, x1 is said to weakly Pareto dominate x2 if fi(x1) ≤ fi(x2) for all i ∈ {1, . . . , m}. A +solution x∗ is a Pareto-optimal solution if x∗ is not dominated by any solution in Ω. The set of all +Pareto-optimal solutions in Ω is called the Pareto-optimal set (PS) X ∗ = {x∗ ∈ Ω | ∄x ∈ Ωs.t.x ≺ x∗}. +The image of the PS in Rm is also called the PF F = F(X ∗). The ideal point pideal ∈ Rm consists +of the minimum values of the PF for m objective functions. The nadir point pnadir ∈ Rm consists +of the maximum values of the PF for m objective functions. Thus, for each i ∈ {1, . . . , m}, pideal +i += +minx∈X ∗{fi(x)} and pnadir +i += maxx∈X ∗{fi(x)}. For the sake of simplicity, we refer F(x) as a point +p = (p1, . . . , pm)⊤ ∈ Rm in the rest of this paper. +2.2 +Quality indicators +A quality indicator is a metric I : Rm → R, I : P �→ I(P) that quantitatively evaluates the quality +of a point set P = {pi}µ +i=1 of size µ in terms of at least one of the following four aspects [18]: i) +convergence: the closeness of the points in P to the PF; ii) uniformity: the distribution of the points +in P; iii) spread: the range of the points in P along the PF; and iv) cardinality: the number of non- +dominated points in P. Note that the cardinality has not received much attention in multi-objective +numerical optimization. As discussed in [33], a quality indicator I is said to be Pareto-compliant if +I(P1) < I(P2)1 for any pair of point sets P1 and P2 in Rm, where ∃p ∈ P1, ∀˜p ∈ P2 we have p ≺ ˜p. +Given K > 1 point sets, a unary quality indicator evaluates each one exclusively whereas a K-nary +quality indicator evaluates the K point sets relatively. As discussed in [17] and [18], both unary and +K-nary quality indicators have pros and cons. For example, K-nary quality indicators generally do +not require information about the PF. This is attractive for real-world problems with unknown PFs. +However, K-nary quality indicators only provide information about the relative quality of the K point +sets. That is to say we have to re-calculate the quality indicator values of the K + 1 point sets when +comparing a new point set to the previous K point sets. This might be disadvantageous from the +perspective of sustainable benchmarking of EMO algorithms. +Below, we describe two representative quality indicators widely used in the EMO community. +1In this case, we assume the quality indicator is to be minimized. Otherwise, we have I(P1) > I(P2) instead. +3 + +2.2.1 +Hypervolume (HV) [19] +It measures the volume of the region dominated by the points in P and bounded by the HV-reference +point y ∈ Rm: +HV(P) = Λ +� � +p∈P +{q ∈ Rm | p ≺ q ≺ y} +� +, +(2) +Λ(·) in (2) is the Lebesgue measure. HV(P) can evaluate the quality of P in terms of both convergence +and diversity. HV is to be maximized. +2.2.2 +Inverted generational distance (IGD) [21] +Let S be a set of IGD-reference points uniformly distributed on the PF, IGD measures the average +distance between each IGD-reference point s ∈ S and its nearest point p ∈ P: +IGD(P) = 1 +|S| +�� +s∈S +min +p∈P +� +dist(s, p) +�� +, +(3) +where dist(·, ·) returns the Euclidean distance between two inputs. IGD in (3) is to be minimized. +In general, IGD-reference points in S are uniformly distributed on the PF. Like HV, IGD can also +measure the convergence and diversity of P while it prefers a uniform distribution of points [30]. +Remark 1. The term reference point has been used in various contexts in the EMO literature. To +avoid confusion, we use the term HV-reference point to indicate the reference point for HV. Similarly, +we use the term IGD-reference point to indicate a reference point for IGD. +Remark 2. Note that Pareto-compliant is an important, yet hardly met, characteristic of a quality +indicator. +To the best of our knowledge, HV is the only Pareto-compliant indicator in the EMO +community. This partially explains that HV has been one of the most popular quality indicators. +2.3 +Achievement scalarizing function +Wierzbicki [10] proposed the ASF s : Rm → R, p �→ s(p) in the context of MCDM. Although a number +of scalarizing functions have been proposed for preference-based multi-objective optimization [34], the +ASF is one of the most popular scalarizing functions. Previous studies on PBEMO (e.g., [9, 14, 26]) +used the following two variants of the ASF: +s(p) = +max +i∈{1,...,m} +pi − zi +wi +, +(4) +s(p) = +max +i∈{1,...,m} wi(pi − zi), +(5) +where z ∈ Rm is the reference point specified by the DM. In (4) and (5), w = (w1, . . . , wm)⊤ is the +weight vector that represents the relative importance of each objective function, where �m +i=1 wi = 1 +and wi ≥ 0 for any i. Like in most previous studies, we set w to (1/m, . . . , 1/m)⊤ throughout this +paper. The ASF is order-preserving in terms of the Pareto dominance relation [10], i.e., s(p1) < s(p2) +if p1 ≺ p2. A point with the minimum ASF value is also weakly Pareto optimal with respect to z and +w. +Only the Pareto-optimal point with respect to z and w can be obtained by minimizing the following +augmented version of the ASF (AASF) [34]: +saug(p) = s(p) + ρ +m +� +i=1 +(pi − zi), +(6) +where s in (6) can be either one of the ASFs in (4) and (5). In (6), ρ is a small positive value, e.g, +ρ = 10−6. +4 + +2.4 +PBEMO algorithms +To be self-contained, we give a briefing of six representative PBEMO algorithms considered in our +experiments2: R-NSGA-II [13], r-NSGA-II [24], g-NSGA-II [23], PBEA [14], R-MEAD2 [32], and +MOEA/D-NUMS [15]. As their names suggest, R-NSGA-II, r-NSGA-II, and g-NSGA-II are extended +versions of NSGA-II for preference-based multi-objective optimization. PBEA is a variant of IBEA +while RMEAD2 and MOEA/D-NUMS are scalarizing function-based approaches. Although R-NSGA- +II, r-NSGA-II, and PBEA can handle multiple reference points, we only introduce the case when using +a single reference point. As in [11], we focus on preference-based multi-objective optimization using +a single reference point as the first step. Below, we use the terms “point set” and “population” syn- +onymously. We use the term “preferred region” to describe a sub-region of the PF approximated by a +PBEMO algorithm in the best case. While the ROI is defined by the DM, the preferred region depends +on the PBEMO algorithm. Although some previous studies used these two regions interchangeably, +we strictly distinguish them. +2.4.1 +R-NSGA-II +As in NSGA-II, the primary criterion in environmental selection in R-NSGA-II is based on the non- +domination level of each point p. While the secondary criterion in NSGA-II is based on the crowding +distance, that of R-NSGA-II is based on the following weighted distance to the reference point z: +dR(p) = +� +� +� +� +m +� +i=1 +wi +� +pi − zi +pmax +i +− pmin +i +� +, +(7) +where pmax +i +and pmin +i +are the maximum and minimum values of the i-th objective function fi in the +population P = {pi}µ +i=1 of size µ. The weight vector w in (7) plays a similar role in w in the ASF. +When comparing individuals in the same non-domination level, ties are broken by their dR values. +Thus, non-dominated individuals close to z are likely to survive to the next iteration. +In addition, R-NSGA-II performs ϵ-clearing to maintain the diversity in the population. If the +distance between two individuals in the objective space is less than ϵ, a randomly selected one is +removed from the population. +2.4.2 +r-NSGA-II +It is an extended version of NSGA-II by replacing the Pareto dominance relation with the r-dominance +relation. For two points p1 and p2 in P, p1 is said to r-dominate p2 if one of the following two criteria +is met: 1) p1 ≺ p2; 2) p1 ⊀ p2, p1 ⊁ p2, and dr(p1, p2) < −δ. Here, dr(p1, p2) is defined as follows: +dr(p1, p2) = +dR(p1) − dR(p2) +maxp∈P{dR(p)} − minp∈P{dR(p)}, +(8) +where the definition of dR in (8) can be found in (7). The threshold δ ∈ [0, 1] determines the spread +of individuals in the objective space. When δ = 1, the r-dominance relation is the same as the Pareto +dominance relation. When δ = 0, the r-dominance relation between two non-dominated points p1 and +p2 is determined by their dR values. +2.4.3 +g-NSGA-II +It uses the g-dominance relation instead of the Pareto dominance relation. Let Q be the set of all points +in Rm that dominate the reference point z or are dominated by z, i.e., Q = {p ∈ Rm | p ≺ z or p ≻ z}. +A point p1 is said to g-dominate p2 if one of the following three criteria is met: 1) p1 ∈ Q and p2 /∈ Q; +2) p1, p2 ∈ Q and p1 ≺ p2; 3) p1, p2 /∈ Q and p1 ≺ p2. +Unlike other PBEMO algorithms, g-NSGA-II does not have a control parameter that adjusts the +size of the preferred region. However, g-NSGA-II can obtain only points in a very small region when +2Their behavior was also investigated in [11]. +5 + +z is close to the PF [24]. In contrast, g-NSGA-II is equivalent to NSGA-II when z is very far the +PF [11]. This is because the preferred region Q covers the whole PF when z dominates the ideal point +or is dominated by the nadir point. +2.4.4 +PBEA +It is a variant of IBEA using the binary additive ϵ-indicator (Iϵ+) [17]. For a point set P, the Iϵ+ +value of a point p ∈ P to another point q ∈ P \ {p} is defined as: +Iϵ+(p, q) = +max +i∈{1,...,m}{p′ +i − q′ +i}, +(9) +where p′ and q′ in (9) are normalized versions of p and q based on the maximum and minimum +values of P. The Iϵ+ value is the minimum objective value such that p′ dominates q′. PBEA uses the +following preference-based indicator Ip, which takes into account the AASF value in (6): +Ip(p, q) = Iϵ+(p, q) +s′(p) +, +(10) +s′(p) = saug(p) + δ − min +u∈P{saug(u)}, +(11) +where the previous study [14] used saug with s in (4). In (11), s′(p) is the normalized AASF value of +p by the minimum AASF value of P. In (11), δ controls the extent of the preferred region. A large +Ip value indicates that the corresponding p is preferred. As acknowledged in [14], one drawback of +PBEA is the difficulty in determining the δ value. +2.4.5 +R-MEAD2 +R-MEAD2 [32] is a decomposition-based EMO algorithm using a set of µ weight vectors W = {wi}µ +i=1. +Similar to MOEA/D [4], R-MEAD2 aims to approximate µ Pareto optimal points by simultaneously +minimizing µ scalar optimization problems with W. +R-MEAD2 adaptively adjusts the µ weight +vectors so that the corresponding individuals move toward z. At the beginning of the search, R- +MEAD2 initializes the weight vector set W randomly. +For each iteration, R-MEAD2 selects the +weight vector wc from W, where the corresponding point pc is closest to the reference point z, i.e., +pc = argmin +p∈P +{dist(p, z)}. Then, R-MEAD2 randomly reinitializes W in an m-dimensional hypersphere +of radius r centered at wc. +2.4.6 +MOEA/D-NUMS +It is featured by a nonuniform mapping scheme (NUMS) that shifts µ uniformly distributed weight +vectors toward the reference point z. In particular, the distribution of the µ shifted weight vectors, +denoted as W′, is expected to be biased toward z. In NUMS, a parameter r controls the extent of +W′. In contrast to R-MEAD2, NUMS adjusts the weight vectors in an offline manner. In theory, +NUMS can be incorporated into any decomposition-based EMO algorithm by using W′ instead of the +original W, while MOEA/D-NUMS proposed in [15] is built upon the vanilla MOEA/D. In addition, +MOEA/D-NUMS uses the AASF in (6) with s in (5) instead of a general scalarizing function (e.g., +the Tchebycheff function). +3 +Review of region of interests +Conventional EMO algorithms (e.g., NSGA-II [2]) aim to find a set of µ non-dominated points that +approximate the PF. In contrast, preference-based EMO algorithms (e.g., R-NSGA-II [13]) are de- +signed to search for a set of µ non-dominated points that approximate the ROI. However, as pointed +out in [15], the ROI has been loosely defined in the EMO community. According to the definition +in [15], we define the ROI as a subset of the PF, denoted as R ⊆ F. We assume that the DM is +interested in not only the closest Pareto-optimal point pc∗ to the reference point z but also a set of +6 + +Pareto-optimal points around pc∗. In some cases, the extent of R is defined by a parameter given by +the DM. +Below, Section 3.1 describes three ROIs addressed in previous studies. The reference point z is +said to be feasible if it cannot dominate any Pareto-optimal point. Otherwise, it is said to be infeasible +if z can dominate at least one Pareto-optimal point. Then, Section 3.2 discusses the three ROIs. +3.1 +Definitions of three ROIs +3.1.1 +ROI based on the closest point +This might be the most intuitive ROI that consists of a set of Pareto-optimal points closest to z in +terms of the Euclidean distance (e.g., [27] and [32]). Mathematically, it is defined as: +ROIC = +� +p∗ ∈ F | dist(p∗, pc∗) < ζ +� +, +(12) +where pc∗ = argmin +p∗∈F +{dist(p∗, z)} is the closest Pareto-optimal point to z, and ζ is the radius of the +ROIC. As the example shown in Fig. 1(a), the ROIC is a set of points in a hypersphere of a radius ζ +centered at pc∗ while the extent of the ROIC depends on ζ. We believe that R-NSGA-II, r-NSGA-II, +and R-MEAD2 were designed for the ROIC implicitly. +3.1.2 +ROI based on the ASF +As studied in [14, 15] and [26], this ROI consists of a set of the Pareto-optimal points closest to the +one with the minimum ASF value. Mathematically, it is defined as: +ROIA = {p∗ ∈ F | dist(p∗, pa∗) < ζ}, +(13) +where pa∗ = argmin +p∗∈F +{s(p∗)} is the Pareto-optimal point pa∗ having the minimum ASF value, and +s is the same as in (4). We believe that PBEA and MOEA/D-NUMS were designed for the ROIA +implicitly. +3.1.3 +ROI based on the Pareto dominance relation +This ROI is defined by an extension of the Pareto dominance relation with regard to the DM specified +reference point z (e.g., [9,35–37]). When z is feasible, the ROIP is a set of Pareto-optimal points that +dominate z, i.e., ROIP = {p∗ ∈ F | p∗ ≺ z}. Otherwise, the ROIP is a set of Pareto-optimal points +dominated by z, i.e., ROIP = {p∗ ∈ F | p∗ ≻ z} when z is infeasible. We believe g-NSGA-II was +designed for the ROIP. +3.2 +Discussions +To have an intuitive understanding of these three ROIs, Fig. 1 shows distributions of Pareto-optimal +points in the aforementioned ROIs on the 2-objective DTLZ2 problem with a non-convex PF. In +particular, z0.5 = (0.5, 0.5)⊤ is used as the reference point (denoted as ▲), and we set ζ = 0.1 in the +ROIC and ROIA. As shown in Figs. 1(a) and (b), the ROIC and ROIA are sets of the points in the +hyper-spheres centered at pc∗ and pa∗ (denoted as �), respectively. They are equivalent if the closest +point to z and the point with the minimum ASF value are the same. For this reason, the ROIC and +ROIA may have been considered as the same ROI in the literature. In contrast, as shown in Fig. 1(c), +the ROIP is a set of points dominated by z0.5 and its extent is larger than that of the ROIC and ROIA. +However, it is worth noting that the ROIP does not have any parameter to control its extent as done +in the ROIC and ROIA. Instead, the size of the ROIP depends on the position of z. If it is too close +to the PF, the size of the ROIP can be very small; otherwise it can be very large if z is too far away +from the PF. In the extreme case, if z dominates the ideal point or is dominated by the nadir point, +the ROIP is the same as the PF. Since a DM has little knowledge of the shape of the PF a priori, it +is not recommended to use the ROIP in real-world black-box applications. +7 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(a) ROIC +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(b) ROIA +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(c) ROIP +Figure 1: Distributions of Pareto-optimal points in the three ROIs on the DTLZ2 problem when using +z0.5. +Table 1: Properties of the 14 quality indicators for PBEMO, including the number of point sets K, the type +of target ROI, the convergence to the PF (C-PF), the convergence to the reference point z (C-z), the diversity +(Div), the ability to handle point sets outside a preferred region (Out), no use of information about the PF +(U-PF), and control parameters. +Indicators +K +ROI +C-PF C-z Div Out U-PF +Param. +MASF [14] +unary +ROIA +� +� +� +� +w +MED [38] +unary +ROIC +� +� +IGD-C [32] +unary +ROIC +� +� +� +� +r, S +IGD-A +unary +ROIA +� +� +� +� +w, r, S +IGD-P [36] +unary +ROIP +� +� +� +� +S +HVz [9] +unary +ROIP +� +� +� +� +PR [39] +unary +ROIP +� +PMOD [28] +unary Unclear +� +� +� +� +r, α +IGD-CF [27] K-nary +ROIC +� +� +� +� +r +HV-CF [27] +K-nary +ROIC +� +� +� +� +r, y +PMDA [31] +K-nary Unclear +� +� +� +� +α, γ +R-IGD [26] +K-nary +ROIA +� +� +� +� +r, zw, w, S +R-HV [26] +K-nary +ROIA +� +� +� +� +� +r, zw, w +EH [29] +K-nary Unclear +� +� +� +� +4 +Review of quality indicators +This section reviews 14 quality indicators proposed in the literature for assessing the performance +of PBEMO algorithms. Their properties are summarized in Table 1. According to the definitions +in Section 3.1, we classify the target ROIs of the 14 quality indicators based on their preferred regions. +In particular since the target ROIs of PMOD, PMDA, and EH do not belong to any of the three +ROIs defined in Section 3.1, their ROIs are labeled as unclear. Note that the target ROIs of R-IGD +and R-HV are slightly different from the ROIA, where they are based on a hypercube, instead of a +hypersphere. As shown in Table 1, the previous studies assumed different ROIs. This suggests that +the ROI has not been standardized in the EMO community. +In the following paragraphs, Section 4.1 first discusses the desirable properties as a quality indicator +for PBEMO. Then, Sections 4.2 to 4.11 delineate the underlying mechanisms of 14 quality indicators, +respectively. In particular, the technical details of some quality indicators, including iIGD [40], F- +HV [41], the referential cluster variance indicator [42], and the hull volume indicator [42], are missing. +In addition, the HV-based indicator developed in [22] and the spread-based indicator proposed in [43] +do not consider the preference information from the DM. Therefore, we do not intend to elaborate +them in this paper. +8 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +P1 +P2 +P3 +P4 +P5 +P6 +P7 +P8 +P9 +(a) Distributions of P1 to P9 +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +P10 +(b) Distribution of P10 +Figure 2: Distributions of the 10 point sets on the PF of the DTLZ2 problem when m = 2. +4.1 +Desirable properties of quality indicators +To facilitate our discussion, we generate 10 synthetic point sets, each of which consists of 20 uni- +formly distributed points, along the PF of the 2-objective DTLZ2 problem as shown in Fig. 2. More +specifically, P1 to P5 are distributed on five different subregions of the PF. P6, P7 and P8 are the +shifted versions of P2, P3 and P4 by adding 0.1 to all elements, respectively. Thus, P6, P7, P8 are +dominated by P2, P3, P4, respectively. P9 is on the PF, but the extent of P9 is worse than that of +P3. Unlike the other point sets, the points in P10 are uniformly distributed on the whole PF. Given +z = (0.5, 0.5)⊤ as the DM specified reference point as shown in Fig. 2, P3 is the best point set in Fig. 2 +with regard to the ROIC, ROIA, and ROIP. In this paper, we argue that a desirable preference-based +quality indicator is required to assess the four aspects including i) the convergence to the PF; ii) the +convergence to z; iii) the diversity of trade-off alternatives in a point set; and iv) the ability to handle +point sets outside an ROI. +Remark 3. The term “convergence” of a point set P has not been specified in the context of PBEMO. +As shown in Table 1, we distinguish the convergence to the PF and the convergence to z. In Fig. 2, +P1, . . . , P5, and P9 have a good convergence to the PF. In contrast, only P3 and P9 have a good +convergence to z. +Remark 4. The diversity of a point set in the preferred region is also an important evaluation cri- +terion. If an ROI contains both P3 and P9, a quality indicator should evaluate P3 as having higher +diversity than P9. +Remark 5. Li et al. [26] pointed out that quality indicators should be able to distinguish point sets +outside the ROI. In Fig. 2, P1 and P2 are outside the ROI. However, P2 is closer to the ROI than P1. +The same is true for the relation between P4 and P5. In this case, a quality indicator should evaluate +P2 and P4 as having better quality than P1 and P5. Mohammadi et al. [27] pointed out the importance +of not using information about the PF, which is generally unavailable in real-world problems. As shown +in Table 1, 9 out of the 14 indicators satisfy this criterion. Note that an approximation of the PF or +the ROI found by PBEMO algorithms is available in practice. We believe that the remaining 5 out of +the 14 indicators can address the issue by simply using the approximation. +4.2 +MASF +As done in some previous studies, e.g., [14,44] and [26], the basic idea of this quality indicator is to +use the minimum ASF (MASF) value of P to evaluate the closeness of P to z: +MASF(P) = min +p∈P {s(p)} , +(14) +9 + +where we use s in (4). MASF can evaluate only the two types of convergence. Since MASF does not +consider the other µ−1 points in P, MASF cannot evaluate the diversity of P. As the example shown +in Fig. 2, MASF prefers P9 to P3. +4.3 +MED +As its name suggests, the mean Euclidean distance (MED) measures the average of Euclidean distance +between each point in P to z [38]: +MED(P) = +1 +|P| +� +p∈P +� +� +� +� +m +� +i=1 +� +pi − zi +pnadir +i +− pideal +i +�2 +. +(15) +MED can evaluate how close all points in P are to z and the PF when z is infeasible. Otherwise, it +cannot evaluate the convergence to the PF if z is feasible. This is because MED prefers the non-Pareto +optimal points close to z than the Pareto optimal points. As the example shown in Fig. 2, MED prefers +the dominated P7 over the non-dominated P3 when z = (1.0, 1.0)⊤. +4.4 +IGD-based indicators +Here, we introduce three quality indicators developed upon the IGD metric. In particular, since they +are designed to deal with the ROIC, ROIA, and ROIP defined in Section 3, respectively, they are thus +denoted as IGD-C, IGD-A, and IGD-P accordingly in this paper. Note that both IGD-C and IGD- +P were used in some previous studies [32, 45], and [36], respectively, whereas IGD-A is deliberately +designed in this paper to facilitate our analysis. +In practice, the only difference between the original IGD and its three extensions is the choice of +the IGD-reference point set S. In IGD, IGD-reference points in S are uniformly distributed on the +whole PF. In contrast, IGD-C, IGD-A, and IGD-P use a subset S′ ⊆ S. S′ can also be a subset of +each ROI. In the example in Fig. 1, S′ of IGD-C, IGD-A, and IGD-P are in the ROIC, ROIA, and +ROIP, respectively. Below, for each indicator, we describe how to select S′ from S. +• For IGD-C, we first find the closest point pc to z from S, i.e., pc = argminp∈S{dist(p, z)}. +Then, S′ is a set of all points in the region of a hypersphere of radius r centered at pc, i.e., +S′ = {p ∈ S | dist(p, pc) < r}. +• The only difference between IGD-C and IGD-A is the choice of the center point. First, a point +with the minimum ASF value pa is selected from S, i.e., pa = argminp∈S{s(p)}. We use the +ASF s in (4) in this study. Then, S′ is a set of all points in the region of a hypersphere of radius +r centered at pa, i.e., S′ = {p ∈ S | dist(p, pa) < r}. +• In IGD-P, S′ is selected from S based on the Pareto dominance relation as in the ROIP. If z +is feasible, S′ = {p ∈ S | p ≺ z}. Otherwise, S′ = {p ∈ S | p ≻ z}. Note that IGD-P does not +require the radius r. +4.5 +HVz +This quality indicator was originally named HVq in [9], where q represents the reference point in [9]. +Since this paper denotes the reference point as z, we use the term “HVz” to make the consistency. It +computes the HV value of P in the ROIP. The only difference between HV and HVz is the choice of the +HV-reference point y ∈ Rm as follows. If z is feasible, y = z. If z is infeasible, yi = maxp∗∈ROIP{p∗ +i } +for each i ∈ {1, . . . , m}. Fig. 3 shows the HV-reference point y in HVz when setting z to (0.9, 0.9)⊤ +and (0.5, 0.5)⊤, where the former is feasible while the latter is infeasible. +HVz can evaluate the convergence and diversity of a point set in terms of the ROIP. However, it +cannot handle the point sets outside the ROIP. This is because HV does not consider points dominated +by the HV-reference point y. In the example in Fig. 2, the HVz values of P1, P2, P4, and P5 are 0. +10 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +y = z +(a) Feasible z = (0.9, 0.9)⊤ +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +z +y +(b) Infeasible z = (0.5, 0.5)⊤ +Figure 3: Examples of the HV-reference point y in HVz. +4.6 +PR +The percentage of points in the ROI (PR) evaluates the cardinality of P [39] lying in the DM specified +ROI: +PR(P) = |{p ∈ P ∩ R}| +|P| +× 100%, +(16) +where R is the ROIP defined in [39], though it can be any type of ROI in principle. Note that PR +is the only cardinality-based indicator considered in our study. A large PR means that many points +in the corresponding P are in the ROIP. Like HVz, it is clear that PR cannot distinguish point sets +outside the ROI. +4.7 +PMOD +PMOD consists of two algorithmic steps [28]. First, it maps each point p ∈ P onto a hyperplane +passing through z as: +p′ = p + ((z − p) · ˆz)ˆz, +(17) +where ˆz is the unit vector of z. Then, PMOD aggregates three measurements including i) the distance +between each mapped point p′ and z, ii) the distance between p and the origin o = (0, . . . , 0)⊤, and +iii) the unbiased standard deviation of all mapped points as: +PMOD(P) = +1 +|P| +� +P′∈P′ +� +dist(p′, z) + α dist(p, o) +� ++ SD +� +{dp′}p′∈P′� +, +(18) +where P′ is a set of |P| mapped points. In (18), α is a penalty parameter for mapped points outside +the preferred region of radius r centered at z, where r is a parameter of PMOD. When p′ is inside +the preferred region (i.e., dist(p′, z) ≤ r), α = 1. Otherwise, α > 1, e.g., α is set to 1.5 in [28]. SD +returns the unbiased standard deviation of input values. For each p′ ∈ P′, dp′ in (18) is the minimum +Manhattan distance between p′ and another point q ∈ P′, i.e., dp′ = +min +q∈P′\{p′} +�m +i=1 |p′ +i − qi|. +The smaller PMOD value is, the better quality P is in terms of i) the convergence to z, ii) the +convergence to the origin (not the PF), and iii) the uniformity. Note that PMOD assumes the ideal +point always does not dominate the origin. When the ideal point dominates the origin, PMOD prefers +points far from the PF in view of above ii). Let us consider the shifted point sets in Fig. 2 again, if +the offset is −100, PMOD is likely to prefer P7 to P3 because P7 is closer to the origin than P3. +4.8 +IGD-CF and HV-CF +A user-preference metric based on a composite front (UPCF) [27] is a framework for evaluating the +quality of P. IGD-CF and HV-CF are the UPCF versions of IGD and HV, respectively. Algorithm 1 +11 + +Algorithm 1: IGD-CF and HV-CF +1 PCF ← Select all non-dominated points from P1 ∪ · · · ∪ PK; +2 pc ← argminp∈PCF{dist(p, z)}; +3 Rpref ← {p ∈ Rm | dist(p, pc) < r}; +4 for i ∈ {1, . . . , K} do +5 +IGD-CF(Pi) ← IGD(Pi ∩ Rpref) using PCF as S; +6 +HV-CF(Pi) ← HV(Pi ∩ Rpref); +shows how to calculate the IGD-CF and HV-CF values of K points sets P1, . . . , PK. First, let PCF +be a set of all non-dominated points in the K point sets (line 1), where PCF was called a composite +front in [27]. Then, the closest point pc to z is selected from PCF (line 2). A preferred region Rpref is +defined as the set of all points in the region of a hypersphere of radius r centered at pc (line 3). Note +that Rpref can include dominated points. If PCF = F, Rpref is equivalent to the ROIC shown in Fig. +1(a). +For each i ∈ {1, . . . , K}, the IGD-CF value of Pi is calculated based only on the points in Pi∩Rpref +(line 5). In other words, points outside Rpref are removed from Pi. For example, in Fig. 1(a), points +outside the large dotted circle are removed from a point set. IGD-CF uses PCF as an approximation +of the IGD-reference point set S. The HV-CF value of Pi is calculated in a similar manner (line 6), +where the previous study [27] did not give a rule of thumb to set the HV-reference point y. When the +trimmed Pi is empty, we set its IGD-CF value to ∞ and its HV-CF value to 0 in this study. +Li et al. [26] pointed out that IGD-CF and HV-CF cannot distinguish point sets outside the +preferred region. This is because IGD-CF and HV-CF do not consider any point outside Rpref. In the +example in Fig. 2, all point sets except for P3, P9, and P10 are equally bad, i.e., IGD-CF(Pi) = ∞ +and HV-CF(Pi) = 0 for i ∈ {1, 2, 4, 5, 6, 7, 8}. +4.9 +PMDA +The preference-based metric based on specified distances and angles (PMDA) [31] is built upon the +concept of light beams [46]. It consists of three algorithmic steps. +Step 1: It lets a set of points Q = {qi}m+1 +i=1 on a hyperplane passing through z while m+1 light beams +pass from the origin (0, . . . , 0)⊤ to q1, . . . , qm+1, respectively. For i ∈ {1, . . . , m}, qi is given +as: +qi = z + α (ei − z), +(19) +where ei is the standard-basis vector for the i-th objective function, e.g., e1 = (1, 0)⊤ and +e2 = (0, 1)⊤ for m = 2. In (19), α controls the spread of the light beams. The remaining +qm+1 in Q is set to z. +Step 2: All the points in Q are further shifted as: +Q′ = βQ, +(20) +where β is the minimum objective value in P′ for all m objectives, i.e., β = min +p∈P′{ +min +i∈{1,...,m}pi}3. +Here, P′ is a set of points in ∪K +i=1Pi that are in a preferred region defined by m light beams, +which pass through q1, . . . , qm but do not pass through qm+1 = z. +Step 3: PMDA measures the distance between each point in P and its closest point in Q′ as: +PMDA(P) = +1 +|P| +� +p∈P +� +min +q∈Q′{dist(p, q)} + γθp +� +, +(21) +3 [31] defined that β is the minimum objective value of P, not P′. Since β can be different for different point sets in +this case, this version of PMDA is not reliable. +12 + +Algorithm 2: R-IGD and R-HV +1 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK; +2 for i ∈ {1, . . . , K} do +3 +Pi ← Pi ∩ Pall; +4 +pa ← argminp∈Pi {s(p)}; +5 +Pi ← +� +p ∈ Pi | |pj − pa +j| ≤ r for j ∈ {1, . . . , m} +� +; +6 +k ← argmaxj∈{1,...,m} +� pa +j−zj +zw +j −zj +� +; +7 +piso ← z + +� +pa +k−zk +zw +k −zk +� +(zw − z); +8 +for p ∈ Pi do +9 +p ← p + (piso − pa); +10 +R-IGD(Pi) ← IGD(Pi) using a trimmed S; +11 +R-HV(Pi) ← HV(Pi) using zw; +where γ is a penalty value and was set to 1/π in [31]. In (21), θp is an angle between p and +z. If p is in the preferred region defined by the m light beams passing through q1, . . . , qm, +θp = 0. Thus, points outside the preferred region are penalized. +A small PMDA value indicates that points in the corresponding P are close to the m + 1 points in +Q′ and the preferred region. Thus, PMDA does not evaluate the diversity of P. Note that all elements +of a point are implicitly assumed to be positive in [31]. +4.10 +R-IGD and R-HV +R-metric [26] is a framework that applies general quality indicators to the performance evaluation +of K PBEMO algorithms. R-metric assumes that the DM prefers points along a line from z to the +worst point zw defined by the DM. As recommended in [26], we set zw = z + 2 × u, where u is a +unit vector. We set u = (1/√m, . . . , 1/√m)⊤ in this study. The previous study [26] considered the +R-metric versions of IGD and HV, denoted as R-IGD and R-HV. +Algorithm 2 gives the pseudo code for calculating R-IGD and R-HV. A set of all non-dominated +points Pall are selected from the union of K point sets (line 1). After that, the following steps are +performed for each point set Pi. First, points dominated by any point in Pall are removed from Pi +(line 3). Then, the best point pa is selected from Pi in terms of the ASF (line 4), where the previous +study [26] used s in (4) as the ASF. R-metric defines a preferred region based on a hypercube of +size 2 × r centered at pa. Points outside the preferred region are removed from Pi (line 5). For the +example shown in Fig. 4(a), only the three points in the dotted box are considered for the R-metric +calculation. This trimming operation can penalize a point set that does not fit the preferred region. +Next, R-metric obtains a projection of pa on the line from z to zw by the ASF (lines 6 and 7). This +projection is called an iso-ASF point piso. R-metric transfers all the points in Pi by the direction +vector from piso to pa (lines 8 and 9). For the example shown in Fig. 4(b), the three points are shifted +horizontally. This transfer operation redefines the convergence to the PF as the convergence to z along +a line based on the DM’s preference information. +Finally, the R-IGD and R-HV values of Pi are calculated (lines 10 and 11). More specifically, for +R-IGD, the same trimming operation (lines 4 and 5) is first applied to the IGD-reference point set S in +R-IGD. Thus, all points in S are inside the preferred region. Then, the IGD value of Pi is calculated +using the trimmed S. For R-HV, zw is used as the HV-reference point y. +4.11 +EH +The expanding hypercube metric (EH) [29] is based on the size of a hypercube centered at z that +contains each point and the fraction of points inside the hypercube. While the former evaluates the +convergence of a point set P to z, the latter tries to evaluate the diversity of P. +The pseudo code of calculating the EH for K point sets P1, . . . , PK is given in Algorithm 3. First, +EH removes duplicated points for each point set (lines 1 and 2). In the meanwhile, it also removes +13 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +z +zw +(a) The trimming operation +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +z +zw +(b) The transfer operation +Figure 4: Examples of the two operations in R-metric. In this example, zw = z + 0.7 × u. +dominated points from each point set (lines 3 and 5). Note that if a point set is empty after these +removal operations, its EH value is set to 0 (line 19). +Then, the following steps are performed for each point set Pi. EH calculates the size of a hypercube +centered at z that contains each point p in Pi (lines 10 and 11). Thereafter, all elements in h are +sorted (line 12). Note that |h| = |Pi|. The maximum size hmax in h is maintained for an adjustment +described later (lines 13 and 14). EH calculates “the area under the trade-off curve” ai between the +hypercube size and the fraction (lines 16 and 17). While l/|Pi| is the fraction of points in the l-th +hypercube, “(hl − hl−1)” is the incremental size of the hypercube. Finally, the EH value of each point +set Pi is calculated by adjusting ai using hmax (lines 18 and 20). +A large EH value means the corresponding P has good convergence to z. Due to the operation +for removing dominated points (line 3), EH also implicitly evaluates the convergence of P to the PF. +Since EH does not define a preferred region, EH fails to evaluate the diversity of P in some cases. Let +us consider a set of non-dominated unduplicated points that are close to z and distributed at intervals +of ∆. EH is maximized when ∆ is a positive value as close to zero as possible. For the example shown +in Fig. 2(a), EH prefers P9 to P3. +5 +Experimental Setup +This section introduces the settings used in our experiments including the quality indicators, the +benchmark problems, and preference-based point sets used in our analysis. +5.1 +Quality Indicators +In our experiments, we empirically analyze the performance and properties of 14 quality indicators +reviewed in Section 4. +As a baseline, we also take the results of HV and IGD into account. +In +particular, the implementations of HV and R-IGD and R-HV are taken from pygmo [47] and pymoo [48], +respectively, while the other quality indicators are implemented by Tanabe in Python. The innate +parameters of the 14 quality indicators are set according to the recommendation in their original +paper. For the IGD-based indicators, we uniformly generated 1 000 IGD-reference points on the PF +of a problem. For those HV-based indicators, we set the HV-reference point y in HV and HV-CF to +(1.1, . . . , 1.1)⊤. We set the radius r of a preferred region to 0.1 for all the quality indicators. We also +set the radius ζ of the ROIC and ROIA to 0.1. +5.2 +Benchmark Test Problems +DTLZ1 [49], DTLZ2 [49], convDTLZ2 [50] are chosen to constitute the benchmark test problems, which +have linear, nonconvex, and convex PFs, respectively. To ensure the fairness of our experiments, the +PF of the DTLZ1 problem is normalized to [0, 1]m. As a first attempt to investigate the properties +14 + +Algorithm 3: EH +1 for i ∈ {1, . . . , K} do +2 +Pi ← {p ∈ P | ̸ ∃pdup ∈ P s.t. pdup = p}; +3 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK; +4 for i ∈ {1, . . . , K} do +5 +Pi ← Pi ∩ Pall; +6 hmax ← ∅; +7 for i ∈ {1, . . . , K} do +8 +h ← ∅; +9 +for p ∈ Pi do +10 +h ← maxj∈{1,...,m}{|pj − zj|}; +11 +h ← h ∪ {h}; +12 +h ← Sort all elements in h in ascending order; +13 +hmax ← maxh∈h{h}; +14 +hmax ← hmax ∪ {hmax}; +15 +ai ← 0; +16 +for l ∈ {1, . . . , |Pi|} do +17 +ai ← ai + +l +|Pi| × (hl − hl−1) ; +// h0 = 0 +18 for i ∈ {1, . . . , K} do +19 +if Pi = ∅ then EH(Pi) ← 0 ; +20 +else EH(Pi) ← ai + (maxh∈hmax{h} − hmax +i +) ; +of preference-based quality indicators, we mainly focus on the two-objective scenarios to facilitate the +analysis and discussion about the impact of the distribution of points on the quality indicators. +Remark 6. We are aware of a previous study [29] evaluated the performance of R-NSGA-II and +g-NSGA-II on the DTLZ problems with m ∈ {3, 5, 8, 10, 15, 20} by EH and R-HV. The previous study +discussed the influence of the distribution of m-dimensional points on EH and R-HV using the parallel +coordinates plot. However, the parallel coordinates plot is likely to lead to a wrong conclusion [51]. In +fact, the results in [29] did not show the undesirable property of EH. +5.3 +Experimental Settings +We conduct two types of experiments. +• One is an experiment using the 10 synthetic point sets as shown in Fig. 2. Fig. S.1 shows the +distributions of the 10 point sets on the PF of the DTLZ1 and convDTLZ2 problems. Fig. S.1 +is similar to Fig. 2. +• The other is an experiment using point sets found by the six PBEMO algorithms introduced +in Section 2.4. Note that comprehensive benchmarking of the PBEMO algorithms is beyond the +scope of this paper. Instead, we focus on an analysis of the behavior of the PBEMO algorithms. +This contributes to the understanding of RQ2. Moreover, we also investigate how the choice +of quality indicators influences the rankings of the PBEMO algorithms. This contributes to +addressing RQ4. +In particular, the source code of the PBEMO algorithms are provided by +Li [11] while the weight vectors used in MOEA/D-NUMS are generated by using the source +code provided by Li [15]. Each PBEMO algorithm is independently run 31 times with different +random seeds. +The population size µ is set to 100. +The parameters associated with these +PBEMO algorithms are set according to the recommendations in their original papers, except +PBEA of which δ is set to 0.01 in this study. +15 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +p1 +p25 +p50 +p75 +p100 +(0.1, 0.1) +(-0.1, -0.1) +(a) 100 points +20 +40 +60 +80 +100 +Point IDs +0 +20 +40 +60 +80 +100 +Rankings +z = (0.1, 0.1) +z = (-0.1, -0.1) +(b) Rankings (distance) +20 +40 +60 +80 +100 +Point IDs +0 +20 +40 +60 +80 +100 +Rankings +z = (0.1, 0.1) +z = (-0.1, -0.1) +(c) Rankings (ASF) +-3 -2 -1 0 1 2 3 +f1 +3 +2 +1 +0 +-1 +-2 +-3 +f2 +-1 +0.5 +0 +0.5 +1 +(d) Kendall τ +Figure 5: (a) Distribution of 100 uniformly distributed points, (b) the rankings of the 100 points by +the distance, (c) the ranking of the 100 points by the ASF, and (d) the Kendall τ values on the DTLZ2 +problem. +6 +Results +This section is dedicated to addressing the four RQs raised in Section 1. First, Section 6.1 analyzes the +relation between the distance to the reference point z and the ASF value. Then, Section 6.2 investigates +differences in the three ROIs and the behavior of EMO algorithms. Thereafter, Section 6.3 examines +the properties of the 14 quality indicators using the synthetic point sets shown in Figs. 2 and S.1. +Finally, Section 6.4 analyzes the influence of quality indicators on the rankings of EMO algorithms. +6.1 +Relation between the distance to z and the ASF value +Fig. 5(a) shows 100 uniformly distributed points p1, . . . , p100 on the PF of the DTLZ2 problem. +Fig. 5(a) also shows the two reference points z0.1 = (0.1, 0.1)⊤ and z−0.1 = (−0.1, −0.1)⊤. While z0.1 +is dominated by the ideal point, z−0.1 dominates the ideal point. +As shown in Fig. 5(a), intuitively, the 50-th point p50 on the center of the PF is closest to both +z0.1 and z−0.1. However, this intuition is incorrect. Figs. 5(b) and 5(c) show the rankings of the 100 +points by the Euclidean distance to z and the ASF in (4), respectively. A low ranking means that the +corresponding point is close to z or obtains a small ASF value. As seen from Fig. 5(b), p50 is closest +to z0.1. In contrast, p50 is farthest from z−0.1. The two extreme points (p1 and p100) are closest to +z−0.1. Thus, the closest points to z0.1 and z−0.1 are different. As shown in Fig. 5(c), the rankings by +the ASF are consistent when using either one of z0.1 and z−0.1. This is because both z0.1 and z−0.1 +are in the same direction. +Fig. 5(d) shows the Kendall rank correlation τ value of the distance to z and the ASF value, where +τ ∈ [−1, 1]. In Fig. 5(d), we uniformly generated z from (−3, 3)⊤ to (3, −3)⊤ at intervals of 0.01. Then, +we calculated the τ value for each z. The τ value quantifies the consistency of the two rankings, where +one is based on the distance to the corresponding z, and the other is based on the ASF value. Positive +and negative τ values indicate that the two rankings are consistent and inconsistent, respectively. +As seen from Fig. 5(d), the rankings by the distance to z and the ASF value are inconsistent when +setting z close to the line passing through (0, 0)⊤ and (−3, −3)⊤. We can also see that the rankings +are weakly inconsistent when setting z to other positions. +Note that the inconsistency between the distance to z and the ASF value depends on not only +the position of z, but also the shape of the PF. Figs. S.2 and S.3 show the results on the DTLZ1 +and convDTLZ2 problems, respectively. +As shown in Fig. +S.3(a), we set z to (2, 2)⊤ instead of +(−0.1, −0.1)⊤ for the convDTLZ2 problem. As shown in Figs. S.2(b) and (c), the rankings by the +distance to z and the ASF value on the DTLZ1 problem are always consistent regardless of the +position of z. In contrast, as seen from Figs. S.3(b) and (c), the inconsistency of the rankings can +be observed on the convDTLZ2 problem. While Fig. S.3(b) is similar to Fig. 5(b), Fig. S.3(d) is +opposite from Fig. 5(d). Unlike Fig. 5(d), Fig. S.3(d) indicates that the inconsistency between the +two rankings occurs when z is dominated by the nadir point pnadir on the convex PF. +16 + +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(a) ROIC (z0.1) +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(b) ROIA (z0.1) +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(c) ROIP (z0.1) +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(d) ROIC (z−0.1) +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(e) ROIA (z−0.1) +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(f) ROIP (z−0.1) +Figure 6: Distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using +z0.1 and z−0.1. +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(a) R-NSGA-II +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(b) MOEA/D-NUMS +0 +0.5 +1 +f1 +0 +0.5 +1 +f2 +(c) g-NSGA-II +Figure 7: Distributions of points found by three PBEMO algorithms on the DTLZ2 problem when +using z−0.1. +Answers to RQ1: Our results show that the closest Pareto-optimal point to the reference point z +does not always minimize the ASF. Although it has been believed that minimizing the ASF means +moving closer to z, this is not always correct. We observed that the inconsistency between the +distance to z and the ASF value depends on the position of z and the shape of the PF. Roughly +speaking, the inconsistency can be observed when z dominates the ideal point on a problem with a +nonconvex PF, and z is dominated by the nadir point on a problem with the convex PF. +6.2 +Analysis of the three ROIs +First, this section investigates the differences between the three ROIs. Similar to Fig. 1, Fig. 6 shows +the distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using +z0.1 = (0.1, 0.1)⊤ and z−0.1 = (−0.1, −0.1)⊤. Figs. S.6 and S.7 show the results on the DTLZ1 and +convDTLZ2 problems, respectively. Figs. 6(a) and (b) are exactly the same as Figs. 1(a) and (b), +respectively. Thus, the ROIC and ROIA are the same even when using either one of z0.1 and z0.5. In +contrast, as shown in Figs. 6(d) and (e), the ROIC and ROIA are totally different when using z−0.1. +While the ROIA is on the center of the PF, the ROIC is on either one of the two extreme points (1, 0)⊤ +and (0, 1)⊤. Since the two extreme points (1, 0)⊤ and (0, 1)⊤ are equally close to z−0.1, Fig. 1(d) shows +two possible ROIs. This strange result is due to the inconsistency between the distance to z and the +17 + +ASF value reported in Section 6.1. As seen from Fig. S.7(g), a similar result can be observed on the +convDTLZ2 problem. Results similar to those in Fig. 6(d) can be obtained by using z with a small +Kendall τ value in Fig. 5(d). +Fig. 6(c) significantly differs from Fig. 1(c). The extent and cardinality of the ROIP in Fig. 6(c) +are much larger than those in Fig. 1(c). As shown in Fig. 6(f), the ROIP and the PF are identical +when the reference point dominates the ideal point. The same is true when the reference point is +dominated by the nadir point. In this case, preference-based multi-objective optimization is the same +as general one. The size of the ROIP increases as z moves away from the PF. This undesirable property +of the ROIP is similar to that of the g-dominance relation pointed out in [11]. As seen from Figs. S.6 +and S.7, this undesirable property of the ROIP can be observed on other problems. Since the DM +does not know any information about the PF in practice, it is difficult to set a reference point that is +neither too close nor too far from the PF. +Next, we point out that the differences in the target ROIs caused the unexpected behavior of some +PBEMO algorithms in [11]. Fig. 7 shows the final point sets found by R-NSGA-II, MOEA/D-NUMS, +and g-NSGA-II on the DTLZ2 problem when using z−0.1. The results in Fig. 7 are consistent with the +results in [11]. Figs. S.8–S.16 show the results of the six PBEMO algorithms on the three problems. +As discussed in Section 3.1, R-NSGA-II, MOEA/D-NUMS, and g-NSGA-II aim to approximate the +ROIC, ROIA, and ROIP, respectively. As demonstrated here, the three ROIs can also be different. +For these reasons, the three EMO algorithms found different point sets, as shown in Fig. 7. +The previous study [11] concluded that R-NSGA-II and g-NSGA-II failed to approximate the +“ROI” when z is far from the PF. However, this conclusion is not very correct. Correctly speaking, +as shown in Figs. 7(a) and (c), R-NSGA-II and g-NSGA-II failed to approximate the “ROIA” but +succeeded in approximating the “ROIC” and “ROIP”, respectively. +Answers to RQ2: There are two takeaways generated from the analysis in this subsection. First, +our results showed that the three ROIs can be significantly different depending on the position of the +reference point z and the shape of the PF. We demonstrated that the ROIA is not always a subregion +of the PF closest to z due to the inconsistency observed in Section 6.1. In addition, we found that +the size of the ROIP significantly depends on the position of z. Unless the DM knows the shape +of the PF in advance, it would be better not to use the ROIP. Second, we also demonstrated that +the differences in the three ROIs could cause the unexpected behavior of PBEMO algorithms. For +this reason, we argue the importance to clearly define a target ROI when benchmarking PBEMO +algorithms and performing a practical decision-making. +18 + +Table 2: Rankings of the 10 synthetic point sets on the DTLZ2 problem by the 16 quality indicators when using z0.5 and z−0.1. +(a) z0.5 = (0.5, 0.5)⊤ +MASF +MED +IGD-C +IGD-A +IGD-P +HVz +PR +PMOD +IGD-CF +HV-CF +PMDA +R-IGD +R-HV +EH +HV +IGD +P1 +9 +10 +9 +9 +9 +5 +7 +7 +4 +4 +10 +6 +6 +6 +7 +9 +P2 +5 +5 +5 +5 +5 +5 +7 +3 +4 +4 +5 +4 +4 +4 +3 +4 +P3 +2 +2 +1 +1 +2 +1 +1 +1 +1 +1 +2 +1 +1 +2 +2 +2 +P4 +5 +4 +6 +6 +5 +5 +7 +5 +4 +4 +4 +5 +4 +4 +3 +4 +P5 +9 +9 +10 +10 +9 +5 +7 +9 +4 +4 +9 +7 +6 +6 +7 +9 +P6 +7 +7 +7 +7 +7 +5 +4 +4 +4 +4 +8 +8 +8 +8 +9 +7 +P7 +4 +3 +4 +4 +4 +4 +1 +2 +4 +4 +3 +8 +8 +8 +6 +3 +P8 +7 +7 +8 +8 +7 +5 +4 +8 +4 +4 +7 +8 +8 +8 +10 +8 +P9 +1 +1 +3 +3 +3 +3 +1 +6 +3 +3 +1 +2 +3 +1 +5 +6 +P10 +3 +6 +2 +2 +1 +2 +6 +10 +2 +2 +6 +3 +2 +3 +1 +1 +(b) z−0.1 = (−0.1, −0.1)⊤ +MASF +MED +IGD-C +IGD-A +IGD-P +HVz +PR +PMOD +IGD-CF +HV-CF +PMDA +R-IGD +R-HV +EH +HV +IGD +P1 +9 +2 +1 +9 +9 +8 +1 +7 +1 +1 +10 +6 +7 +6 +7 +9 +P2 +5 +4 +3 +5 +4 +4 +1 +3 +3 +3 +5 +4 +4 +4 +3 +4 +P3 +2 +6 +5 +1 +2 +2 +1 +1 +3 +3 +2 +1 +1 +2 +2 +2 +P4 +5 +4 +8 +6 +4 +4 +1 +5 +3 +3 +4 +5 +4 +4 +3 +4 +P5 +9 +1 +10 +10 +9 +7 +1 +9 +3 +3 +9 +7 +6 +6 +7 +9 +P6 +7 +8 +4 +7 +7 +9 +1 +4 +3 +3 +8 +8 +8 +8 +9 +7 +P7 +4 +10 +6 +4 +3 +6 +1 +2 +3 +3 +3 +8 +8 +8 +6 +3 +P8 +7 +9 +9 +8 +8 +9 +1 +8 +3 +3 +7 +8 +8 +8 +10 +8 +P9 +1 +7 +7 +3 +6 +3 +1 +6 +3 +3 +1 +2 +3 +1 +5 +6 +P10 +3 +3 +2 +2 +1 +1 +1 +10 +2 +2 +6 +3 +2 +3 +1 +1 +19 + +6.3 +Analysis of quality indicators +Table 2 shows the rankings of the 10 point sets P1 to P10 in Fig. 2 by each quality indicator when +using z0.5 = (0.5, 0.5)⊤ and z−0.1 = (−0.1, −0.1)⊤. For the sake of reference, we show the results of +HV and IGD. Table 2 shows which point set is preferred by each quality indicator. For example, P9 +obtains the best MASF value in the 10 point sets. Tables S.1 and S.2 show the results on the DTLZ1 +and convDTLZ2 problems, respectively. We do not intend to elaborate Tables S.1 and S.2, as they +are similar to Table 2. +6.3.1 +Results for z0.5 = (0.5, 0.5)⊤ +First, we discuss the results shown in Table 2(a). P3 is the best in terms of i) the convergence to the +PF, ii) convergence to the reference point z, and iii) diversity. Thus, quality indicators should give +P3 the highest ranking. P3 is ranked highest by 9 out of the 16 quality indicators. However, four +quality indicators (MASF, MED, PMDA, and EH) prefer P9 with the poorest diversity to P3. This is +because they do not take into account the diversity of points as shown in Table 1. Since HV and IGD +do not handle the preference information, they prefer P10 that covers the whole PF. Interestingly, +IGD-P also prefers P10 the most. This is because the IGD-reference points of IGD-P are relatively +widely distributed around the center of PF. +Since PR evaluates only the cardinality, PR cannot distinguish the quality of P3, P7, and P9. +Since IGD is Pareto non-compliant, IGD prefers P7 to P9, where all the points in P7 are dominated +by those in P9. Similarly, PMOD gives P7 the second highest ranking. Since PMOD does not take +into account the convergence to the PF, PMOD can evaluate the quality of point sets inaccurately. +Since IGD-CF and HV-CF cannot distinguish point sets outside their preferred regions, most point +sets obtain the same ranking. Although this undesirable property was already pointed out in [26], this +is the first time to demonstrate that. The same is true for HVz and PR. Since R-IGD, R-HV, and +EH remove dominated points from point sets, they cannot distinguish the three dominated point sets +(P6, P7, and P8). +6.3.2 +Results for z−0.1 = (−0.1, −0.1)⊤ +Next, we discuss the results shown in Table 2(b). In this setting, P1 and P5 are the best in terms of +all three criteria i), ii), and iii). Thus, quality indicators should give P1 or P5 the highest ranking. +However, the rankings by four ASF-based quality indicators (MASF, IGD-A, R-IGD, and R-HV) +are the same in Tables 2(a) and (b). This is because the point with the minimum ASF value and the +ROIA are the same regardless of whether z0.5 or z−0.1 is used, as demonstrated in Sections 6.1 and +6.2. The same is true for PMOD, PMDA, and EH. Thus, these quality indicators fail to evaluate the +convergence of the point sets to the reference point. +In contrast, the rankings by other quality indicators based on the ROIC and ROIP are different in +Tables 2(a) and (b). As demonstrated in Section 6.2, the ROIC is on either one of the two extreme +points when using z−0.1. For this reason, four quality indicators based on the ROIC (MED, IGD-C, +IGD-CF, and HV-CF) prefer P1 or P5 to P3. While IGD-C and IGD-A are perfectly consistent for the +results of z0.5, they are inconsistent for the results of z−0.1. This is due to the inconsistency revealed +in Section 6.1. +As discussed in Section 6.2, when z dominates the ideal point or is dominated by the nadir point, +the ROIP is equivalent to the PF. For this reason, three quality indicators based on the ROIP (IGD-P, +HVz, and PR) cannot handle the DM’s preference information. Thus, like HV and IGD, IGD-P, HVz, +and PR prefer P10 the most. Since IGD-P and IGD use the same IGD-reference point set S, their +rankings are perfectly consistent. Although the position of the HV-reference point y is different in +HVz and HV, their rankings are almost the same. PR cannot distinguish all the 10 point sets. +20 + +Answers to RQ3: Our results indicated that most quality indicators have some undesirable prop- +erties, which have not been noticed even in their corresponding papers. We demonstrated that the +quality indicators based on the ROIA cannot evaluate the convergence of a point set to the reference +point accurately in some cases. We also demonstrated that the quality indicators based on the ROIP +cannot take into account the DM’s preference information. Our results imply that IGD-C may be +the most reliable quality indicator when considering the practical ROIC. However, IGD-C is Pareto +non-compliant. +21 + +Table 3: Rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality indicators when using z0.5 = (0.5, 0.5)⊤. “NUMS”stands for MOEA/D-NUMS. +MASF +MED +IGD-C +IGD-A +IGD-P +HVz +PR +PMOD +IGD-CF +HV-CF +PMDA +R-IGD +R-HV +EH +HV +IGD +R-NSGA-II +3 +2 +6 +6 +5 +6 +4 +4 +6 +6 +3 +5 +5 +1 +5 +5 +r-NSGA-II +4 +3 +3 +3 +4 +4 +1 +3 +3 +4 +2 +3 +3 +3 +4 +4 +g-NSGA-II +5 +5 +1 +1 +1 +1 +1 +1 +1 +1 +5 +2 +1 +6 +2 +2 +PBEA +1 +6 +2 +2 +2 +2 +6 +5 +2 +2 +6 +1 +2 +5 +1 +1 +R-MEAD2 +6 +4 +5 +5 +3 +3 +5 +6 +4 +3 +4 +6 +6 +4 +3 +3 +NUMS +2 +1 +4 +4 +6 +5 +1 +2 +5 +5 +1 +4 +4 +2 +6 +6 +22 + +6.4 +On the rankings of PBEMO algorithms by quality indicators +Table 3 shows the rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality +indicators, where z = (0.5, 0.5)⊤. We calculated the rankings based on the average quality indicator +values of the PBEMO algorithms over 31 runs. Tables S.3–S.5 show the rankings on the DTLZ1, +DTLZ2, and convDTLZ2 problems when using various reference points. Note that we are interested +in the influence of quality indicators on the rankings of the PBEMO algorithms rather than the +rankings themselves. +As shown in Table 3, the rankings of the PBEMO algorithms are different depending on the choice +of the quality indicator. For example, R-NSGA-II performs the best in terms of EH but the worst +in terms of five quality indicators including IGD-C, IGD-A, HVz, IGD-CF, and HV-CF. Likewise, g- +NSGA-II is the worst performer in terms of EH but it is the best algorithm when considering the other +nine quality indicators. As shown in Table. S.5(b), R-MEAD2 performs the best on the convDTLZ2 +problem in terms of EH. In summary, our results suggest that any PBEMO algorithm can obtain the +best ranking depending on the choice of the quality indicator. +These observations can be explained as the coupling relationship between innate mechanism of +the PBEMO algorithms and the quality indicators. As demonstrated in Section 6.2, each PBEMO +algorithm approximates its target ROI embedded by its designer. As investigated in Section 6.3, each +quality indicator prefers a point set that approximates its target ROI well. Thus, when benchmarking +PBEMO algorithms, it is important to clarify which type of ROI the DM wants to approximate +and select a suitable quality indicator. For example, if the DM wants to approximate the ROIA, +she/he should select either of IGD-A, R-IGD, and R-HV. Otherwise, the DM can overestimate or +underestimate the performance of PBEMO algorithms. +Answers to RQ4: Our results showed that the choice of the quality indicator significantly influ- +ences the performance rankings of EMO algorithms. For example, as seen from Table 3, PBEA +performs the worst in terms of PMDA but the best in terms of R-IGD. This means that any PBEMO +algorithm can possibly be ranked as the best (or the worst) depending on the choice of the quality +indicator. We also discussed how to conduct meaningful benchmarking of PBEMO algorithms. +7 +Conclusion +In this paper, we first reviewed the 3 ROIs and 14 existing quality indicators for PBEMO algorithms +using the reference point. Different from the descriptions in their corresponding papers, we classified +the properties of the quality indicators from the perspective of their working principle. As a result, we +found that some quality indicators have undesirable properties. For example, PMDA and EH cannot +evaluate the diversity of a point set. We also discussed the target ROI of each quality indicator. +Next, we empirically analyzed the performance and properties of those 14 quality indicators to +address 4 RQs (RQ1 to RQ4). Our findings are helpful for benchmarking PBEMO algorithms and +decision-making in real-world problems. In any case, we argue the importance of determining a target +ROI first of all. Our results suggested the use of the ROIC. Afterward, a researcher and the DM should +select a PBEMO algorithm and quality indicator based on their target ROI. For example, R-NSGA-II +aims to approximate the ROIC. In contrast, HVz is to evaluate how a point set approximates the +ROIP. Thus, HVz is not suitable for evaluating the performance of R-NSGA-II. +As demonstrated in the three IGD variants (IGD-C, IGD-A, and IGD-P), we believe that the +target ROI of some quality indicators can be changed easily. +For example, the target ROI of R- +IGD can be changed from the ROIA to the ROIC by revising the line 4 in Algorithm 2, i.e., pc = +argminp∈Pi{dist(p, z)}. An investigation of this concept is an avenue for future work. Note that +the analysis conducted in this paper focused on quality indicators for PBEMO algorithms using the +reference point. It is questionable and important to extend our analysis for other preference-based +optimization (e.g., a value function) in future research. There is room for discussion about a systematic +benchmarking methodology for PBEMO. +23 + +Acknowledgment +Tanabe was supported by JSPS KAKENHI Grant Number 21K17824 and LEADER, MEXT, Japan. +Li was supported by UKRI Future Leaders Fellowship (MR/S017062/1, MR/X011135/1), NSFC +(62076056), EPSRC (2404317), Royal Society (IES/R2/212077) and Amazon Research Award. +References +[1] K. Miettinen, Nonlinear Multiobjective Optimization. +Springer, 1998. +[2] K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, “A fast and elitist multiobjective genetic +algorithm: NSGA-II,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, 2002. +[3] E. Zitzler and S. K¨unzli, “Indicator-based selection in multiobjective search,” in Parallel Problem +Solving from Nature (PPSN), 2004, pp. 832–842. +[4] Q. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposi- +tion,” IEEE Trans. Evol. Comput., vol. 11, no. 6, pp. 712–731, 2007. +[5] K. 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Mag., vol. 12, no. 4, pp. 88–100, 2017. +27 + diff --git a/79FLT4oBgHgl3EQfsi_P/content/tmp_files/load_file.txt b/79FLT4oBgHgl3EQfsi_P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b415b0e009377a38693ba43b1c548882c41f98e --- /dev/null +++ b/79FLT4oBgHgl3EQfsi_P/content/tmp_files/load_file.txt @@ -0,0 +1,1931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf,len=1930 +page_content='Quality Indicators for Preference-based Evolutionary Multi-objective Optimization Using a Reference Point: A Review and Analysis∗ Ryoji Tanabe1 and Ke Li2 1Faculty of Environment and Information Sciences, Yokohama National University, Yokohama, Japan 2Department of Computer Science, University of Exeter, EX4 4QF, Exeter, UK ∗Email: rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='ryoji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='tanabe@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='com, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='li@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='uk Abstract: Some quality indicators have been proposed for benchmarking preference-based evolu- tionary multi-objective optimization algorithms using a reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although a systematic review and analysis of the quality indicators are helpful for both benchmarking and practical decision-making, neither has been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this context, first, this paper reviews existing regions of interest and quality indicators for preference-based evolutionary multi-objective optimization using the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We point out that each quality indicator was designed for a different region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, this paper investigates the properties of the quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We demonstrate that an achievement scalarizing function value is not always consistent with the distance from a solution to the reference point in the objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We observe that the regions of interest can be significantly different depending on the position of the reference point and the shape of the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We identify un- desirable properties of some quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We also show that the ranking of preference-based evolutionary multi-objective optimization algorithms significantly depends on the choice of quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Keywords: Preference-based evolutionary multi-objective optimization, quality indicators, benchmarking 1 Introduction The ultimate goal of multi-objective optimization is to facilitate multi-criterion decision-making (MCDM) that finds the Pareto-optimal solution(s) satisfying the decision maker’s aspirations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Partially due to the population-based property, evolutionary algorithms (EAs) have been widely rec- ognized as an effective approach for multi-objective optimization, as known as evolutionary multi- objective optimization (EMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Conventional EMO algorithms, such as NSGA-II [2], IBEA [3], and MOEA/D [4], are designed to search for a set of trade-off alternatives that approximate the Pareto- optimal front (PF) without considering any preference information [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thereafter, this solution set is handed over to the decision maker (DM) for an a posteriori MCDM to choose the solution(s) of interest (SOI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' On the other hand, if the DM’s preference information is available a priori, it can be used to navigate an EMO algorithm, also known as preference-based EMO (PBEMO) algorithm [6–8], to search for a set of “preferred” trade-off solutions lying in a region of interest (ROI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', a subregion of the PF specified according to the DM’s preference information [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' From the perspective of EMO, approximating an ROI can be relatively easier than approximating the complete PF, especially when having many objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' From the perspective of MCDM, using the preference information can reduce the DM’s workload since she/he is only asked to investigate her/his potentially preferred solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Reference point, also known as an aspiration level vector [10], which consists of desirable objective values specified by the DM, is one of the most popular approaches for expressing the preference in- formation in the EMO literature [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Comparing to the other preference formats [7,8], specifying ∗This manuscript is submitted for potential publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Reviewers can use this version in peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='12148v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='NE] 28 Jan 2023 a reference point is relatively more intuitive and easier for the DM to elicit her/his preference infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Many conventional EMO algorithms have been extended to PBEMO using a reference point, such as R-NSGA-II [13], PBEA [14], and MOEA/D-NUMS [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Besides algorithm development, in the EMO literature, quality indicators play a vital role in quantitatively benchmarking EMO algorithms for approximating the whole PF [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Representative quality indicators are the hypervolume (HV) [19], the additive ϵ-indicator (Iϵ+) [17], the generational distance (GD) [20], and the inverted GD (IGD) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' It is worth noting that none of these quality indicators take any preference information into account in quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, they are not suitable for evaluating the performance of PBEMO algorithms for approximating the ROI(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In fact, quality assessment on PBEMO algorithms have not received significant attention in the EMO community until [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Early studies mainly relied on visual comparisons which are neither reliable nor scalable to many objectives [13,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' On the other hand, some studies around 2010 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [24,25]) directly applied conventional quality indicators thus are likely to lead to some misleading conclusions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' To the best of our knowledge, the first quality indicator for PBEMO was proposed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although it has several technical flaws, this quality indicator had a significant impact on the quality assessment for PBEMO as discussed in [26] and [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Motivation for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although there have been a number of preference-based quality indicators proposed since [22] in 2010, there is no systematic survey along this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Some survey papers on quality indicators [16–18] are available, but they are hardly about preference-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Afsar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [12] conducted a survey on how to evaluate the performance of interactive preference- based multi-objective optimizers, but they focused on experimental conditions rather than quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Bechikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [8] presented an exhaustive review of PBEMO algorithms yet on quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, some previous studies implicitly proposed quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [9] proposed WASF-GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In [9], they also designed a new quality indicator called HVz to evaluate the performance of WASF-GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, they did not clearly state that the design of HVz was their contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For this reason, most previous studies on preference-based quality indicators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [26,28,29]) overlooked HVz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Motivation for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The properties of quality indicators are not obvious, including which point set a quality indicator prefers and which quality indicators are consistent/inconsistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, it is likely to incorrectly evaluate the performance of EMO algorithms when using a particular quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' To address this issue, some previous studies analyzed quality indicators in various ways [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Nevertheless, little is known about the properties of quality indicators for PBEMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although some previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [29,31]) analyzed a few quality indicators for PBEMO, the scale of their experiments is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Apart from this issue of quality indicators, Li et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [11] reported the pathological behavior of some PBEMO algorithms when setting the reference point far from the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' They showed that R- NSGA-II [13], r-NSGA-II [24], and R-MEAD2 [32] unexpectedly obtain points on the edge of the PF, which are far from the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' They also showed that only MOEA/D-NUMS [15] works expectedly even in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since the DM does not know any information about the PF in real-world applications, these undesirable behavior can be observed in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, the previous study [11] could not determine what caused these undesirable behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Motivated by the above discussion, first, we review ROIs and preference-based quality indicators proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We clarify the quality indicators based on their target ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, we analyze the quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Through an analysis, we address the following four research questions: RQ1: Does a Pareto-optimal point with the minimum achievement scalarizing function (ASF) value always minimize the distance from the reference point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' RQ2: What are the differences of the definitions of ROIs considered in previous studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' How do these differences influence the behavior of EMO algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' RQ3: What are the properties of existing quality indicators for PBEMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' RQ4: How does the choice of quality indicator affect the ranking of PBEMO algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Section 2 provides some preliminary knowledge pertinent to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Section 3 reviews and analyzes three ROIs considered in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Section 4 reviews 14 preference-based quality indicators developed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Our experimental settings are provided in Section 5 while the results are analyzed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Section 7 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This paper has a supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figure S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='∗ and Table S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='∗ indicate a figure and a table in the supplementary file, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Code availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The Python implementation of all preference-based quality indicators investigated in this work is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='com/ryojitanabe/prefqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Multi-objective optimization The multi-objective optimization problem (MOP) considered in this paper is formulated as: minimize F(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , fm(x))⊤ subject to x ∈ Ω , (1) where x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , xn)⊤ is an n-dimensional decision vector, and F(x) is an m-dimensional objective vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Ω is the feasible set in the decision space Rn and F : Ω → Rm is the corresponding attainable set in the objective space Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A solution x1 is said to Pareto dominate x2 if and only if fi(x1) ≤ fi(x2) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m} and fi(x1) < fi(x2) for at least one index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We denote x1 ≺ x2 when x1 dominates x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, x1 is said to weakly Pareto dominate x2 if fi(x1) ≤ fi(x2) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A solution x∗ is a Pareto-optimal solution if x∗ is not dominated by any solution in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The set of all Pareto-optimal solutions in Ω is called the Pareto-optimal set (PS) X ∗ = {x∗ ∈ Ω | ∄x ∈ Ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='x ≺ x∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The image of the PS in Rm is also called the PF F = F(X ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The ideal point pideal ∈ Rm consists of the minimum values of the PF for m objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The nadir point pnadir ∈ Rm consists of the maximum values of the PF for m objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m}, pideal i = minx∈X ∗{fi(x)} and pnadir i = maxx∈X ∗{fi(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the sake of simplicity, we refer F(x) as a point p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , pm)⊤ ∈ Rm in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Quality indicators A quality indicator is a metric I : Rm → R, I : P �→ I(P) that quantitatively evaluates the quality of a point set P = {pi}µ i=1 of size µ in terms of at least one of the following four aspects [18]: i) convergence: the closeness of the points in P to the PF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' ii) uniformity: the distribution of the points in P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' iii) spread: the range of the points in P along the PF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' and iv) cardinality: the number of non- dominated points in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that the cardinality has not received much attention in multi-objective numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As discussed in [33], a quality indicator I is said to be Pareto-compliant if I(P1) < I(P2)1 for any pair of point sets P1 and P2 in Rm, where ∃p ∈ P1, ∀˜p ∈ P2 we have p ≺ ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Given K > 1 point sets, a unary quality indicator evaluates each one exclusively whereas a K-nary quality indicator evaluates the K point sets relatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As discussed in [17] and [18], both unary and K-nary quality indicators have pros and cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, K-nary quality indicators generally do not require information about the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is attractive for real-world problems with unknown PFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, K-nary quality indicators only provide information about the relative quality of the K point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' That is to say we have to re-calculate the quality indicator values of the K + 1 point sets when comparing a new point set to the previous K point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This might be disadvantageous from the perspective of sustainable benchmarking of EMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Below, we describe two representative quality indicators widely used in the EMO community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1In this case, we assume the quality indicator is to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, we have I(P1) > I(P2) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Hypervolume (HV) [19] It measures the volume of the region dominated by the points in P and bounded by the HV-reference point y ∈ Rm: HV(P) = Λ � � p∈P {q ∈ Rm | p ≺ q ≺ y} � , (2) Λ(·) in (2) is the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' HV(P) can evaluate the quality of P in terms of both convergence and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' HV is to be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Inverted generational distance (IGD) [21] Let S be a set of IGD-reference points uniformly distributed on the PF, IGD measures the average distance between each IGD-reference point s ∈ S and its nearest point p ∈ P: IGD(P) = 1 |S| �� s∈S min p∈P � dist(s, p) �� , (3) where dist(·, ·) returns the Euclidean distance between two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' IGD in (3) is to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In general, IGD-reference points in S are uniformly distributed on the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Like HV, IGD can also measure the convergence and diversity of P while it prefers a uniform distribution of points [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The term reference point has been used in various contexts in the EMO literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' To avoid confusion, we use the term HV-reference point to indicate the reference point for HV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Similarly, we use the term IGD-reference point to indicate a reference point for IGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that Pareto-compliant is an important, yet hardly met, characteristic of a quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' To the best of our knowledge, HV is the only Pareto-compliant indicator in the EMO community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This partially explains that HV has been one of the most popular quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 Achievement scalarizing function Wierzbicki [10] proposed the ASF s : Rm → R, p �→ s(p) in the context of MCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although a number of scalarizing functions have been proposed for preference-based multi-objective optimization [34], the ASF is one of the most popular scalarizing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Previous studies on PBEMO (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [9, 14, 26]) used the following two variants of the ASF: s(p) = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m} pi − zi wi , (4) s(p) = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m} wi(pi − zi), (5) where z ∈ Rm is the reference point specified by the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (4) and (5), w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , wm)⊤ is the weight vector that represents the relative importance of each objective function, where �m i=1 wi = 1 and wi ≥ 0 for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Like in most previous studies, we set w to (1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , 1/m)⊤ throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The ASF is order-preserving in terms of the Pareto dominance relation [10], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', s(p1) < s(p2) if p1 ≺ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A point with the minimum ASF value is also weakly Pareto optimal with respect to z and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Only the Pareto-optimal point with respect to z and w can be obtained by minimizing the following augmented version of the ASF (AASF) [34]: saug(p) = s(p) + ρ m � i=1 (pi − zi), (6) where s in (6) can be either one of the ASFs in (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (6), ρ is a small positive value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g, ρ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4 PBEMO algorithms To be self-contained, we give a briefing of six representative PBEMO algorithms considered in our experiments2: R-NSGA-II [13], r-NSGA-II [24], g-NSGA-II [23], PBEA [14], R-MEAD2 [32], and MOEA/D-NUMS [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As their names suggest, R-NSGA-II, r-NSGA-II, and g-NSGA-II are extended versions of NSGA-II for preference-based multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' PBEA is a variant of IBEA while RMEAD2 and MOEA/D-NUMS are scalarizing function-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although R-NSGA- II, r-NSGA-II, and PBEA can handle multiple reference points, we only introduce the case when using a single reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As in [11], we focus on preference-based multi-objective optimization using a single reference point as the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Below, we use the terms “point set” and “population” syn- onymously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We use the term “preferred region” to describe a sub-region of the PF approximated by a PBEMO algorithm in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While the ROI is defined by the DM, the preferred region depends on the PBEMO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although some previous studies used these two regions interchangeably, we strictly distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 R-NSGA-II As in NSGA-II, the primary criterion in environmental selection in R-NSGA-II is based on the non- domination level of each point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While the secondary criterion in NSGA-II is based on the crowding distance, that of R-NSGA-II is based on the following weighted distance to the reference point z: dR(p) = � � � � m � i=1 wi � pi − zi pmax i − pmin i � , (7) where pmax i and pmin i are the maximum and minimum values of the i-th objective function fi in the population P = {pi}µ i=1 of size µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The weight vector w in (7) plays a similar role in w in the ASF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When comparing individuals in the same non-domination level, ties are broken by their dR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, non-dominated individuals close to z are likely to survive to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, R-NSGA-II performs ϵ-clearing to maintain the diversity in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If the distance between two individuals in the objective space is less than ϵ, a randomly selected one is removed from the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 r-NSGA-II It is an extended version of NSGA-II by replacing the Pareto dominance relation with the r-dominance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For two points p1 and p2 in P, p1 is said to r-dominate p2 if one of the following two criteria is met: 1) p1 ≺ p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2) p1 ⊀ p2, p1 ⊁ p2, and dr(p1, p2) < −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Here, dr(p1, p2) is defined as follows: dr(p1, p2) = dR(p1) − dR(p2) maxp∈P{dR(p)} − minp∈P{dR(p)}, (8) where the definition of dR in (8) can be found in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The threshold δ ∈ [0, 1] determines the spread of individuals in the objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When δ = 1, the r-dominance relation is the same as the Pareto dominance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When δ = 0, the r-dominance relation between two non-dominated points p1 and p2 is determined by their dR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 g-NSGA-II It uses the g-dominance relation instead of the Pareto dominance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Let Q be the set of all points in Rm that dominate the reference point z or are dominated by z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', Q = {p ∈ Rm | p ≺ z or p ≻ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A point p1 is said to g-dominate p2 if one of the following three criteria is met: 1) p1 ∈ Q and p2 /∈ Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2) p1, p2 ∈ Q and p1 ≺ p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3) p1, p2 /∈ Q and p1 ≺ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Unlike other PBEMO algorithms, g-NSGA-II does not have a control parameter that adjusts the size of the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, g-NSGA-II can obtain only points in a very small region when 2Their behavior was also investigated in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5 z is close to the PF [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, g-NSGA-II is equivalent to NSGA-II when z is very far the PF [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because the preferred region Q covers the whole PF when z dominates the ideal point or is dominated by the nadir point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4 PBEA It is a variant of IBEA using the binary additive ϵ-indicator (Iϵ+) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For a point set P, the Iϵ+ value of a point p ∈ P to another point q ∈ P \\ {p} is defined as: Iϵ+(p, q) = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m}{p′ i − q′ i}, (9) where p′ and q′ in (9) are normalized versions of p and q based on the maximum and minimum values of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The Iϵ+ value is the minimum objective value such that p′ dominates q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' PBEA uses the following preference-based indicator Ip, which takes into account the AASF value in (6): Ip(p, q) = Iϵ+(p, q) s′(p) , (10) s′(p) = saug(p) + δ − min u∈P{saug(u)}, (11) where the previous study [14] used saug with s in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (11), s′(p) is the normalized AASF value of p by the minimum AASF value of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (11), δ controls the extent of the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A large Ip value indicates that the corresponding p is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As acknowledged in [14], one drawback of PBEA is the difficulty in determining the δ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 R-MEAD2 R-MEAD2 [32] is a decomposition-based EMO algorithm using a set of µ weight vectors W = {wi}µ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Similar to MOEA/D [4], R-MEAD2 aims to approximate µ Pareto optimal points by simultaneously minimizing µ scalar optimization problems with W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' R-MEAD2 adaptively adjusts the µ weight vectors so that the corresponding individuals move toward z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' At the beginning of the search, R- MEAD2 initializes the weight vector set W randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For each iteration, R-MEAD2 selects the weight vector wc from W, where the corresponding point pc is closest to the reference point z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', pc = argmin p∈P {dist(p, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, R-MEAD2 randomly reinitializes W in an m-dimensional hypersphere of radius r centered at wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='6 MOEA/D-NUMS It is featured by a nonuniform mapping scheme (NUMS) that shifts µ uniformly distributed weight vectors toward the reference point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, the distribution of the µ shifted weight vectors, denoted as W′, is expected to be biased toward z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In NUMS, a parameter r controls the extent of W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast to R-MEAD2, NUMS adjusts the weight vectors in an offline manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In theory, NUMS can be incorporated into any decomposition-based EMO algorithm by using W′ instead of the original W, while MOEA/D-NUMS proposed in [15] is built upon the vanilla MOEA/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, MOEA/D-NUMS uses the AASF in (6) with s in (5) instead of a general scalarizing function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', the Tchebycheff function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3 Review of region of interests Conventional EMO algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', NSGA-II [2]) aim to find a set of µ non-dominated points that approximate the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, preference-based EMO algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', R-NSGA-II [13]) are de- signed to search for a set of µ non-dominated points that approximate the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, as pointed out in [15], the ROI has been loosely defined in the EMO community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' According to the definition in [15], we define the ROI as a subset of the PF, denoted as R ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We assume that the DM is interested in not only the closest Pareto-optimal point pc∗ to the reference point z but also a set of 6 Pareto-optimal points around pc∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In some cases, the extent of R is defined by a parameter given by the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Below, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 describes three ROIs addressed in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The reference point z is said to be feasible if it cannot dominate any Pareto-optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, it is said to be infeasible if z can dominate at least one Pareto-optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 discusses the three ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Definitions of three ROIs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 ROI based on the closest point This might be the most intuitive ROI that consists of a set of Pareto-optimal points closest to z in terms of the Euclidean distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [27] and [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Mathematically, it is defined as: ROIC = � p∗ ∈ F | dist(p∗, pc∗) < ζ � , (12) where pc∗ = argmin p∗∈F {dist(p∗, z)} is the closest Pareto-optimal point to z, and ζ is the radius of the ROIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(a), the ROIC is a set of points in a hypersphere of a radius ζ centered at pc∗ while the extent of the ROIC depends on ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We believe that R-NSGA-II, r-NSGA-II, and R-MEAD2 were designed for the ROIC implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 ROI based on the ASF As studied in [14, 15] and [26], this ROI consists of a set of the Pareto-optimal points closest to the one with the minimum ASF value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Mathematically, it is defined as: ROIA = {p∗ ∈ F | dist(p∗, pa∗) < ζ}, (13) where pa∗ = argmin p∗∈F {s(p∗)} is the Pareto-optimal point pa∗ having the minimum ASF value, and s is the same as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We believe that PBEA and MOEA/D-NUMS were designed for the ROIA implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 ROI based on the Pareto dominance relation This ROI is defined by an extension of the Pareto dominance relation with regard to the DM specified reference point z (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [9,35–37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When z is feasible, the ROIP is a set of Pareto-optimal points that dominate z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', ROIP = {p∗ ∈ F | p∗ ≺ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, the ROIP is a set of Pareto-optimal points dominated by z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', ROIP = {p∗ ∈ F | p∗ ≻ z} when z is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We believe g-NSGA-II was designed for the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Discussions To have an intuitive understanding of these three ROIs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1 shows distributions of Pareto-optimal points in the aforementioned ROIs on the 2-objective DTLZ2 problem with a non-convex PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ is used as the reference point (denoted as ▲), and we set ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 in the ROIC and ROIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(a) and (b), the ROIC and ROIA are sets of the points in the hyper-spheres centered at pc∗ and pa∗ (denoted as �), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' They are equivalent if the closest point to z and the point with the minimum ASF value are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For this reason, the ROIC and ROIA may have been considered as the same ROI in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(c), the ROIP is a set of points dominated by z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 and its extent is larger than that of the ROIC and ROIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, it is worth noting that the ROIP does not have any parameter to control its extent as done in the ROIC and ROIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Instead, the size of the ROIP depends on the position of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If it is too close to the PF, the size of the ROIP can be very small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' otherwise it can be very large if z is too far away from the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the extreme case, if z dominates the ideal point or is dominated by the nadir point, the ROIP is the same as the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since a DM has little knowledge of the shape of the PF a priori, it is not recommended to use the ROIP in real-world black-box applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (a) ROIC 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (b) ROIA 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (c) ROIP Figure 1: Distributions of Pareto-optimal points in the three ROIs on the DTLZ2 problem when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Table 1: Properties of the 14 quality indicators for PBEMO, including the number of point sets K, the type of target ROI, the convergence to the PF (C-PF), the convergence to the reference point z (C-z), the diversity (Div), the ability to handle point sets outside a preferred region (Out), no use of information about the PF (U-PF), and control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Indicators K ROI C-PF C-z Div Out U-PF Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' MASF [14] unary ROIA � � � � w MED [38] unary ROIC � � IGD-C [32] unary ROIC � � � � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S IGD-A unary ROIA � � � � w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S IGD-P [36] unary ROIP � � � � S HVz [9] unary ROIP � � � � PR [39] unary ROIP � PMOD [28] unary Unclear � � � � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' α IGD-CF [27] K-nary ROIC � � � � r HV-CF [27] K-nary ROIC � � � � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' y PMDA [31] K-nary Unclear � � � � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' γ R-IGD [26] K-nary ROIA � � � � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' zw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S R-HV [26] K-nary ROIA � � � � � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' zw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' w EH [29] K-nary Unclear � � � � 4 Review of quality indicators This section reviews 14 quality indicators proposed in the literature for assessing the performance of PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Their properties are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' According to the definitions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, we classify the target ROIs of the 14 quality indicators based on their preferred regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular since the target ROIs of PMOD, PMDA, and EH do not belong to any of the three ROIs defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, their ROIs are labeled as unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that the target ROIs of R-IGD and R-HV are slightly different from the ROIA, where they are based on a hypercube, instead of a hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Table 1, the previous studies assumed different ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This suggests that the ROI has not been standardized in the EMO community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the following paragraphs, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 first discusses the desirable properties as a quality indicator for PBEMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='11 delineate the underlying mechanisms of 14 quality indicators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, the technical details of some quality indicators, including iIGD [40], F- HV [41], the referential cluster variance indicator [42], and the hull volume indicator [42], are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, the HV-based indicator developed in [22] and the spread-based indicator proposed in [43] do not consider the preference information from the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Therefore, we do not intend to elaborate them in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 P1 P2 P3 P4 P5 P6 P7 P8 P9 (a) Distributions of P1 to P9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 P10 (b) Distribution of P10 Figure 2: Distributions of the 10 point sets on the PF of the DTLZ2 problem when m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Desirable properties of quality indicators To facilitate our discussion, we generate 10 synthetic point sets, each of which consists of 20 uni- formly distributed points, along the PF of the 2-objective DTLZ2 problem as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' More specifically, P1 to P5 are distributed on five different subregions of the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' P6, P7 and P8 are the shifted versions of P2, P3 and P4 by adding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 to all elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, P6, P7, P8 are dominated by P2, P3, P4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' P9 is on the PF, but the extent of P9 is worse than that of P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Unlike the other point sets, the points in P10 are uniformly distributed on the whole PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Given z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ as the DM specified reference point as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, P3 is the best point set in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 with regard to the ROIC, ROIA, and ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this paper, we argue that a desirable preference-based quality indicator is required to assess the four aspects including i) the convergence to the PF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' ii) the convergence to z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' iii) the diversity of trade-off alternatives in a point set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' and iv) the ability to handle point sets outside an ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The term “convergence” of a point set P has not been specified in the context of PBEMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Table 1, we distinguish the convergence to the PF and the convergence to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , P5, and P9 have a good convergence to the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, only P3 and P9 have a good convergence to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The diversity of a point set in the preferred region is also an important evaluation cri- terion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If an ROI contains both P3 and P9, a quality indicator should evaluate P3 as having higher diversity than P9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [26] pointed out that quality indicators should be able to distinguish point sets outside the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, P1 and P2 are outside the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, P2 is closer to the ROI than P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The same is true for the relation between P4 and P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this case, a quality indicator should evaluate P2 and P4 as having better quality than P1 and P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Mohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [27] pointed out the importance of not using information about the PF, which is generally unavailable in real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Table 1, 9 out of the 14 indicators satisfy this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that an approximation of the PF or the ROI found by PBEMO algorithms is available in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We believe that the remaining 5 out of the 14 indicators can address the issue by simply using the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 MASF As done in some previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', [14,44] and [26], the basic idea of this quality indicator is to use the minimum ASF (MASF) value of P to evaluate the closeness of P to z: MASF(P) = min p∈P {s(p)} , (14) 9 where we use s in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' MASF can evaluate only the two types of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since MASF does not consider the other µ−1 points in P, MASF cannot evaluate the diversity of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, MASF prefers P9 to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 MED As its name suggests, the mean Euclidean distance (MED) measures the average of Euclidean distance between each point in P to z [38]: MED(P) = 1 |P| � p∈P � � � � m � i=1 � pi − zi pnadir i − pideal i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' (15) MED can evaluate how close all points in P are to z and the PF when z is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, it cannot evaluate the convergence to the PF if z is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because MED prefers the non-Pareto optimal points close to z than the Pareto optimal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, MED prefers the dominated P7 over the non-dominated P3 when z = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='0)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4 IGD-based indicators Here, we introduce three quality indicators developed upon the IGD metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, since they are designed to deal with the ROIC, ROIA, and ROIP defined in Section 3, respectively, they are thus denoted as IGD-C, IGD-A, and IGD-P accordingly in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that both IGD-C and IGD- P were used in some previous studies [32, 45], and [36], respectively, whereas IGD-A is deliberately designed in this paper to facilitate our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In practice, the only difference between the original IGD and its three extensions is the choice of the IGD-reference point set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In IGD, IGD-reference points in S are uniformly distributed on the whole PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, IGD-C, IGD-A, and IGD-P use a subset S′ ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S′ can also be a subset of each ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1, S′ of IGD-C, IGD-A, and IGD-P are in the ROIC, ROIA, and ROIP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Below, for each indicator, we describe how to select S′ from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For IGD-C, we first find the closest point pc to z from S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', pc = argminp∈S{dist(p, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, S′ is a set of all points in the region of a hypersphere of radius r centered at pc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', S′ = {p ∈ S | dist(p, pc) < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The only difference between IGD-C and IGD-A is the choice of the center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, a point with the minimum ASF value pa is selected from S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', pa = argminp∈S{s(p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We use the ASF s in (4) in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, S′ is a set of all points in the region of a hypersphere of radius r centered at pa, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', S′ = {p ∈ S | dist(p, pa) < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In IGD-P, S′ is selected from S based on the Pareto dominance relation as in the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If z is feasible, S′ = {p ∈ S | p ≺ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, S′ = {p ∈ S | p ≻ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that IGD-P does not require the radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 HVz This quality indicator was originally named HVq in [9], where q represents the reference point in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since this paper denotes the reference point as z, we use the term “HVz” to make the consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' It computes the HV value of P in the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The only difference between HV and HVz is the choice of the HV-reference point y ∈ Rm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If z is feasible, y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If z is infeasible, yi = maxp∗∈ROIP{p∗ i } for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3 shows the HV-reference point y in HVz when setting z to (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='9)⊤ and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤, where the former is feasible while the latter is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' HVz can evaluate the convergence and diversity of a point set in terms of the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, it cannot handle the point sets outside the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because HV does not consider points dominated by the HV-reference point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, the HVz values of P1, P2, P4, and P5 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 y = z (a) Feasible z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='9)⊤ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 z y (b) Infeasible z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ Figure 3: Examples of the HV-reference point y in HVz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='6 PR The percentage of points in the ROI (PR) evaluates the cardinality of P [39] lying in the DM specified ROI: PR(P) = |{p ∈ P ∩ R}| |P| × 100%, (16) where R is the ROIP defined in [39], though it can be any type of ROI in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that PR is the only cardinality-based indicator considered in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A large PR means that many points in the corresponding P are in the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Like HVz, it is clear that PR cannot distinguish point sets outside the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='7 PMOD PMOD consists of two algorithmic steps [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, it maps each point p ∈ P onto a hyperplane passing through z as: p′ = p + ((z − p) · ˆz)ˆz, (17) where ˆz is the unit vector of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, PMOD aggregates three measurements including i) the distance between each mapped point p′ and z, ii) the distance between p and the origin o = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , 0)⊤, and iii) the unbiased standard deviation of all mapped points as: PMOD(P) = 1 |P| � P′∈P′ � dist(p′, z) + α dist(p, o) � + SD � {dp′}p′∈P′� , (18) where P′ is a set of |P| mapped points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (18), α is a penalty parameter for mapped points outside the preferred region of radius r centered at z, where r is a parameter of PMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When p′ is inside the preferred region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', dist(p′, z) ≤ r), α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, α > 1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', α is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' SD returns the unbiased standard deviation of input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For each p′ ∈ P′, dp′ in (18) is the minimum Manhattan distance between p′ and another point q ∈ P′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', dp′ = min q∈P′\\{p′} �m i=1 |p′ i − qi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The smaller PMOD value is, the better quality P is in terms of i) the convergence to z, ii) the convergence to the origin (not the PF), and iii) the uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that PMOD assumes the ideal point always does not dominate the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When the ideal point dominates the origin, PMOD prefers points far from the PF in view of above ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Let us consider the shifted point sets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 again, if the offset is −100, PMOD is likely to prefer P7 to P3 because P7 is closer to the origin than P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='8 IGD-CF and HV-CF A user-preference metric based on a composite front (UPCF) [27] is a framework for evaluating the quality of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' IGD-CF and HV-CF are the UPCF versions of IGD and HV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Algorithm 1 11 Algorithm 1: IGD-CF and HV-CF 1 PCF ← Select all non-dominated points from P1 ∪ · · · ∪ PK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 pc ← argminp∈PCF{dist(p, z)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3 Rpref ← {p ∈ Rm | dist(p, pc) < r};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 5 IGD-CF(Pi) ← IGD(Pi ∩ Rpref) using PCF as S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6 HV-CF(Pi) ← HV(Pi ∩ Rpref);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' shows how to calculate the IGD-CF and HV-CF values of K points sets P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , PK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, let PCF be a set of all non-dominated points in the K point sets (line 1), where PCF was called a composite front in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, the closest point pc to z is selected from PCF (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A preferred region Rpref is defined as the set of all points in the region of a hypersphere of radius r centered at pc (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that Rpref can include dominated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If PCF = F, Rpref is equivalent to the ROIC shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K}, the IGD-CF value of Pi is calculated based only on the points in Pi∩Rpref (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In other words, points outside Rpref are removed from Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(a), points outside the large dotted circle are removed from a point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' IGD-CF uses PCF as an approximation of the IGD-reference point set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The HV-CF value of Pi is calculated in a similar manner (line 6), where the previous study [27] did not give a rule of thumb to set the HV-reference point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' When the trimmed Pi is empty, we set its IGD-CF value to ∞ and its HV-CF value to 0 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' [26] pointed out that IGD-CF and HV-CF cannot distinguish point sets outside the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because IGD-CF and HV-CF do not consider any point outside Rpref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2, all point sets except for P3, P9, and P10 are equally bad, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', IGD-CF(Pi) = ∞ and HV-CF(Pi) = 0 for i ∈ {1, 2, 4, 5, 6, 7, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='9 PMDA The preference-based metric based on specified distances and angles (PMDA) [31] is built upon the concept of light beams [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' It consists of three algorithmic steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Step 1: It lets a set of points Q = {qi}m+1 i=1 on a hyperplane passing through z while m+1 light beams pass from the origin (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , 0)⊤ to q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , qm+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m}, qi is given as: qi = z + α (ei − z), (19) where ei is the standard-basis vector for the i-th objective function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', e1 = (1, 0)⊤ and e2 = (0, 1)⊤ for m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (19), α controls the spread of the light beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The remaining qm+1 in Q is set to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Step 2: All the points in Q are further shifted as: Q′ = βQ, (20) where β is the minimum objective value in P′ for all m objectives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', β = min p∈P′{ min i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m}pi}3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Here, P′ is a set of points in ∪K i=1Pi that are in a preferred region defined by m light beams, which pass through q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , qm but do not pass through qm+1 = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Step 3: PMDA measures the distance between each point in P and its closest point in Q′ as: PMDA(P) = 1 |P| � p∈P � min q∈Q′{dist(p, q)} + γθp � , (21) 3 [31] defined that β is the minimum objective value of P, not P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since β can be different for different point sets in this case, this version of PMDA is not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 12 Algorithm 2: R-IGD and R-HV 1 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 3 Pi ← Pi ∩ Pall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4 pa ← argminp∈Pi {s(p)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5 Pi ← � p ∈ Pi | |pj − pa j| ≤ r for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , m} � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6 k ← argmaxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m} � pa j−zj zw j −zj � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 piso ← z + � pa k−zk zw k −zk � (zw − z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 8 for p ∈ Pi do 9 p ← p + (piso − pa);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 10 R-IGD(Pi) ← IGD(Pi) using a trimmed S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 11 R-HV(Pi) ← HV(Pi) using zw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' where γ is a penalty value and was set to 1/π in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In (21), θp is an angle between p and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' If p is in the preferred region defined by the m light beams passing through q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , qm, θp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, points outside the preferred region are penalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A small PMDA value indicates that points in the corresponding P are close to the m + 1 points in Q′ and the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, PMDA does not evaluate the diversity of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that all elements of a point are implicitly assumed to be positive in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='10 R-IGD and R-HV R-metric [26] is a framework that applies general quality indicators to the performance evaluation of K PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' R-metric assumes that the DM prefers points along a line from z to the worst point zw defined by the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As recommended in [26], we set zw = z + 2 × u, where u is a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We set u = (1/√m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , 1/√m)⊤ in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The previous study [26] considered the R-metric versions of IGD and HV, denoted as R-IGD and R-HV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Algorithm 2 gives the pseudo code for calculating R-IGD and R-HV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A set of all non-dominated points Pall are selected from the union of K point sets (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' After that, the following steps are performed for each point set Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, points dominated by any point in Pall are removed from Pi (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, the best point pa is selected from Pi in terms of the ASF (line 4), where the previous study [26] used s in (4) as the ASF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' R-metric defines a preferred region based on a hypercube of size 2 × r centered at pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Points outside the preferred region are removed from Pi (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4(a), only the three points in the dotted box are considered for the R-metric calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This trimming operation can penalize a point set that does not fit the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Next, R-metric obtains a projection of pa on the line from z to zw by the ASF (lines 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This projection is called an iso-ASF point piso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' R-metric transfers all the points in Pi by the direction vector from piso to pa (lines 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4(b), the three points are shifted horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This transfer operation redefines the convergence to the PF as the convergence to z along a line based on the DM’s preference information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Finally, the R-IGD and R-HV values of Pi are calculated (lines 10 and 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' More specifically, for R-IGD, the same trimming operation (lines 4 and 5) is first applied to the IGD-reference point set S in R-IGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, all points in S are inside the preferred region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, the IGD value of Pi is calculated using the trimmed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For R-HV, zw is used as the HV-reference point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='11 EH The expanding hypercube metric (EH) [29] is based on the size of a hypercube centered at z that contains each point and the fraction of points inside the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While the former evaluates the convergence of a point set P to z, the latter tries to evaluate the diversity of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The pseudo code of calculating the EH for K point sets P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , PK is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, EH removes duplicated points for each point set (lines 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In the meanwhile, it also removes 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 z zw (a) The trimming operation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 z zw (b) The transfer operation Figure 4: Examples of the two operations in R-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this example, zw = z + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='7 × u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' dominated points from each point set (lines 3 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that if a point set is empty after these removal operations, its EH value is set to 0 (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, the following steps are performed for each point set Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' EH calculates the size of a hypercube centered at z that contains each point p in Pi (lines 10 and 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thereafter, all elements in h are sorted (line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that |h| = |Pi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The maximum size hmax in h is maintained for an adjustment described later (lines 13 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' EH calculates “the area under the trade-off curve” ai between the hypercube size and the fraction (lines 16 and 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While l/|Pi| is the fraction of points in the l-th hypercube, “(hl − hl−1)” is the incremental size of the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Finally, the EH value of each point set Pi is calculated by adjusting ai using hmax (lines 18 and 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A large EH value means the corresponding P has good convergence to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Due to the operation for removing dominated points (line 3), EH also implicitly evaluates the convergence of P to the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since EH does not define a preferred region, EH fails to evaluate the diversity of P in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Let us consider a set of non-dominated unduplicated points that are close to z and distributed at intervals of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' EH is maximized when ∆ is a positive value as close to zero as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2(a), EH prefers P9 to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5 Experimental Setup This section introduces the settings used in our experiments including the quality indicators, the benchmark problems, and preference-based point sets used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Quality Indicators In our experiments, we empirically analyze the performance and properties of 14 quality indicators reviewed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As a baseline, we also take the results of HV and IGD into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, the implementations of HV and R-IGD and R-HV are taken from pygmo [47] and pymoo [48], respectively, while the other quality indicators are implemented by Tanabe in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The innate parameters of the 14 quality indicators are set according to the recommendation in their original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the IGD-based indicators, we uniformly generated 1 000 IGD-reference points on the PF of a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For those HV-based indicators, we set the HV-reference point y in HV and HV-CF to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We set the radius r of a preferred region to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 for all the quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We also set the radius ζ of the ROIC and ROIA to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Benchmark Test Problems DTLZ1 [49], DTLZ2 [49], convDTLZ2 [50] are chosen to constitute the benchmark test problems, which have linear, nonconvex, and convex PFs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' To ensure the fairness of our experiments, the PF of the DTLZ1 problem is normalized to [0, 1]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As a first attempt to investigate the properties 14 Algorithm 3: EH 1 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 2 Pi ← {p ∈ P | ̸ ∃pdup ∈ P s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' pdup = p};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 3 Pall ← Select all non-dominated points from P1 ∪ · · · ∪ PK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 4 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 5 Pi ← Pi ∩ Pall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6 hmax ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 8 h ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 9 for p ∈ Pi do 10 h ← maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=',m}{|pj − zj|};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 11 h ← h ∪ {h};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 12 h ← Sort all elements in h in ascending order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 13 hmax ← maxh∈h{h};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 14 hmax ← hmax ∪ {hmax};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 15 ai ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 16 for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , |Pi|} do 17 ai ← ai + l |Pi| × (hl − hl−1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' // h0 = 0 18 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , K} do 19 if Pi = ∅ then EH(Pi) ← 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 20 else EH(Pi) ← ai + (maxh∈hmax{h} − hmax i ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' of preference-based quality indicators, we mainly focus on the two-objective scenarios to facilitate the analysis and discussion about the impact of the distribution of points on the quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We are aware of a previous study [29] evaluated the performance of R-NSGA-II and g-NSGA-II on the DTLZ problems with m ∈ {3, 5, 8, 10, 15, 20} by EH and R-HV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The previous study discussed the influence of the distribution of m-dimensional points on EH and R-HV using the parallel coordinates plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, the parallel coordinates plot is likely to lead to a wrong conclusion [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In fact, the results in [29] did not show the undesirable property of EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 Experimental Settings We conduct two types of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' One is an experiment using the 10 synthetic point sets as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 shows the distributions of the 10 point sets on the PF of the DTLZ1 and convDTLZ2 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The other is an experiment using point sets found by the six PBEMO algorithms introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that comprehensive benchmarking of the PBEMO algorithms is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Instead, we focus on an analysis of the behavior of the PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This contributes to the understanding of RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Moreover, we also investigate how the choice of quality indicators influences the rankings of the PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This contributes to addressing RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In particular, the source code of the PBEMO algorithms are provided by Li [11] while the weight vectors used in MOEA/D-NUMS are generated by using the source code provided by Li [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Each PBEMO algorithm is independently run 31 times with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The population size µ is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The parameters associated with these PBEMO algorithms are set according to the recommendations in their original papers, except PBEA of which δ is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='01 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 p1 p25 p50 p75 p100 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) (a) 100 points 20 40 60 80 100 Point IDs 0 20 40 60 80 100 Rankings z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) z = (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) (b) Rankings (distance) 20 40 60 80 100 Point IDs 0 20 40 60 80 100 Rankings z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) z = (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) (c) Rankings (ASF) 3 -2 -1 0 1 2 3 f1 3 2 1 0 1 2 3 f2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 (d) Kendall τ Figure 5: (a) Distribution of 100 uniformly distributed points, (b) the rankings of the 100 points by the distance, (c) the ranking of the 100 points by the ASF, and (d) the Kendall τ values on the DTLZ2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6 Results This section is dedicated to addressing the four RQs raised in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 analyzes the relation between the distance to the reference point z and the ASF value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 investigates differences in the three ROIs and the behavior of EMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thereafter, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 examines the properties of the 14 quality indicators using the synthetic point sets shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Finally, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4 analyzes the influence of quality indicators on the rankings of EMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Relation between the distance to z and the ASF value Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(a) shows 100 uniformly distributed points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' , p100 on the PF of the DTLZ2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(a) also shows the two reference points z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤ and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 is dominated by the ideal point, z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 dominates the ideal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(a), intuitively, the 50-th point p50 on the center of the PF is closest to both z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, this intuition is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(b) and 5(c) show the rankings of the 100 points by the Euclidean distance to z and the ASF in (4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' A low ranking means that the corresponding point is close to z or obtains a small ASF value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(b), p50 is closest to z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, p50 is farthest from z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The two extreme points (p1 and p100) are closest to z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, the closest points to z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(c), the rankings by the ASF are consistent when using either one of z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because both z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 are in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d) shows the Kendall rank correlation τ value of the distance to z and the ASF value, where τ ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d), we uniformly generated z from (−3, 3)⊤ to (3, −3)⊤ at intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Then, we calculated the τ value for each z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The τ value quantifies the consistency of the two rankings, where one is based on the distance to the corresponding z, and the other is based on the ASF value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Positive and negative τ values indicate that the two rankings are consistent and inconsistent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d), the rankings by the distance to z and the ASF value are inconsistent when setting z close to the line passing through (0, 0)⊤ and (−3, −3)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We can also see that the rankings are weakly inconsistent when setting z to other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that the inconsistency between the distance to z and the ASF value depends on not only the position of z, but also the shape of the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 show the results on the DTLZ1 and convDTLZ2 problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3(a), we set z to (2, 2)⊤ instead of (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤ for the convDTLZ2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2(b) and (c), the rankings by the distance to z and the ASF value on the DTLZ1 problem are always consistent regardless of the position of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, as seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3(b) and (c), the inconsistency of the rankings can be observed on the convDTLZ2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3(b) is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3(d) is opposite from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Unlike Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3(d) indicates that the inconsistency between the two rankings occurs when z is dominated by the nadir point pnadir on the convex PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 16 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (a) ROIC (z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (b) ROIA (z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (c) ROIP (z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (d) ROIC (z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (e) ROIA (z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (f) ROIP (z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1) Figure 6: Distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (a) R-NSGA-II 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (b) MOEA/D-NUMS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 1 f2 (c) g-NSGA-II Figure 7: Distributions of points found by three PBEMO algorithms on the DTLZ2 problem when using z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Answers to RQ1: Our results show that the closest Pareto-optimal point to the reference point z does not always minimize the ASF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although it has been believed that minimizing the ASF means moving closer to z, this is not always correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We observed that the inconsistency between the distance to z and the ASF value depends on the position of z and the shape of the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Roughly speaking, the inconsistency can be observed when z dominates the ideal point on a problem with a nonconvex PF, and z is dominated by the nadir point on a problem with the convex PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Analysis of the three ROIs First, this section investigates the differences between the three ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6 shows the distributions of Pareto optimal points in the three ROIs on the DTLZ2 problem when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤ and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='6 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='7 show the results on the DTLZ1 and convDTLZ2 problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(a) and (b) are exactly the same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, the ROIC and ROIA are the same even when using either one of z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(d) and (e), the ROIC and ROIA are totally different when using z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While the ROIA is on the center of the PF, the ROIC is on either one of the two extreme points (1, 0)⊤ and (0, 1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since the two extreme points (1, 0)⊤ and (0, 1)⊤ are equally close to z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(d) shows two possible ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This strange result is due to the inconsistency between the distance to z and the 17 ASF value reported in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='7(g), a similar result can be observed on the convDTLZ2 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Results similar to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(d) can be obtained by using z with a small Kendall τ value in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(c) significantly differs from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The extent and cardinality of the ROIP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(c) are much larger than those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6(f), the ROIP and the PF are identical when the reference point dominates the ideal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The same is true when the reference point is dominated by the nadir point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this case, preference-based multi-objective optimization is the same as general one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The size of the ROIP increases as z moves away from the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This undesirable property of the ROIP is similar to that of the g-dominance relation pointed out in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='6 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='7, this undesirable property of the ROIP can be observed on other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since the DM does not know any information about the PF in practice, it is difficult to set a reference point that is neither too close nor too far from the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Next, we point out that the differences in the target ROIs caused the unexpected behavior of some PBEMO algorithms in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 shows the final point sets found by R-NSGA-II, MOEA/D-NUMS, and g-NSGA-II on the DTLZ2 problem when using z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 are consistent with the results in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='8–S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='16 show the results of the six PBEMO algorithms on the three problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, R-NSGA-II, MOEA/D-NUMS, and g-NSGA-II aim to approximate the ROIC, ROIA, and ROIP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As demonstrated here, the three ROIs can also be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For these reasons, the three EMO algorithms found different point sets, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The previous study [11] concluded that R-NSGA-II and g-NSGA-II failed to approximate the “ROI” when z is far from the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, this conclusion is not very correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Correctly speaking, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7(a) and (c), R-NSGA-II and g-NSGA-II failed to approximate the “ROIA” but succeeded in approximating the “ROIC” and “ROIP”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Answers to RQ2: There are two takeaways generated from the analysis in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' First, our results showed that the three ROIs can be significantly different depending on the position of the reference point z and the shape of the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We demonstrated that the ROIA is not always a subregion of the PF closest to z due to the inconsistency observed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In addition, we found that the size of the ROIP significantly depends on the position of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Unless the DM knows the shape of the PF in advance, it would be better not to use the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Second, we also demonstrated that the differences in the three ROIs could cause the unexpected behavior of PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For this reason, we argue the importance to clearly define a target ROI when benchmarking PBEMO algorithms and performing a practical decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 18 Table 2: Rankings of the 10 synthetic point sets on the DTLZ2 problem by the 16 quality indicators when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' (a) z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='MASF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='MED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='IGD-C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='IGD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='IGD-P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='HVz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='PR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='PMOD ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3 Analysis of quality indicators Table 2 shows the rankings of the 10 point sets P1 to P10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 2 by each quality indicator when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ and z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For the sake of reference, we show the results of HV and IGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Table 2 shows which point set is preferred by each quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, P9 obtains the best MASF value in the 10 point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Tables S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 show the results on the DTLZ1 and convDTLZ2 problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We do not intend to elaborate Tables S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2, as they are similar to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 Results for z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤ First, we discuss the results shown in Table 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' P3 is the best in terms of i) the convergence to the PF, ii) convergence to the reference point z, and iii) diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, quality indicators should give P3 the highest ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' P3 is ranked highest by 9 out of the 16 quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, four quality indicators (MASF, MED, PMDA, and EH) prefer P9 with the poorest diversity to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because they do not take into account the diversity of points as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since HV and IGD do not handle the preference information, they prefer P10 that covers the whole PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Interestingly, IGD-P also prefers P10 the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because the IGD-reference points of IGD-P are relatively widely distributed around the center of PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since PR evaluates only the cardinality, PR cannot distinguish the quality of P3, P7, and P9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since IGD is Pareto non-compliant, IGD prefers P7 to P9, where all the points in P7 are dominated by those in P9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Similarly, PMOD gives P7 the second highest ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since PMOD does not take into account the convergence to the PF, PMOD can evaluate the quality of point sets inaccurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since IGD-CF and HV-CF cannot distinguish point sets outside their preferred regions, most point sets obtain the same ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although this undesirable property was already pointed out in [26], this is the first time to demonstrate that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The same is true for HVz and PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since R-IGD, R-HV, and EH remove dominated points from point sets, they cannot distinguish the three dominated point sets (P6, P7, and P8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2 Results for z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1)⊤ Next, we discuss the results shown in Table 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In this setting, P1 and P5 are the best in terms of all three criteria i), ii), and iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, quality indicators should give P1 or P5 the highest ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, the rankings by four ASF-based quality indicators (MASF, IGD-A, R-IGD, and R-HV) are the same in Tables 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is because the point with the minimum ASF value and the ROIA are the same regardless of whether z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 or z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 is used, as demonstrated in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' The same is true for PMOD, PMDA, and EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, these quality indicators fail to evaluate the convergence of the point sets to the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, the rankings by other quality indicators based on the ROIC and ROIP are different in Tables 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As demonstrated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2, the ROIC is on either one of the two extreme points when using z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For this reason, four quality indicators based on the ROIC (MED, IGD-C, IGD-CF, and HV-CF) prefer P1 or P5 to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' While IGD-C and IGD-A are perfectly consistent for the results of z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, they are inconsistent for the results of z−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This is due to the inconsistency revealed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2, when z dominates the ideal point or is dominated by the nadir point, the ROIP is equivalent to the PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For this reason, three quality indicators based on the ROIP (IGD-P, HVz, and PR) cannot handle the DM’s preference information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, like HV and IGD, IGD-P, HVz, and PR prefer P10 the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Since IGD-P and IGD use the same IGD-reference point set S, their rankings are perfectly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Although the position of the HV-reference point y is different in HVz and HV, their rankings are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' PR cannot distinguish all the 10 point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 20 Answers to RQ3: Our results indicated that most quality indicators have some undesirable prop- erties, which have not been noticed even in their corresponding papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We demonstrated that the quality indicators based on the ROIA cannot evaluate the convergence of a point set to the reference point accurately in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We also demonstrated that the quality indicators based on the ROIP cannot take into account the DM’s preference information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Our results imply that IGD-C may be the most reliable quality indicator when considering the practical ROIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' However, IGD-C is Pareto non-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 21 Table 3: Rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality indicators when using z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' “NUMS”stands for MOEA/D-NUMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' MASF MED IGD-C IGD-A IGD-P HVz PR PMOD IGD-CF HV-CF PMDA R-IGD R-HV EH HV IGD R-NSGA-II 3 2 6 6 5 6 4 4 6 6 3 5 5 1 5 5 r-NSGA-II 4 3 3 3 4 4 1 3 3 4 2 3 3 3 4 4 g-NSGA-II 5 5 1 1 1 1 1 1 1 1 5 2 1 6 2 2 PBEA 1 6 2 2 2 2 6 5 2 2 6 1 2 5 1 1 R-MEAD2 6 4 5 5 3 3 5 6 4 3 4 6 6 4 3 3 NUMS 2 1 4 4 6 5 1 2 5 5 1 4 4 2 6 6 22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='4 On the rankings of PBEMO algorithms by quality indicators Table 3 shows the rankings of the six PBEMO algorithms on the DTLZ2 problem by the 16 quality indicators, where z = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We calculated the rankings based on the average quality indicator values of the PBEMO algorithms over 31 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Tables S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3–S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5 show the rankings on the DTLZ1, DTLZ2, and convDTLZ2 problems when using various reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that we are interested in the influence of quality indicators on the rankings of the PBEMO algorithms rather than the rankings themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Table 3, the rankings of the PBEMO algorithms are different depending on the choice of the quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, R-NSGA-II performs the best in terms of EH but the worst in terms of five quality indicators including IGD-C, IGD-A, HVz, IGD-CF, and HV-CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Likewise, g- NSGA-II is the worst performer in terms of EH but it is the best algorithm when considering the other nine quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='5(b), R-MEAD2 performs the best on the convDTLZ2 problem in terms of EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In summary, our results suggest that any PBEMO algorithm can obtain the best ranking depending on the choice of the quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' These observations can be explained as the coupling relationship between innate mechanism of the PBEMO algorithms and the quality indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As demonstrated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='2, each PBEMO algorithm approximates its target ROI embedded by its designer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As investigated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='3, each quality indicator prefers a point set that approximates its target ROI well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, when benchmarking PBEMO algorithms, it is important to clarify which type of ROI the DM wants to approximate and select a suitable quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, if the DM wants to approximate the ROIA, she/he should select either of IGD-A, R-IGD, and R-HV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Otherwise, the DM can overestimate or underestimate the performance of PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Answers to RQ4: Our results showed that the choice of the quality indicator significantly influ- ences the performance rankings of EMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, as seen from Table 3, PBEA performs the worst in terms of PMDA but the best in terms of R-IGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' This means that any PBEMO algorithm can possibly be ranked as the best (or the worst) depending on the choice of the quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We also discussed how to conduct meaningful benchmarking of PBEMO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 7 Conclusion In this paper, we first reviewed the 3 ROIs and 14 existing quality indicators for PBEMO algorithms using the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Different from the descriptions in their corresponding papers, we classified the properties of the quality indicators from the perspective of their working principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As a result, we found that some quality indicators have undesirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, PMDA and EH cannot evaluate the diversity of a point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' We also discussed the target ROI of each quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Next, we empirically analyzed the performance and properties of those 14 quality indicators to address 4 RQs (RQ1 to RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Our findings are helpful for benchmarking PBEMO algorithms and decision-making in real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In any case, we argue the importance of determining a target ROI first of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Our results suggested the use of the ROIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Afterward, a researcher and the DM should select a PBEMO algorithm and quality indicator based on their target ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, R-NSGA-II aims to approximate the ROIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' In contrast, HVz is to evaluate how a point set approximates the ROIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Thus, HVz is not suitable for evaluating the performance of R-NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' As demonstrated in the three IGD variants (IGD-C, IGD-A, and IGD-P), we believe that the target ROI of some quality indicators can be changed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' For example, the target ROI of R- IGD can be changed from the ROIA to the ROIC by revising the line 4 in Algorithm 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', pc = argminp∈Pi{dist(p, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' An investigation of this concept is an avenue for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Note that the analysis conducted in this paper focused on quality indicators for PBEMO algorithms using the reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' It is questionable and important to extend our analysis for other preference-based optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=', a value function) in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' There is room for discussion about a systematic benchmarking methodology for PBEMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' 23 Acknowledgment Tanabe was supported by JSPS KAKENHI Grant Number 21K17824 and LEADER, MEXT, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfsi_P/content/2301.12148v1.pdf'} +page_content=' Li was supported by UKRI Future Leaders Fellowship (MR/S017062/1, MR/X011135/1), NSFC (62076056), EPSRC (2404317), Royal 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a/BNE3T4oBgHgl3EQfswvl/content/tmp_files/2301.04671v1.pdf.txt b/BNE3T4oBgHgl3EQfswvl/content/tmp_files/2301.04671v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9e7ffa86b0ee1c7c67ddc379178ea72ecdb4315 --- /dev/null +++ b/BNE3T4oBgHgl3EQfswvl/content/tmp_files/2301.04671v1.pdf.txt @@ -0,0 +1,1760 @@ +Circuit Complexity through phase transitions: +consequences in quantum state preparation +Sebasti´an Roca-Jerat,1 Teresa Sancho-Lorente,1 Juan Rom´an-Roche,1 and David Zueco1 +1Instituto de Nanociencia y Materiales de Arag´on (INMA) and Departamento de F´ısica de la Materia Condensada, +CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain +(Dated: January 13, 2023) +In this paper, we analyze the circuit complexity for preparing ground states of quantum many- +body systems. In particular, how this complexity grows as the ground state approaches a quantum +phase transition. We discuss different definitions of complexity, namely the one following the Fubini- +Study metric or the Nielsen complexity. We also explore different models: Ising, ZZXZ or Dicke. +In addition, different forms of state preparation are investigated: analytic or exact diagonalization +techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigen- +solvers. +We find that the divergence (or lack thereof) of the complexity near a phase transition depends on +the non-local character of the operations used to reach the ground state. For Fubini-Study based +complexity, we extract the universal properties and their critical exponents. +In practical algorithms, we find that the complexity depends crucially on whether or not the system +passes close to a quantum critical point when preparing the state. While in the adiabatic case it is +difficult not to cross a critical point when the reference and target states are in different phases, for +VQE the algorithm can find a way to avoid criticality. +CONTENTS +I. Introduction +2 +A. Complexity overview +2 +1. Complexity `a la Nielsen +3 +2. Circuit Complexity from the Fubini-Study metric +4 +3. Some remarks comparing both approaches +5 +B. Main results and manuscript organization +5 +II. Complexity and the geometry of states close to a quantum phase transition. +5 +A. Complexity and its derivative when crossing a QPT +6 +B. Finite size scaling +6 +III. Solvable Hamiltonians +7 +A. Quantum Ising model +7 +1. Complexity through QPTs +7 +2. Relation between CN and CFS +8 +B. The Dicke model +8 +IV. Complexity in a quantum computer, the case of ground state preparation +10 +A. Adiabatic algorithms +10 +1. Complexity in adiabatic algorithms +11 +B. Circuit Complexity in VQEs +13 +1. Local VQE ansatz +15 +2. VQE complexity through QPTs +15 +V. Discussion +17 +Acknowledgments +18 +A. Complexity associated to the VQE +18 +B. Other paths in the adiabatic algorithm +19 +References +19 +arXiv:2301.04671v1 [quant-ph] 11 Jan 2023 + +2 +I. +INTRODUCTION +How much does it cost to generate a target quantum state from another reference state? This is a rather general +question that has been discussed in quantum information for obvious reasons. In quantum computation it is desirable +to obtain the result with the minimum set of gates. This number is, roughly speaking, the computational cost and it +is called circuit complexity (C) [1–3]. It is, let us say, the quantum analog of the concept of computational complexity +in computer science. Importantly enough, this cost builds upon a concrete physical architecture, i.e the available +set of gates. Therefore, C not only depends on the reference and target states but on the restrictions for reaching +the latter. This is quite natural if one thinks of an actual quantum computer where the possible operations have +restrictions. Note that, if any unitary were allowed, a simple rotation would achieve the goal and every quantum +state would be easily prepared, so that (essentially) the complexity would be a trivial quantity. Therefore, also in +analytic calculations, the path between the reference and target is restricted to a set, e.g. gaussian states [4–8] . +Beyond quantum computation, circuit complexity is a relevant concept in quantum gravity. In particular, for its +consequences in holography [9–11]. For those who are not experts (like us), we can say that holography describes +quantum gravity within a region of space by looking at the boundary of that region, that is described by a non +gravitational theory. Then, any bulk quantity in the gravitational theory is equivalent or dual to another quantity +in the boundary of the non gravitational theory. One of the main problems of this duality is that the volume behind +the black hole horizon keeps growing for a very long time while the entanglement at the boundary saturates at much +shorter times. One possible solution is to conjecture that the dual of volume is not entanglement but complexity, +via the identification Complexity = Volume. This is because we expect that volume is an extensive quantity, while +entanglement (typically) fulfils an area law. +Therefore, the calculations of complexity are beyond the quantum +information community and different calculations in field theories have been discussed in the literature [12, 13]. +The notion of complexity (C) is much related to the geometry of states (or operators). It is a measure of the +distance between two of them. Therefore, one possible choice for C is finding the geodesics in the Fubini-Study metric +in the projected Hilbert space for the case of pure states. For mixed states different measures have been introduced +via state purification [14] or distance measures for mixed states as the Bures distance [15]. This geometric background +is a powerful way to understand complexity, since it allows us to know how much it will cost to prepare a state by +solving a geodesic equation. It is true, however, that the metric, in principle, can only be obtained in some cases: +surely in exactly solvable models. And there we know how to prepare states. Thus, it is interesting to be able to +predict the typical behaviour in general models. Here, we move in this direction. +In this article we are interested in a quite generic situation, i.e. +when a critical point is crossed to reach the +target state. In particular, we investigate what general statements about the behaviour of the circuit complexity +we can make. We are not the first to calculate C in a quantum phase transition (QPT) [16]. Recent papers discuss +exactly solvable models as the topological Kitaev, Bose-Hubbard and Lipkin-Meshkov-Glick ones [17–23]. Importantly +enough, complexity has been shown to be a useful probe of topological phase transitions. Complementary to these +calculations, in this work, we use that close to a transition point, the concept of universality emerges naturally, so +we expect these universal properties to be inherited by complexity. If so, we can argue for its scaling laws or how +complexity behaves regardless of model details or even on the particular chosen definition of complexity. In addition, +we apply our theory for state preparation in quantum computers. This is a key and hard task [24]. It is within +the QMA complexity class [25], roughly speaking the NP-complete analogue for quantum computers. Nevertheless, +quantum computers are expected to be better than classical methods such as density functional theory [26], density +normalization group [27], tensor networks [28], quantum Montecarlo [29] or even ML-inspired techniques [30], in some +instances. For a recent discussion of these issues, see [31]. Heuristic quantum algorithms as adiabatic [32] or varational +ones [33, 34] can outperform classical calculations and serve for the generation of quantum states as quantum data, +e.g. phase classification [35]. Motivated by all of this, we discuss how useful the concept of complexity is and how +much one can anticipate the difficulty of state preparation in variational quantum eigensolvers (VQEs) or adiabatic +quantum algorithms (with and without shortcuts to adiabaticity). To challenge our theory we tackle both integrable +and non-integrable models using numerical simulations and computing C. +A. +Complexity overview +We find it convenient to discuss first the different notions of circuit complexity that we will use in this paper and +the relationship between them. + +3 +1. +Complexity `a la Nielsen +The original notion of complexity is due to the works of Nielsen and collaborators [1–3]. See [13] for a recent review. +Restricting ourselves to unitary operations, target and reference states are related as +|ψT ⟩ = U(t, 0)|ψR⟩ = T e−i +� t +0 H(τ) dτ|ψR⟩ . +(1) +T stands for time ordering. Notice that, +H(τ) = i(∂τU)U † . +(2) +A Cost function is formally defined as: +CN := min{U} +� t +0 +dτ F[U, ˙U] +(3) +with F some functional fulfilling some basic properties as continuity, homogeneity (F[U, λ ˙U] = λF[U, ˙U] for λ ≥ 0), +positivity and the triangular inequality 1. If, in addition to these, smoothness is assumed and the Hessian of F as a +function of U is strictly positive, the functional is a Finsler metric. Thus, CN is nothing but the geodesics. The suffix +N stands for Nielsen. +Being a little more explicit, we can write that the evolution is given by +H(τ) = +� +n +Y (n)(τ)On . +(4) +with On some operators and Y (n)(τ) parameters. A usual functional is then given by, +Fk(τ) ∼ +�� +n +|Y (n)(τ)|k +�1/k +. +(5) +If we restrict ourselves to two level systems (qubits), Fk(τ) is a natural distance in SU(2n), such that d = +� t +0 dτ Fk(τ), +Cf. with Eq. (3). What has been explained so far is the continuous version of complexity, that provides a lower +bound for the number of gates needed to approximate |ψT ⟩ from |ψR⟩ [1]. The discrete version of CN can be computed +introducing the functional (using the same notation as in the original [1]): +F(τ) = +� +� +� +� +′ +� +σ +hσ(τ)2 + p2 +′′ +� +σ +hσ(τ)2 +(6) +where the Hamiltonian in this case is H(τ) = �′ +σ hσ(τ)σ + �′′ +σ hσ(τ)σ. In the first sum, σ ranges over all possible +one- and two-body interactions, that is, over all products of either one or two qubit gates. In the second sum, instead, +the sum is over other tensor products of Pauli matrices and the identity. The factor p > 0 penalizes three, four, ... +-body interactions. All put together, finding the geodesics in the continuum version is a good estimate of the resources +needed to prepare a state. +At this point, we think it is necessary to emphasise something. If any unitary is possible, the complexity has a +narrow utility, since its value is given by C = arccos(|⟨ψR|ψt⟩|), i.e. of the order of one (it doesn’t matter which state +reference and destination are chosen). This can be verified by noting that the target state can always be written as +|ψT ⟩ = cos θ|ψR⟩ + eiγ sin θ|ψ⊥ +R⟩ with ⟨ψR|ψ⊥ +R⟩ = 0. A rotation in the subspace generated by {|ψR⟩, |ψ⊥ +R⟩} does the +job. Therefore, some restrictions on the possible unitaries or Hamiltonian (4) will be imposed. We will discuss this +point in some depth later. +1 Notice that due to homogeneity, w.l.o.g. we can always set t = 1. + +4 +2. +Circuit Complexity from the Fubini-Study metric +The functionals F discussed so far, see Eqs. (4) and (5), are not unique. Others can be chosen satisfying continuity, +homogeneity, positivity and the triangular property. We want to discuss next another possibility where the distance +between the reference and target states is given by the Fubini-Study metric. Originally proposed for Quantum Field +Theories in [5], we prefer to study it here from a quantum information perspective. Let us time-slice the unitary (1) +such that +|ψT (λ)⟩ = Uλ(t, tN−1)...Uλ(t1, t0)|ψR(λ)⟩ → |ψ(λ; tn)⟩ = U(tn, tn−1)|ψ(λ; tn−1)⟩ +(7) +We have assumed that the unitaries and so the wave functions depend on the parameters λ. Then, for sufficiently +small time step δτ := tn − tn−1, the fidelity between two contiguous states is +Fn,n+1 ≡ |⟨ψ(λ; tn)|ψ(λ; tn−1)⟩| = 1 − χF δτ 2 + O(δ4) +(8) +where χF is denoted the fidelity susceptibility [36–41], see Ref. [42] for a review. Interestingly enough, we can relate +χF with the geometric tensor, in fact [Cf. Eq. (11)] +χF = gµν ˙λµ ˙λν +(9) +with [5, 43] +gµν = Re (Tµν) . +(10) +Here, Tµν is the quantum geometric tensor which is nothing but the Fubini-Study metric (FSM) on the CP n manifold, +namely: +Tµν = ⟨∂λµψ|P|∂λνψ⟩ +(11) +with P = 1 − |ψ⟩⟨ψ| 2. +Another useful way of writing the metric tensor is as follows. Using a formal Taylor expansion for the states, the +metric tensor can be written as, +gµν = 1 +2 (⟨∂µψ|∂νψ⟩ − ⟨∂µψ|ψ⟩⟨ψ|∂νψ⟩ + c.c.) +(12) +Setting now in the Hamiltonian (4), ˙λν ≡ Y (ν) then |∂ν⟩ = Oν|ψ⟩, it is straightforward to see that [6], +gµν = 1 +2 ⟨ψ |{Oµ, Oν}| ψ⟩ − ⟨ψ) |Oµ| ψ⟩ ⟨ψ |Oν| ψ⟩ = 1 +2 +��ψ|⟨ +� +Oµ − ⟨Oµ⟩λ , Oν − ⟨Oν⟩λ +��� ψ⟩ , +(13) +i.e. the fluctuations of the Hamiltonian operators Oν. +Using the fact that 1−Fn,n+1 is a distance, thus satisfying the properties we imposed for the F-functional, we have +that we can understand C as the distance defined through the Fubini-Study metric: +CFS := min{U} +� t +0 +� +gµν ˙λµ ˙λν dτ . +(14) +The suffix stands for Fubini-Study metric and the notion of distance is quite explicit. This is an alternative definition +to that given by Eq. (3) that has some remarkable properties. The first one is that knowing the metric tensor the +geodesics can be found, at least in principle, by solving the differential equation: +d2λµ +dτ 2 + Γµ +νρ +dλν +dτ +dλρ +dτ = 0 +(15) +Here, Γ are the Christoffel symbols: +Γµ +νρ = 1 +2gµξ (∂ρgξν + ∂νgξρ − ∂ξgνρ) . +(16) +The second property of CFS is that, from its relation to the fidelity between states, F, its properties close to a QPT +can be used when discussing the complexity, C, see also Eq. (13). +2 Notice that the imaginary part of T is nothing but the Berry phase. + +5 +3. +Some remarks comparing both approaches +The complexity according to Nielsen estimates the minimum number of gates needed to reach the target state. For +this purpose, a metric in the space of quantum circuits or unitary transformations is defined. The optimization of +the trajectory in this space thus minimizes the number of accessible gates. On the other hand, with the Fubini-Study +metric, the complexity is computed by monitoring the state changes along the preparation of the target state. The +Fubini-Study metric defines a geometry in the space of states. A key difference is that in the latter geometry a variable +cost is assigned to specific gates as it depends on the states they act on, while in the former, each gate is assigned a +fixed cost. On top of that, with CN there will be degenerate operations that leave the state unchanged (e.g. adding a +global phase). Therefore and in general, it is found that the space of unitaries has a higher dimension. +In general, different results are obtained using both approaches [6, 44]. Depending on the application, one form is +preferred over the other. We believe that due to the equations of the geodesics given the metric, CFS is ideal for doing +analytical calculations while, from the point of view of quantum computation and cost estimation to prepare states, +CN will prevail. In any case, in some particular cases it has been shown that both methods give identical results, such +as the preparation of Gaussian fundamental states [6]. +As a final remark, let us note that typically, the fidelity, no matter how close two quantum states are, drops +exponentially with system size. This is nothing but the Anderson orthogonality catastrophe. Therefore, in numerical +studies a variant of CFS would be to use the fidelity per site instead, +log f ≡ 1 +N log F , +(17) +in Eq. (8). On top of that, this allows to extract the extensive part for the complexity which, on the other hand, is +what seems to matter [5]. +B. +Main results and manuscript organization +For the exactly solvable systems that we discuss in this work, we find that CFS ≥ CN when crossing a phase +transition. We understand this inequality as a consequence of the fact that the unitary space is larger, see previous +subsection I A 3. In any case, C does not diverge at the critical point, its derivative does. For CFS we can characterize +this divergence and its critical exponents in rather general circumstances. Let us remark, again, that throughout the +paper we focus on the extensive part of C. Two models are studied in detail, namely the one-dimensional quantum +Ising and Dicke models. +After this general discussion, we focus on calculating the complexity when preparing a fundamental state in a quantum +computer. Here, obviously, we compute CN in its discrete version. We explore two algorithms in detail. First, we +discuss the circuit complexity in adiabatic algorithms (with and without shortcuts to adiabaticity). Here, the adiabatic +path crosses a QPT explicitly and the complexity grows around it. There is not much difference (with respect to the +CN) in using shortcuts. Then, we discuss the circuit complexity using VQEs. These algorithms are variational and +do not need to cross the critical point even if the reference and target are in different phases. In such a case, CN is +not necessarily aware of the QPT. On the other hand, if the target state is close enough to a phase transition, also in +VQEs, the complexity grows. +The rest of the manuscript is organized as follows. In the next section, II, we discuss the relation between circuit +complexity, in this case CFS from Eq. (14), and the geometry of quantum states that allows extracting the critical +exponents for the derivative of C. This is our first result that emphasizes that through phase transitions CFS has +universal properties. In section III we perform explicit calculations for CFS and CN in two solvable systems, namely +the one dimensional XY-anisotropic and Dicke models. We extract the critical exponents. Then, in section IV, we +perform numerical simulations where CN is computed in two types of algorithms: variational and adiabatic ones. +Concretely we benchmark with exactly solvable models as the Ising model, and we complement our discussion with +non-integrable Hamiltonians as the ZZXZ model. Lastly, we discuss these results and conclude the paper in V. Some +technical issues are left for the Appendices. The code used to obtain the numerical results is available upon request. +II. +COMPLEXITY AND THE GEOMETRY OF STATES CLOSE TO A QUANTUM PHASE +TRANSITION. +In this section, we discuss general aspects for the complexity close to a QPT. To be as general as possible, we find +it convenient to focus on CFS, Eq. (14). Within this geometric formalism, we see that, in general, the complexity is + +6 +finite, but not its derivative, which can diverge when crossing a QPT. We study its finite size scaling obtaining the +corresponding critical exponents. +A. +Complexity and its derivative when crossing a QPT +We have already argued in section I A 1 that if we are allowed to use any unitary, C is of the order of one. In the liter- +ature, several unitary restrictions have been used: considering one and two qubit gates or considering gaussian states +when moving from reference to target states. In this subsection, we consider another kind of restriction, quite natural +when talking about a QPT. We will consider that one (and only one) parameter, say λ, of the Hamiltonian model is +varied to pass through the QPT, keeping other variables or parameters fixed. Thus the metric is unidimensional. We +know, that in this case, the geodesic is given by: +gλλ ˙λ2 = cte +(18) +Therefore, +C = min +λ(τ) +� T +0 +√gλλ ˙λ dτ ∼ T . +(19) +Below, we will work some examples and we will see that T does not diverge at the QPT. However, if we compute the +derivative instead: +∂C +∂λ = √gλλ . +(20) +It is known that some components of the metric tensor can diverge, thus diverging the derivative of C. Equation (20) +has two consequences. The first one is that, under quite general circumstances, the derivative of C close to a QPT is +related to the metric tensor and inherits its universal properties. The second one is that this derivative can be used +to witness and characterize QPTs. +B. +Finite size scaling +Close to a critical point correlation length diverges as, +ξ ∼ |λ − λc|−1/a , +(21) +with a a critical exponent. Similar relations occur for other thermodynamic quantities. In particular, and for what +interests us, the metric tensor can be written as [45], +gµν ∼ |λ − λc|∆µν/a , +(22) +with ∆µν the corresponding critical exponent. Notice that, for the reasons already explained in section I A 3, from +now on we will be interested in the intensive part of the metric tensor gµν → gµν/Ld, with d the spatial dimensions. +Near a phase transition, finite-size scaling dictates how quantities behave after scale transformations. Very briefly, +after a length scale transformation x′ = αx, time scales as τ ′ = αzτ, with z its critical exponent. This fixes the energy +fluctuations ∆E∆τ ∼ 1 → ∆E′ = α−z∆E. Putting it all together, it is interesting to extract the value of the critical +exponent ∆µν above, which controls how the metric tensor behaves, in terms of other critical exponents that dictate +more fundamental quantities. Looking at equations (4) and (12) and (13) and writing the scaling for the derivatives +of the Hamiltonian as ∂µ′H′ = α−∆µ∂µH we arrive to [45], +∆µν = ∆ν + ∆µ − 2z − d . +(23) +Finally, merging, (21) and (22), we find that close enough to the transition, where the relevant length is given by the +system size, L, we arrive to +gµν ∼ L−∆µν . +(24) +As a consequence of all of this and using (20), when a single parameter is varied across the QPT we have the scaling: +∂C +∂λ ∼ L−∆λλ/2 . +(25) + +7 +It is remarkable that the complexity derivative scaling is dictated by universal exponents, whenever one parameter is +varied to cross a critical point. In particular, if ∆λλ > −2 the derivative is sub-extensive. If ∆λλ = −2 it is extensive +and if ∆λλ < −2 is superextensive. +III. +SOLVABLE HAMILTONIANS +Let us test the above ideas on a couple of solvable models: the one dimensional quantum Ising model [46] and the +Dicke [47–49] model. +A. +Quantum Ising model +The transverse field Ising model (Periodic Boundary Conditions will be assumed) is +H = −J +L +� +j=1 +σz +j σz +j+1 + +L +� +j=1 +σx +j . +(26) +Hamiltonian (26) can be solved via the Jordan-Wigner transformation [46]. This Ising model has a second order phase +transition occurring at Jc = 1(−1) in the N → ∞ limit. For Jc > 1(Jc < −1) the Z2 symmetry is spontaneously broken +and the g.s. is ferromagnetically (antiferromagnetically) ordered. W.l.o.g. we fix our attention in the paramagnetic- +ferromagnetic transition occurring at Jc = 1. On top of that, the ground state can be written in terms of fermionic +excitations (after the Jordan-Wigner transformation) as, +|ψgs⟩ = +� +k>0 +� +cos(θk/2) + ieiφ sin(θk/2) a† +ka† +−k +� +|0⟩ . +(27) +with k = (2m−1)π +L +3 and, +tan θk = +−J sin k +1 + J cos k . +(28) +For the rest of the section the metric tensor (12) is needed. It has been computed several times already [50, 51] +gJJ = 1 +4 +� +k +�∂θk +∂h +�2 +. +(29) +In the thermodynamic limit, the k-sum is an integral � +k → N/π +� +and it can be computed explicitly, yielding +gJJ = +−π(J2 − 1) + i +� +J2 + 1 +� � +log +� +− 2i(J+1) +J−1 +� +− log +� +2i(J+1) +J−1 +�� +32J2(J2 − 1) +. +(30) +1. +Complexity through QPTs +From Eq. (30) we see that gJJ diverges at J = Jc. This is the reason behind the divergence in the derivative of the +complexity at the QPT, Cf. Eq. (20). In figure 6, we plot CFS both in the continuum and for N-finite using either +(30) or the sum (29). In both cases, the integral (19) is computed. It is evident that the complexity does not diverge +at the QPT, but its derivative does, inheriting this behaviour from the metric tensor, Cf. Figs. 6a and b. For the +Ising transition, the exponent a = 1, Cf. Eq. (21). We know that ∆hh/a = 1, so the complexity derivative diverges +as ∼ L1/2 at the Ising transition. +3 We have used even L and periodic boundary conditions. + +8 +0.9 +1.0 +1.1 +J +0.0 +0.2 +0.4 +0.6 +FS +(a) +L = 100 +L = 1000 +L = 2000 +0.9 +1.0 +1.1 +J +0 +50 +100 +150 +FS +(b) +L = 100 +L = 1000 +L = 2000 +log(L) +log|( +FS)max| += 0.499 +(c) +( +FS)max data +fit to ( +FS)max(L) = A L + B +0.9 +1.0 +1.1 +J +0.0 +0.1 +0.2 +0.3 +N +(d) +L = 100 +L = 1000 +L = 2000 +0.9 +1.0 +1.1 +J +0.0 +1.5 +3.0 +4.5 +N +(e) +L = 100 +L = 1000 +L = 2000 +L +exp (( +N)max) +(f) +( +N)max data +fit to ( +N)max(L) = A log(L) + B +FIG. 1. Study of the complexity for the Transverse Field Ising model. (a) Complexity for different sizes of the chain computed +using the Fubini-Study metric. The discretization in J used is δJ = 1e−3. (b) Derivative of the Fubini-Study complexity for +different L, δJ = 2e−3. (c) Finite size scaling of the maximum in the derivative of the Fubini-Study complexity. See that this +maximum diverges polynomially with the size of the chain. (d) Study of the Nielsen complexity, δJ = 3e−4. (e) Derivative +of the Nielsen complexity for different L, δJ = 3e−4. (f) Finite size scaling of the maximum in the derivative of the Nielsen +complexity. See that this maximum diverges logarithmically. +2. +Relation between CN and CFS +Formula (27) is formally equivalent to the ground state for the 1D-Kitaev model. For the latter, CN has been +computed in [18]. If the reference, target and intermediate states have the same form (27), CN reads: +CN = +� +k +|∆θk|2 +(31) +where ∆θk = θT +k − θR +k and θT +k (θR +k ) are the angles (28) at the target (reference) states. Following the same procedure +as in [18] we checked that ∂JCN ∼ log N, i.e. it diverges logarithmically. This must be confronted with the divergence +(with critical exponent 1/2) for the case of ∂JCFS. This is an important difference. While using the FS distance the +complexity is associated with the fluctuations, cf. Eq. (13), the CN is more related to the angles difference and its +divergence is therefore smoothed. +B. +The Dicke model +The Hamiltonian for the ground state sector of the N-spin Dicke model can be written in terms of total spin +operators of spin S = N/2 as [52] +H = ωca†a + ωsSz + +λ +√ +2S +� +a† + a +� +(S+ + S−) , +(32) + +9 +where the spin and ladder operators obey the canonical commutation relations [Sz, S±] = ±S±, [S+, S−] = 2Sz. This +model can be solved in the thermodynamic limit, S → ∞, with a Holstein-Primakoff transformation on the spins +S+ → +√ +2Sb† +� +1 − b†b +2S , +(33) +S− → +√ +2S +� +1 − b†b +2S b , +(34) +Sz → b†b − S , +(35) +(36) +yielding +H = ωca†a + ω† +sa + λ +� +a† + a +� +� +b† +� +1 − b†b +2S + +� +1 − b†b +2S b +� +− ωcS . +(37) +In the normal phase of the Dicke model we can obtain an effective Hamiltonian for S → ∞ by neglecting terms +with 2S in the denominator in the Hamiltonian of Eq. (37), resulting in a completely symmetric model of coupled +harmonic oscillators, one corresponding to the physical oscillator and the other corresponding to the spins within the +Holstein-Primakoff transformation +H = ωca†a + ω† +sa + λ +� +a† + a +� � +b† + b +� +− ωcS . +(38) +In the superradiant phase, the bosonic modes must be displaced to accommodate the macroscopic occupations that +the spins and field develop in this phase. Once the displacements are introduced, terms with powers of 2S in the +denominator are again neglected in the thermodynamic limit, yielding +H = ωc¯a†¯a + ωs +2µ(1 + µ)¯b†¯b + ωs(1 − µ)(3 + µ) +8µ(1 + µ) +�¯b† + ¯b +�2 + λµ +� +2 +1 + µ +� +¯a† + ¯a +� �¯b† + ¯b +� +, +(39) +where µ = ωzΩ/ +� +4λ2� +and ¯a,¯b are the displaced bosonic operators [53]. We omit the expressions of the displacement +as they are irrelevant in the following. Both the normal and superradiant effective Hamiltonians can be diagonalized +in the real space basis, where they present a gaussian profile given by +g(x, y) = +�ϵ+ϵ− +π2 +�1/4 +e− (R,AR) +2 +, +(40) +where R = (x, y), x and y are the real-space coordinates associated to the modes a(¯a) and b(¯b), A = U −1MU with +U a unitary matrix, M = diag [ϵ−, ϵ+] and ϵ± are the eigenmodes of the system [54]. The overlap of two different +ground states is given by +⟨g|g′⟩ = 2 [det M det M ′]1/4 +[det (M + M ′)]1/2 . +(41) +This allows us to compute the components of the quantum metric tensor for the Dicke model exactly in the thermody- +namic limit. We combine this with finite size results from exact diagonalization of Hamiltonian (32). The results are +shown in Fig. 2. Just like we showed for the case of the Ising model, there is no divergence in CFS, the only signature +of the phase transition is a non-analiticity that is only noticeable in the N → ∞ case. This non-analiticity, or its +precursor in the case of finite N is best revealed as a divergence in the derivative of the complexity, which is naturally +the square root of the metric tensor. Here we are considering the complexity along a λ-path and the divergence is +revealed in ∂CFS = √gλλ. We perform a finite-size scaling analysis of the metric tensor by fitting the maximal values +(∂CFS)max(N) and critical parameters at said maxima λmax(N) to their respective scaling laws +|(∂CFS)max(N) − B| = C · N δ , +(42) +|λmax(N) − λc| = A · N −ν . +(43) +The resulting critical exponents ν = 0.655(22) ≊ 2/3 and δ = 0.6711(15) ≊ 2/3 are in agreement with values reported +in the literature [55]. + +10 +0.4 +0.6 +0 +1 +2 +3 +4 +FS(0 +) +(a) +N +50 +100 +150 +200 +N +0.4 +0.6 +100 +101 +102 +103 +FS +(b) +log| +max(N) +c| += 0.655(22) +(c) +max(N) data +fit to | +max(N) +c| = A N +log(N) +log|( +FS)max(N) +B| += 0.6711(15) +(d) +( +FS)max(N) data +fit to |( +FS)max(N) +B| = C N +FIG. 2. Fubini-Study complexity (a) and its derivative with respect to λ (b) across the phase transition of the Dicke model +as a function of the system size (numerical results) and in the thermodynamic limit (analytical results). Plots on the right +showcase the fits of λmax(N) (c) and (∂CFS)max(N) (d) (extracted from center plot) to their respective finite size scaling laws. +All results are at resonance ωc = ωs = 1 and with a discretization of dλ = 10−3. Numerical results were obtained with a cutoff +for bosonic excitations of Nexc = 30 . +IV. +COMPLEXITY IN A QUANTUM COMPUTER, THE CASE OF GROUND STATE PREPARATION +In this section, we compute CN when preparing ground states in a quantum computer. We study both adiabatic +algorithms and variational quantum eigensolvers (VQEs). Two versions of the former algorithms are discussed: with +and without shortcuts to adiabaticity. +In both cases, the initial state is the “trivial zero” |0⟩ ≡ |00 · · · 0⟩4. Some gates are applied to prepare the ground +state of a given Hamiltonian. Here, we are especially interested when this initial state (that can be understood as the +ground state in the paramagnetic phase) is in a different phase than the final one. In addition, we discuss whether or +not a QPT is crossed during the algorithm. Finally, notice that in quantum computing applications it seems natural +to compute CN and, in particular, its discrete version (the number of gates needed), Cf. Sec. I A 1. Thus, through +this section, we compute CN. +A. +Adiabatic algorithms +A systematic way of finding the ground state of a given Hamiltonian is by adiabatic passage or annealing. Let us +consider the time-dependent Hamiltonian: +H(t) = (1 − λ(t))H0 + λ(t)HT . +(44) +Here, H0 has a trivial ground state (easy to prepare), and HT is the hamiltonian from which we want to obtain its +ground state. Consider that the time-dependent function λ(t) runs from λ(t = 0) = 0 to λ(t = tf) = 1, where tf is +the final time of the algorithm. At t = 0 the state is prepared in the ground state of H0. If ˙λ is sufficiently small +compared to the instantaneous gap, by means of the adiabatic theorem the final state is the ground state of HT +[32]. On the other hand, the adiabatic condition alerts us that as the gap closes, for example in continuous phase +transitions, the execution time, i.e. the circuit depth, scales with the inverse of this gap, thus also C. +Importantly enough the adiabatic condition can be relaxed by introducing counter-diabatic terms. +Generally +speaking, instead of H(τ) (whose ground states are |ψ(tn)⟩) what is evolved is the “modified” Hamiltonian [56, 57]: +H′(τ) = H(τ) + HCD(τ) +(45) +4 In fact, in almost all algorithms the initial state seems to be |00 · · · 0⟩. + +11 +The last term ensures that the time evolution exactly matches the instantaneous ground state of H(τ) no matter how +fast the evolution is. This is known in the literature as shortcuts to adiabaticity and HCD is called counter-diabatic +Hamiltonian. There are different ways of writing HCD. In its original form we can write: +HCD(τ) = i ˙λµ � ⟨m|∂µH|n⟩ +En − Em +|m⟩⟨n| + h.c. +(46) +with ∂µH ≡ ∂H/∂λµ. We emphasize that at times 0 and t, |ψR⟩ and |ψT ⟩ are ground states of H(0) and H(t) +respectively. Explicitly |ψT ⟩ = T e− +� t +0 H′(τ) dτ|ψR⟩. Here τ means time, cf. with Eq (1). To connect this evolution +with the previous sections, we note that the fidelity susceptibility can be written in terms of the HCD(τ) fluctuations +[Cf. Eq. (9) and (13)]: +χF = ⟨(H(τ) + HCD(τ))2⟩ − ⟨(H(τ) + HCD(τ))⟩2 = ⟨HCD(τ)2⟩ = ˙λµ ˙λν gµν . +(47) +In practice HCD is difficult to find. Therefore, a systematic although approximate way of writing is convenient. +Following [58] it can be rewritten as, +HCD(τ) = ˙λµAµ +(48) +Here, A is the adiabatic gauge potential that can be approximated as: +A(ℓ) +µ += i +ℓ +� +k=1 +αk [H, [H, . . . [H +� +�� +� +2k−1 +, ∂µH]]] +(49) +where (l) is the “degree of approximation”. On top of that, the {αk} are found variationally by minimising the action +[59]: +Sℓ = Tr +� +G2 +ℓ +� +, +Gℓ = ˙λµ � +∂µH − i +� +H, A(ℓ) +µ +�� +(50) +In many cases of interest, in the adiabatic protocol, H(τ) is expected to be a local Hamiltonian, in particular a +two body one. Notice that due to nested commutators, the higher the order (l), the longer the range of interaction. +Following the functional (6) three, four, or higher order body interactions will be highly penalised. Thus, in what +follows, we will restrict ourselves to l = 1 that introduces two body interactions at most. This, in turn, provides a +systematic way of preparing, via trotterization, quantum states. +1. +Complexity in adiabatic algorithms +As has been previously discussed, in order to compute the complexity as defined by Nielsen [Cf. Sec.(I A 1)], we +only need to express our unitary operation as the time evolution of some Hamiltonian. In the present case, it is +straightforward, with and without shortcuts, as the Hamiltonian (45) is given explicitly. We study the Ising model in +transverse field and the ZZXZ model. For both, we use the function λ(t) = sin2 � π +2 sin2 � πt +2T +�� +to drive the evolution +from the initial Hamiltonian, H0, to the target one, HT. +For the Ising model, we start from H0 = hx +� +i σx +i , leaving the transverse field fixed and switching on the spin-spin +interaction until we reach Hint = J � +i σz +i σz +i+1. The counter-diabatic Hamiltonian follows from equations (46), (49) +and (50) with l = 1 yielding HCD(t) = ˙λ(t)α(t) � +i(σy +i σz +i+1 + σz +i σy +i+1), where α(t) is the variational parameter in Eqs. +(49) and (50). The full-time-dependent Hamiltonian reads +H′(t) = H0 + λ(t)Hint + HCD(t) . +(51) +Since we need to implement our unitary evolution via Trotter decomposition, we have to split the total time T in +T/δT steps, where δT is the time discretization employed. The smaller δT, the more precise the implementation +will be but more gates will be needed, increasing the complexity of the operation. In the simulations presented here +δT = min(0.1, T/30). +Therefore, following the prescription given for the Nielsen complexity, CN, given in (6), it is straightforward to obtain5: +CN = +� T +0 +dt +� T +δT +�1/2 � +N + (N − 1)λ(t)2J2 + 2(N − 1) ˙λ2(t)α2(t) +�1/2 +(52) +5 For the adiabatic evolution without shortcuts, the expression would be the same but with α = 0. + +12 +10 +1 +100 +101 +102 +T +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +|J| = 0.25 +|J| = 0.75 +|J| = 1.00 +|J| = 1.25 +|J| = 1.75 +0.00 +0.25 +0.50 +0.75 +1.00 +t/T +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +|E1 +E0| +(b) +0.25 +1.00 +1.75 +|J| +50 +100 +150 +200 +250 +/L +(c) +L = 6 +L = 8 +L = 10 +L = 12 +L = 14 +FIG. 3. Complexity study for the Transverse Field Ising model using the adiabatic algorithm. (a) Evolution of the fidelity +obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing time lengths of the full +algorithm and different target Js for L = 12. At shorter times the shortcuts provide better results, being identical to the +simple case (without shortcuts) for the longest times. (b) Evolution of the gap between the ground state and the first excited +state during the algorithm for the same values of J as in (a). The gap closes with an increasing value of |J|, explaining why +longer times are needed for the larger |J| to obtain the same fidelity. (c) Complexity per spin computed for different sizes +with shortcuts (solid lines) and without shortcuts (dashed). As the gap closes, more gates are needed to achieve the fidelity +threshold (0.9 in this case) but we do not find relevant differences between applying shortcuts or not in the final result for the +complexity. +Figure 3 summarises our results for the Ising model with adiabatic algorithms. The transverse field was fixed to +hx = 1 and different values of J were studied. In Figure 3a we plot the fidelity between the final state obtained +adiabatically and the target state. As expected, the longer the time the better. We also confirm that at lower times +higher fidelities are achieved thanks to the counter-diabatic term. Figure 3b shows the gap evolution within the +adiabatic algorithm, giving insights about why as |J| is greater, it takes more time to achieve a high fidelity: the +gap becomes smaller. Finally, the last panel 3c shows the actual Nielsen complexity values. Reflecting the fidelity +behaviour, the complexity jumps around the transition as the gap is closing. As the adiabatic theorem states, crossing +a QPT is hard for this kind of algorithms and complexity serves the purpose of quantifying such difficulty. +In order to check if this holds in other models, we also study the so-called antiferromagnetic ZZXZ model: +HT = J +� +i +σz +i σz +i+1 + hx +� +i +σx +i + hz +� +i +σz +i . +(53) +Due to the combination of longitudinal and transverse fields, this is a non-integrable model. It is ideal, then, to +explore the phenomenology of complexity beyond the exactly solvable models considered so far. In Fig. 4 we draw +the phase diagram of the model at zero temperature as a function of the fields applied to the spins and the exchange +constant [60]. The critical line separates paramagnetic and antiferromagnetic phases. For our particular purposes, +keeping the same initial Hamiltonian, H0, we set the transverse field, hx = 1 and the target longitudinal field to +hz = 0.75. We thus study the quantum phase transition appearing when moving to different target values of J. This +path is shown as the red line in Fig. 4, where the final point marks the maximum value simulated for the target +J. Therefore, the transverse field is going to be fixed while we turn on both the longitudinal field and the magnetic +interaction. The counter-diabatic term can be computed in the same fashion as before, getting the same result as in +[58]. The time-dependent Hamiltonian reads +H′(t) = H0 + λ(t) +� +i +� +Jσz +i σz +i+1 + hzσz +i +� ++ HCD(t) +(54) +and the complexity acquires the following expression +CN = +� T +0 +dt +� T +δT +�1/2 � +N +� +1 + h2 +zλ2(t) +� ++ (N − 1)λ(t)2J2 + ˙λ2(t) +� +Nα2(t) + 2(N − 1) +� +β2(t) + γ2(t) +���1/2 +. +(55) + +13 +0.0 +0.5 +1.0 +1.5 +hx / J +0.0 +1.0 +2.0 +hz / J +AFM +PM +FIG. 4. Phase diagram of the ZZXZ model for zero temperature. The black dotted line signals the critical region between +phases for different ratios of the fields (hx, hz) to the magnitude of the exchange interaction (J). The coloured lines depict the +path followed for the adiabatic algorithm (red) and the values computed in the VQE (blue) [Cf. Sec. IV B]. +In figure 5 we show the results obtained for the different values of J and the chain sizes, N. The behaviour is equiva- +lent to the previous model except that for sufficiently large values of J, the gap decreases sharply, closing completely +(see figure 5b), causing the counter-diabatic terms to cause more error than the simple evolution itself, as we can see +in panel (a) of the same figure. This is a consequence of the fact that our expression for the counter-diabatic term is +not exact, but a first-order approximation of a general expression [cf Eq. (49)]. The smaller the gap, the more careful +we will have to be with the design of the CD term. +Putting all together, we can conclude that, due to the gap closing at the QPT the CN with adiabatic algorithms +diverges with system size. The inclusion of shortcuts does not provide any significant advantage in terms of complexity +reduction. This is because we have constrained these shortcuts to be as local as possible, in our case l = 1 in (50), +introducing two body interactions at much. It is expected that by introducing long-range terms in (46) the complexity +decreases as the system approaches to the QPT. This can be compared to the previous section II, where there was no +restriction to local operations, so both CF and CN remained finite despite crossing the QPT. Other paths investigated +in this work are sent to App. (B). +B. +Circuit Complexity in VQEs +VQEs, introduced in [34], use the fact that any quantum state can be written in terms of a unitary operation as +|φ(⃗θ)⟩ = U(⃗θ)|0⟩ , +(56) +where U(⃗θ) is a parameterized unitary that transforms the initial state into the desired wave function |φ(⃗θ)⟩. This +unitary can be implemented in a quantum circuit as a set of quantum gates. The expectation value of the Hamiltonian +where we encode our problem (H) results +⟨H⟩ = ⟨0|U †(⃗θ)HU(⃗θ)|0⟩ ≥ E0 . +(57) +The optimization process consists on minimizing the average energy of the parameterized state: +EVQE = min +θ +⟨0|U(⃗θ)†HU(⃗θ)|0⟩ ≥ E0 . +(58) + +14 +10 +1 +100 +101 +102 +T +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +J = 0.25 +J = 0.75 +J = 1.00 +J = 2.00 +J = 5.00 +0.00 +0.25 +0.50 +0.75 +1.00 +t/T +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +|E1 +E0| +(b) +10 +2 +10 +1 +100 +J +102 +103 +/L +(c) +L = 6 +L = 8 +L = 10 +L = 12 +L = 14 +FIG. 5. Complexity study for the ZZXZ model using the adiabatic algorithm. The phenomenology is essentially the same as +for the TFI model. (a) Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed +lines) for increasing time lengths of the full algorithm and L = 12. In this case we see that, for sufficiently large values of J, no +applying shortcuts works better than applying them. This is explained by the gap closing much more abruptly than in the TFI +model, as can be seen in (b). (c) Complexity per qubit computed for different sizes with shortcuts (solid lines) and without +shorcuts (dashed). As the gap closes, more gates are needed to achieve the fidelity threshold (0.9 in this case). +The algorithm can be divided into three different stages. First, we need to choose the trial wave function (see +Eq.(56)). Choosing the unitary U(⃗θ) is equivalent to constructing the quantum circuit that transforms the initial +state into the parameterized wave function. +The circuit used to achieve |φ(⃗θ)⟩ is called the ansatz and can be +represented as, +q0 : +U(⃗θ) +q1 : +q2 : +q3 : +q4 : +(59) +Choosing an appropriate ansatz is crucial for the optimization process. This choice depends completely on the +model we are simulating and the set of gates available. We will dig into our choice of unitary below. The next step +is constructing the Hamiltonian of the problem. Since this Hamiltonian is going to be evaluated later, Eq. (57), it +must be written in terms of Pauli strings {I, σx, σy, σz}⊗N. Pauli operators are related to spin observables, which +are suitable for direct measurement in quantum devices [61]. With the Hamiltonian and the wave function defined, +we can measure the energy of the state, which is the cost function. To compute this cost function, the expectation +values of the Pauli observables are measured determining the value of the energy. Since the technique uses quantum +and classical processors, VQEs are cast as hybrid algorithms. Our results are numerical and our Python code simply +computes the product of the matrices U(⃗θ)†HU(⃗θ) previously defined and then projects onto the zero state obtaining +⟨0|U(⃗θ)†HU(⃗θ)|0⟩. We will not discuss its measurement overhead. Here, we are interested in the circuit complexity +for reaching the desired ground state. +The final step is to minimize this cost function through the variation of the parameters θ in the wave function. At +the end of each iteration we obtain the value of the energy (58). Then, a classical optimizer determines the best +direction of variation of the parameter vector ⃗θ to minimize this value. We use as many iterations as needed until we +converge to a final solution for the coordinates of the parameter vector. Ideally, this solution is the absolute minimum +in the space of parameters. Still, obtaining this minimum is not an easy task. The optimizer can get trapped in +local minima which will imply serious limitations in the minimization process. This problem and others have been +previously discussed in the literature [61, 62] and are out of scope for this work. + +15 +Summarizing, we assume a given ansatz, the set of available gates in U(⃗θ) in (56) and the hybrid algorithm finds the +optimal solution. CN counts the number of gates, and once the VQE circuit is chosen, it can be done systematically. +1. +Local VQE ansatz +We focus on a fixed geometry that is suitable for one-dimensional systems with single and two-qubit gates, besides +the two-qubit gates act only on contiguous qubits. This ansatz can be interpreted as a Trotter approximation of +continuous evolution by a local 1D Hamiltonian [63]. In this case, we can separate the terms of the Hamiltonian that +act on even and odd links and obtain two sets, each made of mutually commuting gates. In particular, the circuit is +given by +q0 : +RY (θ[0]) +• +• +RZ (−π/2) +q1 : +RY (θ[1]) +RZ (π/2) +RZ (−π/2) +RY (θ[5]) +• +• +RZ (−π/2) +q2 : +RY (θ[2]) +• +• +RZ (−π/2) +RY (θ[6]) +RZ (π/2) +RZ (−π/2) +q3 : +RY (θ[3]) +RZ (π/2) +RZ (−π/2) +RY (θ[7]) +• +• +RZ (−π/2) +q4 : +RY (θ[4]) +RZ (π/2) +RZ (−π/2) +(60) +i.e. +it consists of fundamental blocks (or layers) (separated by dashed lines above). +Each layer is made out of +single-qubit rotations Ry(θ) and control-Z gates (CZ). At the end of the circuit, we add a final column of rotations +(Ry). +For computing CN we rewrite the CZ gates in terms of Pauli operators, count the gates and use equation (6). This +is a routine process that we send to Appendix A. Here, we just give the final result: +CN = +d +� +j=1 +� +� +� +� +2(L−1) +� +i +� +θj +i +2 +�2 ++ 3(L − 1) +�π +4 +�2 +. +(61) +2. +VQE complexity through QPTs +As before, we focus on Ising and ZZXZ models, Eqs. (26) and (53). In figure 6 we summarize our results for the +Ising Hamiltonian. In panel a) we plot the complexity using the local VQE ansatz for obtaining the ground state at +a given J. We see that CN grows when the ground state approaches the QPT, that in this case is given by Jc ∼= 1 6. +In fact, close enough to the transition the VQE cannot reach an acceptable ground state for a maximum depth of 8 +(in our simulations). This can be checked in panel b) where the fidelity between the state obtained within the VQE +algorithm and the exact ground state falls below 0.8 in the gray region of panel b). Therefore, all indications are that, +also with VQE, complexity increases as QPT is approached. With what has been said so far this should not be a +surprise. Perhaps, the most remarkable thing here is that the complexity is only high near the transition. When the +target state is far from the critical point the complexity drops, even though the latter and the reference state may be +in different phases. This is due to the fact that, contrary to the adiabatic algorithm, the VQE does not necessarily +need to visit states in the transition region to go from |ψR⟩ to |ψT ⟩, it can circumvent criticality and go directly from +one phase to another. This is easy to understand in the Ising model, because in the paramagnetic phase the ground +state is approximately given by |+, ..., +⟩ (|+⟩ = 1/ +√ +2(|0⟩ + |1⟩)), Cf. Eq. (26). This is easy to prepare: it can be +obtained with single qubit rotations from the reference state |ψR⟩ = |0, ..., 0⟩. +Since we are dealing with finite simulations, deep in the ferromagnetic phase, the Z2 symmetry is not broken so +the ground state manifold found by exact diagonalization is spanned by the states 1 +2 (|0, ..., 0⟩ ± |1, ..., 1⟩). The VQE +6 We say Jc ∼ += 1 since our simulations are done in finite systems. Jc = 1 in the thermodynamic limit. + +16 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +J +1.0 +2.0 +3.0 +4.0 +/L +(a) +1 +2 +3 +4 +5 +6 +7 +8 +Layers +0.5 +0.75 +1.0 (b) +J = 0.90 +J = 0.92 +J = 0.94 +J = 0.96 +J = 0.98 +J = 1.00 +J = 1.02 +J = 1.04 +J = 1.06 +J = 1.08 +J = 1.10 +J = 1.30 +J = 1.50 +J = 1.70 +FIG. 6. Transverse Field Ising model with bias, ϵ = 0.001 and size N = 12. (a) Complexity per size as a function of J. The +grey zone indicates that the VQE does not converge for points inside that region in a reasonable number of layers to the fidelity +threshold (0.9). (b) Fidelity obtained for different numbers of layers for points inside the grey box in (a) and in its vicinity. +For those points whose fidelity is above the threshold (0.9) it has only been plotted the best result for clarity’s sake. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +J +1.0 +2.0 +3.0 +4.0 +/L +(a) +1 +2 +3 +4 +5 +Layers +0.5 +0.75 +1.0 (b) +J = 0.90 +J = 0.92 +J = 0.94 +J = 0.96 +J = 0.98 +J = 1.00 +J = 1.02 +J = 1.04 +J = 1.06 +J = 1.08 +J = 1.10 +J = 1.30 +J = 1.50 +J = 1.70 +J = 1.90 +J = 2.10 +J = 2.30 +J = 2.50 +FIG. 7. ZZXZ Ising model for size N = 12. (a) Complexity per size as a function of J. The grey zone, as in the TFI model, +indicates that the algorithm fails to achieve fidelity over 0.9 for points within that region. (b) The fidelity behaviour with the +depth of the ansatz shows that, again, once the QPT is crossed the algorithm cannot reach fidelities over 0.9. In contrast to +the TFI model, here we don’t recover high fidelity once we are fully in the antiferromagnetic phase, reaching a maximum value +of 0.5 for the highest values of J. +reaches instead one of the fully polarized states, either |0, ..., 0⟩ or |1, ..., 1⟩, given that they are degenerate with the +symmetric ground state. Our convergence criterion is based on reaching a fidelity of 0.9 between the state generated +by the VQE and the result of exact diagonalization. Because of the discrepancy in the ground states obtained by +both methods, in the ferromagnetic phase the fidelity is capped at 0.5 and the convergence criterion is never satisfied. +Driven by the physics of actual QPTs in the thermodynamic limit, where the symmetry is (spontaneously) broken, +we decide to add a small bias, ϵ � σz +i in (26). In doing so, the VQE should a priori be able to reach full convergence. +This is indeed the case as can be seen in Fig. 6. Additionally, convergence is reached in very few layers, equivalently +to what is observed in the PM phase. This low complexity can be explained by noticing that the symmetry broken +ferromagnetic ground state is either the reference state or can be obtained from it by means of single-qubit rotations. +We now consider the ZZXZ model, Hamiltonian (53)7. Here, we are not going to explicitly break the symmetry +7 The parameters employed in the simulations are depicted as the blue line in Fig. 4, namely hx = 1, hz = 0.75 and J ∈ (0., 2.5] + +17 +0 +1 +2 +J +-0.5 +0.0 +0.5 +1.0 +Magnetization +(a) +Total +Even sites +Odd sites +0 +1 +2 +J +0.5 +0.75 +1.0 (b) +Single state +Subspace +0 +1 +2 +J +0.994 +1.0 +Energy accuracy +(c) +FIG. 8. VQE state characterization in the ZZXZ model for N = 12. (a) Magnetization of the spin chain as a function of J +obtained from the states generated by VQE. The solid black line represents the total magnetization per site that the spin chain +should have (obtained via exact diagonalization) whereas the dashed black line sets the magnetization per site in even/odd +sites. (b) Evolution of the best fidelity obtained as a function of J. In blue it is computed the fidelity as the overlap between +the state generated by the VQE and the exact ground state; in red it is computed as the projection onto the subspace generated +by the ground state and the first excited state. (c) Energy accuracy obtained for the same configurations displayed in the other +panels computed as 1 − Erel, being Erel the relative error between the energy obtained from VQE and the exact value. +in order to discuss the scenario in which the symmetric ground state is sought. +In the ZZXZ model, the QPT +separates paramagnetic (PM) and antiferromagnetic (AFM) phases. In the PM phase, the behavior is analogous to +the Ising model, Cf. Figs. 6 and 7. Deep in the AFM phase, the ground state manifold is spanned by the states +|ψAFM⟩ ∼= +1 +√ +2(|1, 0, 1, 0, ...⟩ ± |0, 1, 0, 1, ...⟩). Following the previous discussion, the VQE does not reach the symmetric +ground state. Therefore, we see that CN grows as it approaches the phase transition (with our parameters Jc ∼= 1, +see Fig. 4) but does not decrease afterwards. At some point near criticality, the VQE cannot produce a ground state +with a fidelity larger than 0.9, see panel b), similar to the Ising model case. Here, however, the state remains difficult +for the VQE after the near-transition region is surpassed. This is further confirmed in figure 8. There, we can see +that although the total magnetization is well reproduced by the VQE (also the energy, in panel c), once we enter +the antiferromagnetic phase the VQE generates either |1, 0, 1, 0, ...⟩ or |0, 1, 0, 1, ...⟩, as can be seen by computing the +magnetization per site, which should be close to 1/2 in the exact ground state. However, the VQE gives 0 (1) for the +even (odd) sites. To conclude our characterization, we see that all this is consistent with obtaining a F = 0.5, as well +as a F ∼= 1 if we compare the state generated by the VQE with the projection onto the subspace generated by the +ground state and the first excited state. +V. +DISCUSSION +Knowing in advance how much a computation will cost, even if only approximately, is of great help. Unfortunately, +this estimation can pose a great challenge. Computer science has traditionally categorized problems into different +complexity classes, allowing one to know whether a given problem is tractable on a classical computer. +For a +quantum computer, we can ask a similar question to know if the task we want to tackle is going to be feasible with +the architecture we have at hand. For this purpose, the concept of circuit complexity was invented. Again, knowing +the complexity of each task in any architecture seems too general to be able to give a concrete answer. On the other +hand, we can shed some light on generic situations where some kind of general statement can be made. This is the +idea that motivated us to write this manuscript. We have studied the situation in which a critical region is crossed in +the process of preparing a state. +Our work has shown that, regardless of the type of complexity one chooses, and for diverse models, it appears +that complexity grows if the algorithm visits states near a phase transition. We have further proven that this is a +characteristic trait of typical algorithms for state preparation such as VQE and adiabatic evolution. The degree of +divergence does depend on the definition of complexity used and on the allowed gates. In the case of local ans¨atze +or evolutions, C tends to diverge as the system size grows. Importantly, we have shown that VQEs, to the extent +that they can go “directly” from the reference to the target state, can potentially avoid the divergence in complexity + +18 +even if the reference and target states lie in different classes. Whether this is possible depends on the model, as it +is determined by the degree of entanglement of the target and reference states. In the case of adiabatic algorithms, +keeping the complexity down seems to be a matter of allowing non-local gates in the evolution, to fully exploit +shortcuts to adiabaticity. This is supported analytically in Sec. III. Here, the Ising critical point is traversed along +a restricted path of states of the form (27). Despite this restriction, these states are sufficiently non-local for CN to +remain finite. +The impact of our work on the preparation of states in a quantum machine seems straightforward. What our +results mean in the field of holography is another matter. Unfortunately, we do not have the knowledge to anticipate +anything, but it would be interesting to think in this direction. Other ideas not discussed here would be the use of +other types of complexity such as Krylov [22, 64–66] or mixed states and their behavior in thermal phase transitions. +We leave this for future work. +Note Added in Proof.- While we were finishing writing this manuscript, the paper [67], which discusses the impor- +tance of local and non-local gates in the computation of complexity, appeared in the arXiv. +ACKNOWLEDGMENTS +The authors thank Fernando Luis for his helpful comments and insights during the preparation of this manuscript. +The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), +the Spanish Government Grants PID2020-115221GB-C41/AEI/10.13039/501100011033 and TED2021-131447B-C21 +funded by MCIN/AEI/10.13039/501100011033 and the EU “NextGenerationEU”/PRTR, the Gobierno de Arag´on +(Grant E09-17R Q-MAD) and the CSIC Quantum Technologies Platform PTI-001. This work has been financially +supported by the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through +the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, +Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda”. J +R-R acknowledges support from the Ministry of Universities of the Spanish Government through the grant FPU2020- +07231. +Appendix A: Complexity associated to the VQE +To compute F we must express our ansatz as a unitary of the form U = T e−i +� T +0 H(τ) dτ where H is written in terms +of Pauli matrices {σx, σy, σz} and tensor products of these matrices. To do so, recall that the local VQE ansatz only +contains one and two qubit gates (between nearest neighbors). To construct the effective Hamiltonian, notice that +Ry(θi) = e−i θi +2 σy . +(A1) +Now, the C-Z gate, can be decomposed +q0 : +• += +q0 : +• +• +RZ (−π/2) +q1 : +• +q1 : +RZ (π/2) +RZ (−π/2) +Therefore +C-Z = e−i π +4 (σ0 +zσ1 +z−σ0 +z−σ1 +z) = e−i π +4 σ0 +zσ1 +zei π +4 σ0 +zei π +4 σ1 +z . +(A2) +If we substitute in the representation of a layer of the ansatz, we find that each one of the building blocks marked +with a dashed line in the main text is represented by a unitary of the form +U = e−i � +j +θj +2 σj +ye−i π +4 (σ0 +zσ1 +z−� +j σj +z) ≈ e−i(� +j +θj +2 σj +y+ π +4 σ0 +zσ1 +z− π +4 +� +j σj +z) , +(A3) +Finally, +H = +� +j +θj +2 σj +y + π +4 σ0 +zσ1 +z − π +4 +� +j +σj +z . +(A4) + +19 +More generally, each layer of the ansatz can be written as an operator of the type +H = Heven + Hodd , +(A5) +where +Heven = 1 +t +�� +i +θi +2 σi +y − π +4 +L−1 +� +i=0 +σi +z + π +4 +� +i=even +σi +zσi+1 +z +� +, +(A6) +Hodd = 1 +t +�L−2 +� +i=0 +θi+L +2 +σi +y − π +4 +L +� +i=1 +σi +z + π +4 +� +i=odd +σi +zσi+1 +z +� +. +(A7) +Now we use a Trotter decomposition to compute the complexity of this circuit. We have fixed the total evolution +time to 1 and each layer is considered a Trotter step. This way, t = T/#steps = 1/d, where d is the number of layers +of the circuit. Now, using Eq. (6) we find +F(U) = +� +� +� +� +2(L−1) +� +i +� +dθi +2 +�2 ++ 3(L − 1) +� +dπ +4 +�2 +. +(A8) +Here, L − 1 corresponds to the number of C-Zs in the layer, with L is the number of qubits. Now, the complexity is +nothing but the integral of this functional across the number of layers in the circuit +CN = +� 1 +0 +F(U)dt ≈ +d +� +j=1 +F(U)1 +d , +(A9) +which leads to Eq. (61) in the main text. +Appendix B: Other paths in the adiabatic algorithm +In Sec. IV A we show an adiabatic evolution for the Transverse Field Ising model where we let the field fixed as we +increase the interaction between the neighbouring spins. However, we could have let the interaction fixed and switched +on the transverse field, going from a classical Ising model to the TFI. In Fig. 9 we show this possible adiabatic path. +The behaviour of the gap between the ground state and the first excited state is qualitatively different, to the point +of even closing. This results in a much worse performance for small values of the field. +Similarly, the gap behaviour also causes a big impact in the ZZXZ model. In Fig. 10 we show that for odd number +of spins in the chain we get a higher complexity as the gap presents a dip at intermediate times which makes necessary +longer times to achieve the fidelity threshold. +[1] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Science 311, 1133 (2006). +[2] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Phys. Rev. A 73, 062323 (2006). +[3] M. Dowling and M. Nielsen, Quantum Inf. Comput. 8, 861 (2008). +[4] R. A. Jefferson and R. C. Myers, J. High Energy Phys. 2017, 1 (2017). +[5] S. Chapman, M. P. Heller, H. Marrochio, and F. Pastawski, Phys. Rev. Lett. 120, 121602 (2018). +[6] M. Guo, J. Hernandez, R. C. 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(a) +Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing +time lengths of the full algorithm and L = 12. We see a clear difference with the plot in the main text, where the field is fixed +and we vary the interaction, J. The gap closes much earlier for small field values (b), making the algorithm need much longer +times to achieve high fidelity. +10-1 +100 +101 +102 +T +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F +(a) +J = 0.25 +J = 0.75 +J = 1.00 +J = 2.00 +J = 5.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t/T +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +|E1 − E0| +(b) +10-2 +10-1 +100 +|J| +102 +103 +104 +105 +C/L +(c) +L = 5 +L = 7 +L = 9 +L = 11 +L = 13 +FIG. 10. Evolution in the ZZXZ model of the fidelity (a), the gap between the ground state and the first excited state (b) +and the complexity (c) for spin chains with odd number of constituents. 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Svensson, arXiv:2211.14106 (2022). + diff --git a/BNE3T4oBgHgl3EQfswvl/content/tmp_files/load_file.txt b/BNE3T4oBgHgl3EQfswvl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..642c6682a2db3f7ea3e740efc4ea8079713949e7 --- /dev/null +++ b/BNE3T4oBgHgl3EQfswvl/content/tmp_files/load_file.txt @@ -0,0 +1,1353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf,len=1352 +page_content='Circuit Complexity through phase transitions: consequences in quantum state preparation Sebasti´an Roca-Jerat,1 Teresa Sancho-Lorente,1 Juan Rom´an-Roche,1 and David Zueco1 1Instituto de Nanociencia y Materiales de Arag´on (INMA) and Departamento de F´ısica de la Materia Condensada, CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain (Dated: January 13, 2023) In this paper, we analyze the circuit complexity for preparing ground states of quantum many- body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, how this complexity grows as the ground state approaches a quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We discuss different definitions of complexity, namely the one following the Fubini- Study metric or the Nielsen complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We also explore different models: Ising, ZZXZ or Dicke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In addition, different forms of state preparation are investigated: analytic or exact diagonalization techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigen- solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We find that the divergence (or lack thereof) of the complexity near a phase transition depends on the non-local character of the operations used to reach the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For Fubini-Study based complexity, we extract the universal properties and their critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In practical algorithms, we find that the complexity depends crucially on whether or not the system passes close to a quantum critical point when preparing the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' While in the adiabatic case it is difficult not to cross a critical point when the reference and target states are in different phases, for VQE the algorithm can find a way to avoid criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Introduction 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity overview 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity `a la Nielsen 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Circuit Complexity from the Fubini-Study metric 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Some remarks comparing both approaches 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Main results and manuscript organization 5 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity and the geometry of states close to a quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity and its derivative when crossing a QPT 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Finite size scaling 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Solvable Hamiltonians 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Quantum Ising model 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity through QPTs 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Relation between CN and CFS 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The Dicke model 8 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity in a quantum computer, the case of ground state preparation 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Adiabatic algorithms 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity in adiabatic algorithms 11 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Circuit Complexity in VQEs 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Local VQE ansatz 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' VQE complexity through QPTs 15 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Discussion 17 Acknowledgments 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity associated to the VQE 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Other paths in the adiabatic algorithm 19 References 19 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='04671v1 [quant-ph] 11 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' INTRODUCTION How much does it cost to generate a target quantum state from another reference state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is a rather general question that has been discussed in quantum information for obvious reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In quantum computation it is desirable to obtain the result with the minimum set of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This number is, roughly speaking, the computational cost and it is called circuit complexity (C) [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is, let us say, the quantum analog of the concept of computational complexity in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Importantly enough, this cost builds upon a concrete physical architecture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e the available set of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, C not only depends on the reference and target states but on the restrictions for reaching the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is quite natural if one thinks of an actual quantum computer where the possible operations have restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Note that, if any unitary were allowed, a simple rotation would achieve the goal and every quantum state would be easily prepared, so that (essentially) the complexity would be a trivial quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, also in analytic calculations, the path between the reference and target is restricted to a set, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' gaussian states [4–8] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Beyond quantum computation, circuit complexity is a relevant concept in quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, for its consequences in holography [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For those who are not experts (like us), we can say that holography describes quantum gravity within a region of space by looking at the boundary of that region, that is described by a non gravitational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Then, any bulk quantity in the gravitational theory is equivalent or dual to another quantity in the boundary of the non gravitational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' One of the main problems of this duality is that the volume behind the black hole horizon keeps growing for a very long time while the entanglement at the boundary saturates at much shorter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' One possible solution is to conjecture that the dual of volume is not entanglement but complexity, via the identification Complexity = Volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is because we expect that volume is an extensive quantity, while entanglement (typically) fulfils an area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, the calculations of complexity are beyond the quantum information community and different calculations in field theories have been discussed in the literature [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The notion of complexity (C) is much related to the geometry of states (or operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is a measure of the distance between two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, one possible choice for C is finding the geodesics in the Fubini-Study metric in the projected Hilbert space for the case of pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For mixed states different measures have been introduced via state purification [14] or distance measures for mixed states as the Bures distance [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This geometric background is a powerful way to understand complexity, since it allows us to know how much it will cost to prepare a state by solving a geodesic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is true, however, that the metric, in principle, can only be obtained in some cases: surely in exactly solvable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' And there we know how to prepare states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Thus, it is interesting to be able to predict the typical behaviour in general models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, we move in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In this article we are interested in a quite generic situation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' when a critical point is crossed to reach the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, we investigate what general statements about the behaviour of the circuit complexity we can make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We are not the first to calculate C in a quantum phase transition (QPT) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Recent papers discuss exactly solvable models as the topological Kitaev, Bose-Hubbard and Lipkin-Meshkov-Glick ones [17–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Importantly enough, complexity has been shown to be a useful probe of topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complementary to these calculations, in this work, we use that close to a transition point, the concept of universality emerges naturally, so we expect these universal properties to be inherited by complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If so, we can argue for its scaling laws or how complexity behaves regardless of model details or even on the particular chosen definition of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In addition, we apply our theory for state preparation in quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is a key and hard task [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is within the QMA complexity class [25], roughly speaking the NP-complete analogue for quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Nevertheless, quantum computers are expected to be better than classical methods such as density functional theory [26], density normalization group [27], tensor networks [28], quantum Montecarlo [29] or even ML-inspired techniques [30], in some instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For a recent discussion of these issues, see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Heuristic quantum algorithms as adiabatic [32] or varational ones [33, 34] can outperform classical calculations and serve for the generation of quantum states as quantum data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' phase classification [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Motivated by all of this, we discuss how useful the concept of complexity is and how much one can anticipate the difficulty of state preparation in variational quantum eigensolvers (VQEs) or adiabatic quantum algorithms (with and without shortcuts to adiabaticity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To challenge our theory we tackle both integrable and non-integrable models using numerical simulations and computing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity overview We find it convenient to discuss first the different notions of circuit complexity that we will use in this paper and the relationship between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity `a la Nielsen The original notion of complexity is due to the works of Nielsen and collaborators [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' See [13] for a recent review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Restricting ourselves to unitary operations, target and reference states are related as |ψT ⟩ = U(t, 0)|ψR⟩ = T e−i � t 0 H(τ) dτ|ψR⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (1) T stands for time ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Notice that, H(τ) = i(∂τU)U † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (2) A Cost function is formally defined as: CN := min{U} � t 0 dτ F[U, ˙U] (3) with F some functional fulfilling some basic properties as continuity, homogeneity (F[U, λ ˙U] = λF[U, ˙U] for λ ≥ 0), positivity and the triangular inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If, in addition to these, smoothness is assumed and the Hessian of F as a function of U is strictly positive, the functional is a Finsler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Thus, CN is nothing but the geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The suffix N stands for Nielsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Being a little more explicit, we can write that the evolution is given by H(τ) = � n Y (n)(τ)On .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (4) with On some operators and Y (n)(τ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A usual functional is then given by, Fk(τ) ∼ �� n |Y (n)(τ)|k �1/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (5) If we restrict ourselves to two level systems (qubits), Fk(τ) is a natural distance in SU(2n), such that d = � t 0 dτ Fk(τ), Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' What has been explained so far is the continuous version of complexity, that provides a lower bound for the number of gates needed to approximate |ψT ⟩ from |ψR⟩ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The discrete version of CN can be computed introducing the functional (using the same notation as in the original [1]): F(τ) = � � � � ′ � σ hσ(τ)2 + p2 ′′ � σ hσ(τ)2 (6) where the Hamiltonian in this case is H(τ) = �′ σ hσ(τ)σ + �′′ σ hσ(τ)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the first sum, σ ranges over all possible one- and two-body interactions, that is, over all products of either one or two qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the second sum, instead, the sum is over other tensor products of Pauli matrices and the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The factor p > 0 penalizes three, four, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' All put together, finding the geodesics in the continuum version is a good estimate of the resources needed to prepare a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At this point, we think it is necessary to emphasise something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If any unitary is possible, the complexity has a narrow utility, since its value is given by C = arccos(|⟨ψR|ψt⟩|), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' of the order of one (it doesn’t matter which state reference and destination are chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This can be verified by noting that the target state can always be written as |ψT ⟩ = cos θ|ψR⟩ + eiγ sin θ|ψ⊥ R⟩ with ⟨ψR|ψ⊥ R⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A rotation in the subspace generated by {|ψR⟩, |ψ⊥ R⟩} does the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, some restrictions on the possible unitaries or Hamiltonian (4) will be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We will discuss this point in some depth later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 1 Notice that due to homogeneity, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' we can always set t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Circuit Complexity from the Fubini-Study metric The functionals F discussed so far, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (4) and (5), are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Others can be chosen satisfying continuity, homogeneity, positivity and the triangular property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We want to discuss next another possibility where the distance between the reference and target states is given by the Fubini-Study metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Originally proposed for Quantum Field Theories in [5], we prefer to study it here from a quantum information perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Let us time-slice the unitary (1) such that |ψT (λ)⟩ = Uλ(t, tN−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='Uλ(t1, t0)|ψR(λ)⟩ → |ψ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' tn)⟩ = U(tn, tn−1)|ψ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' tn−1)⟩ (7) We have assumed that the unitaries and so the wave functions depend on the parameters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Then, for sufficiently small time step δτ := tn − tn−1, the fidelity between two contiguous states is Fn,n+1 ≡ |⟨ψ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' tn)|ψ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' tn−1)⟩| = 1 − χF δτ 2 + O(δ4) (8) where χF is denoted the fidelity susceptibility [36–41], see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [42] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Interestingly enough, we can relate χF with the geometric tensor, in fact [Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (11)] χF = gµν ˙λµ ˙λν (9) with [5, 43] gµν = Re (Tµν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (10) Here, Tµν is the quantum geometric tensor which is nothing but the Fubini-Study metric (FSM) on the CP n manifold, namely: Tµν = ⟨∂λµψ|P|∂λνψ⟩ (11) with P = 1 − |ψ⟩⟨ψ| 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Another useful way of writing the metric tensor is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Using a formal Taylor expansion for the states, the metric tensor can be written as, gµν = 1 2 (⟨∂µψ|∂νψ⟩ − ⟨∂µψ|ψ⟩⟨ψ|∂νψ⟩ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=') (12) Setting now in the Hamiltonian (4), ˙λν ≡ Y (ν) then |∂ν⟩ = Oν|ψ⟩, it is straightforward to see that [6], gµν = 1 2 ⟨ψ |{Oµ, Oν}| ψ⟩ − ⟨ψ) |Oµ| ψ⟩ ⟨ψ |Oν| ψ⟩ = 1 2 ��ψ|⟨ � Oµ − ⟨Oµ⟩λ , Oν − ⟨Oν⟩λ ��� ψ⟩ , (13) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' the fluctuations of the Hamiltonian operators Oν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Using the fact that 1−Fn,n+1 is a distance, thus satisfying the properties we imposed for the F-functional, we have that we can understand C as the distance defined through the Fubini-Study metric: CFS := min{U} � t 0 � gµν ˙λµ ˙λν dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (14) The suffix stands for Fubini-Study metric and the notion of distance is quite explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is an alternative definition to that given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (3) that has some remarkable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The first one is that knowing the metric tensor the geodesics can be found, at least in principle, by solving the differential equation: d2λµ dτ 2 + Γµ νρ dλν dτ dλρ dτ = 0 (15) Here, Γ are the Christoffel symbols: Γµ νρ = 1 2gµξ (∂ρgξν + ∂νgξρ − ∂ξgνρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (16) The second property of CFS is that, from its relation to the fidelity between states, F, its properties close to a QPT can be used when discussing the complexity, C, see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2 Notice that the imaginary part of T is nothing but the Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Some remarks comparing both approaches The complexity according to Nielsen estimates the minimum number of gates needed to reach the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For this purpose, a metric in the space of quantum circuits or unitary transformations is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The optimization of the trajectory in this space thus minimizes the number of accessible gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On the other hand, with the Fubini-Study metric, the complexity is computed by monitoring the state changes along the preparation of the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The Fubini-Study metric defines a geometry in the space of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A key difference is that in the latter geometry a variable cost is assigned to specific gates as it depends on the states they act on, while in the former, each gate is assigned a fixed cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On top of that, with CN there will be degenerate operations that leave the state unchanged (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' adding a global phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore and in general, it is found that the space of unitaries has a higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In general, different results are obtained using both approaches [6, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Depending on the application, one form is preferred over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We believe that due to the equations of the geodesics given the metric, CFS is ideal for doing analytical calculations while, from the point of view of quantum computation and cost estimation to prepare states, CN will prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In any case, in some particular cases it has been shown that both methods give identical results, such as the preparation of Gaussian fundamental states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' As a final remark, let us note that typically, the fidelity, no matter how close two quantum states are, drops exponentially with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is nothing but the Anderson orthogonality catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, in numerical studies a variant of CFS would be to use the fidelity per site instead, log f ≡ 1 N log F , (17) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On top of that, this allows to extract the extensive part for the complexity which, on the other hand, is what seems to matter [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Main results and manuscript organization For the exactly solvable systems that we discuss in this work, we find that CFS ≥ CN when crossing a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We understand this inequality as a consequence of the fact that the unitary space is larger, see previous subsection I A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In any case, C does not diverge at the critical point, its derivative does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For CFS we can characterize this divergence and its critical exponents in rather general circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Let us remark, again, that throughout the paper we focus on the extensive part of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Two models are studied in detail, namely the one-dimensional quantum Ising and Dicke models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' After this general discussion, we focus on calculating the complexity when preparing a fundamental state in a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, obviously, we compute CN in its discrete version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We explore two algorithms in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' First, we discuss the circuit complexity in adiabatic algorithms (with and without shortcuts to adiabaticity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, the adiabatic path crosses a QPT explicitly and the complexity grows around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' There is not much difference (with respect to the CN) in using shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Then, we discuss the circuit complexity using VQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' These algorithms are variational and do not need to cross the critical point even if the reference and target are in different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In such a case, CN is not necessarily aware of the QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On the other hand, if the target state is close enough to a phase transition, also in VQEs, the complexity grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The rest of the manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the next section, II, we discuss the relation between circuit complexity, in this case CFS from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (14), and the geometry of quantum states that allows extracting the critical exponents for the derivative of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is our first result that emphasizes that through phase transitions CFS has universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In section III we perform explicit calculations for CFS and CN in two solvable systems, namely the one dimensional XY-anisotropic and Dicke models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We extract the critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Then, in section IV, we perform numerical simulations where CN is computed in two types of algorithms: variational and adiabatic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Concretely we benchmark with exactly solvable models as the Ising model, and we complement our discussion with non-integrable Hamiltonians as the ZZXZ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Lastly, we discuss these results and conclude the paper in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Some technical issues are left for the Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The code used to obtain the numerical results is available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' COMPLEXITY AND THE GEOMETRY OF STATES CLOSE TO A QUANTUM PHASE TRANSITION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In this section, we discuss general aspects for the complexity close to a QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To be as general as possible, we find it convenient to focus on CFS, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Within this geometric formalism, we see that, in general, the complexity is 6 finite, but not its derivative, which can diverge when crossing a QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We study its finite size scaling obtaining the corresponding critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity and its derivative when crossing a QPT We have already argued in section I A 1 that if we are allowed to use any unitary, C is of the order of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the liter- ature, several unitary restrictions have been used: considering one and two qubit gates or considering gaussian states when moving from reference to target states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In this subsection, we consider another kind of restriction, quite natural when talking about a QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We will consider that one (and only one) parameter, say λ, of the Hamiltonian model is varied to pass through the QPT, keeping other variables or parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Thus the metric is unidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We know, that in this case, the geodesic is given by: gλλ ˙λ2 = cte (18) Therefore, C = min λ(τ) � T 0 √gλλ ˙λ dτ ∼ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (19) Below, we will work some examples and we will see that T does not diverge at the QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' However, if we compute the derivative instead: ∂C ∂λ = √gλλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (20) It is known that some components of the metric tensor can diverge, thus diverging the derivative of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Equation (20) has two consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The first one is that, under quite general circumstances, the derivative of C close to a QPT is related to the metric tensor and inherits its universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The second one is that this derivative can be used to witness and characterize QPTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Finite size scaling Close to a critical point correlation length diverges as, ξ ∼ |λ − λc|−1/a , (21) with a a critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Similar relations occur for other thermodynamic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, and for what interests us, the metric tensor can be written as [45], gµν ∼ |λ − λc|∆µν/a , (22) with ∆µν the corresponding critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Notice that, for the reasons already explained in section I A 3, from now on we will be interested in the intensive part of the metric tensor gµν → gµν/Ld, with d the spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Near a phase transition, finite-size scaling dictates how quantities behave after scale transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Very briefly, after a length scale transformation x′ = αx, time scales as τ ′ = αzτ, with z its critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This fixes the energy fluctuations ∆E∆τ ∼ 1 → ∆E′ = α−z∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Putting it all together, it is interesting to extract the value of the critical exponent ∆µν above, which controls how the metric tensor behaves, in terms of other critical exponents that dictate more fundamental quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Looking at equations (4) and (12) and (13) and writing the scaling for the derivatives of the Hamiltonian as ∂µ′H′ = α−∆µ∂µH we arrive to [45], ∆µν = ∆ν + ∆µ − 2z − d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (23) Finally, merging, (21) and (22), we find that close enough to the transition, where the relevant length is given by the system size, L, we arrive to gµν ∼ L−∆µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (24) As a consequence of all of this and using (20), when a single parameter is varied across the QPT we have the scaling: ∂C ∂λ ∼ L−∆λλ/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (25) 7 It is remarkable that the complexity derivative scaling is dictated by universal exponents, whenever one parameter is varied to cross a critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, if ∆λλ > −2 the derivative is sub-extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If ∆λλ = −2 it is extensive and if ∆λλ < −2 is superextensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' SOLVABLE HAMILTONIANS Let us test the above ideas on a couple of solvable models: the one dimensional quantum Ising model [46] and the Dicke [47–49] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Quantum Ising model The transverse field Ising model (Periodic Boundary Conditions will be assumed) is H = −J L � j=1 σz j σz j+1 + L � j=1 σx j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (26) Hamiltonian (26) can be solved via the Jordan-Wigner transformation [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This Ising model has a second order phase transition occurring at Jc = 1(−1) in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For Jc > 1(Jc < −1) the Z2 symmetry is spontaneously broken and the g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' is ferromagnetically (antiferromagnetically) ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' we fix our attention in the paramagnetic- ferromagnetic transition occurring at Jc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On top of that, the ground state can be written in terms of fermionic excitations (after the Jordan-Wigner transformation) as, |ψgs⟩ = � k>0 � cos(θk/2) + ieiφ sin(θk/2) a† ka† −k � |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (27) with k = (2m−1)π L 3 and, tan θk = −J sin k 1 + J cos k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (28) For the rest of the section the metric tensor (12) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It has been computed several times already [50, 51] gJJ = 1 4 � k �∂θk ∂h �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (29) In the thermodynamic limit, the k-sum is an integral � k → N/π � and it can be computed explicitly, yielding gJJ = −π(J2 − 1) + i � J2 + 1 � � log � − 2i(J+1) J−1 � − log � 2i(J+1) J−1 �� 32J2(J2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (30) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity through QPTs From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (30) we see that gJJ diverges at J = Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is the reason behind the divergence in the derivative of the complexity at the QPT, Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In figure 6, we plot CFS both in the continuum and for N-finite using either (30) or the sum (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In both cases, the integral (19) is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is evident that the complexity does not diverge at the QPT, but its derivative does, inheriting this behaviour from the metric tensor, Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 6a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For the Ising transition, the exponent a = 1, Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We know that ∆hh/a = 1, so the complexity derivative diverges as ∼ L1/2 at the Ising transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 3 We have used even L and periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 FS (a) L = 100 L = 1000 L = 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1 J 0 50 100 150 FS (b) L = 100 L = 1000 L = 2000 log(L) log|( FS)max| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='499 (c) ( FS)max data fit to ( FS)max(L) = A L + B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='3 N (d) L = 100 L = 1000 L = 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 N (e) L = 100 L = 1000 L = 2000 L exp (( N)max) (f) ( N)max data fit to ( N)max(L) = A log(L) + B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Study of the complexity for the Transverse Field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Complexity for different sizes of the chain computed using the Fubini-Study metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The discretization in J used is δJ = 1e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (b) Derivative of the Fubini-Study complexity for different L, δJ = 2e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (c) Finite size scaling of the maximum in the derivative of the Fubini-Study complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' See that this maximum diverges polynomially with the size of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (d) Study of the Nielsen complexity, δJ = 3e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (e) Derivative of the Nielsen complexity for different L, δJ = 3e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (f) Finite size scaling of the maximum in the derivative of the Nielsen complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' See that this maximum diverges logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Relation between CN and CFS Formula (27) is formally equivalent to the ground state for the 1D-Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For the latter, CN has been computed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If the reference, target and intermediate states have the same form (27), CN reads: CN = � k |∆θk|2 (31) where ∆θk = θT k − θR k and θT k (θR k ) are the angles (28) at the target (reference) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Following the same procedure as in [18] we checked that ∂JCN ∼ log N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' it diverges logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This must be confronted with the divergence (with critical exponent 1/2) for the case of ∂JCFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is an important difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' While using the FS distance the complexity is associated with the fluctuations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (13), the CN is more related to the angles difference and its divergence is therefore smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The Dicke model The Hamiltonian for the ground state sector of the N-spin Dicke model can be written in terms of total spin operators of spin S = N/2 as [52] H = ωca†a + ωsSz + λ √ 2S � a† + a � (S+ + S−) , (32) 9 where the spin and ladder operators obey the canonical commutation relations [Sz, S±] = ±S±, [S+, S−] = 2Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This model can be solved in the thermodynamic limit, S → ∞, with a Holstein-Primakoff transformation on the spins S+ → √ 2Sb† � 1 − b†b 2S , (33) S− → √ 2S � 1 − b†b 2S b , (34) Sz → b†b − S , (35) (36) yielding H = ωca†a + ω† sa + λ � a† + a � � b† � 1 − b†b 2S + � 1 − b†b 2S b � − ωcS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (37) In the normal phase of the Dicke model we can obtain an effective Hamiltonian for S → ∞ by neglecting terms with 2S in the denominator in the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (37), resulting in a completely symmetric model of coupled harmonic oscillators, one corresponding to the physical oscillator and the other corresponding to the spins within the Holstein-Primakoff transformation H = ωca†a + ω† sa + λ � a† + a � � b† + b � − ωcS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (38) In the superradiant phase, the bosonic modes must be displaced to accommodate the macroscopic occupations that the spins and field develop in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Once the displacements are introduced, terms with powers of 2S in the denominator are again neglected in the thermodynamic limit, yielding H = ωc¯a†¯a + ωs 2µ(1 + µ)¯b†¯b + ωs(1 − µ)(3 + µ) 8µ(1 + µ) �¯b† + ¯b �2 + λµ � 2 1 + µ � ¯a† + ¯a � �¯b† + ¯b � , (39) where µ = ωzΩ/ � 4λ2� and ¯a,¯b are the displaced bosonic operators [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We omit the expressions of the displacement as they are irrelevant in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Both the normal and superradiant effective Hamiltonians can be diagonalized in the real space basis, where they present a gaussian profile given by g(x, y) = �ϵ+ϵ− π2 �1/4 e− (R,AR) 2 , (40) where R = (x, y), x and y are the real-space coordinates associated to the modes a(¯a) and b(¯b), A = U −1MU with U a unitary matrix, M = diag [ϵ−, ϵ+] and ϵ± are the eigenmodes of the system [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The overlap of two different ground states is given by ⟨g|g′⟩ = 2 [det M det M ′]1/4 [det (M + M ′)]1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (41) This allows us to compute the components of the quantum metric tensor for the Dicke model exactly in the thermody- namic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We combine this with finite size results from exact diagonalization of Hamiltonian (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Just like we showed for the case of the Ising model, there is no divergence in CFS, the only signature of the phase transition is a non-analiticity that is only noticeable in the N → ∞ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This non-analiticity, or its precursor in the case of finite N is best revealed as a divergence in the derivative of the complexity, which is naturally the square root of the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here we are considering the complexity along a λ-path and the divergence is revealed in ∂CFS = √gλλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We perform a finite-size scaling analysis of the metric tensor by fitting the maximal values (∂CFS)max(N) and critical parameters at said maxima λmax(N) to their respective scaling laws |(∂CFS)max(N) − B| = C · N δ , (42) |λmax(N) − λc| = A · N −ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (43) The resulting critical exponents ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='655(22) ≊ 2/3 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6711(15) ≊ 2/3 are in agreement with values reported in the literature [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0 1 2 3 4 FS(0 ) (a) N 50 100 150 200 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 100 101 102 103 FS (b) log| max(N) c| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='655(22) (c) max(N) data fit to | max(N) c| = A N log(N) log|( FS)max(N) B| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6711(15) (d) ( FS)max(N) data fit to |( FS)max(N) B| = C N FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Fubini-Study complexity (a) and its derivative with respect to λ (b) across the phase transition of the Dicke model as a function of the system size (numerical results) and in the thermodynamic limit (analytical results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Plots on the right showcase the fits of λmax(N) (c) and (∂CFS)max(N) (d) (extracted from center plot) to their respective finite size scaling laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' All results are at resonance ωc = ωs = 1 and with a discretization of dλ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Numerical results were obtained with a cutoff for bosonic excitations of Nexc = 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' COMPLEXITY IN A QUANTUM COMPUTER, THE CASE OF GROUND STATE PREPARATION In this section, we compute CN when preparing ground states in a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We study both adiabatic algorithms and variational quantum eigensolvers (VQEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Two versions of the former algorithms are discussed: with and without shortcuts to adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In both cases, the initial state is the “trivial zero” |0⟩ ≡ |00 · · · 0⟩4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Some gates are applied to prepare the ground state of a given Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, we are especially interested when this initial state (that can be understood as the ground state in the paramagnetic phase) is in a different phase than the final one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In addition, we discuss whether or not a QPT is crossed during the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Finally, notice that in quantum computing applications it seems natural to compute CN and, in particular, its discrete version (the number of gates needed), Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' I A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Thus, through this section, we compute CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Adiabatic algorithms A systematic way of finding the ground state of a given Hamiltonian is by adiabatic passage or annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Let us consider the time-dependent Hamiltonian: H(t) = (1 − λ(t))H0 + λ(t)HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (44) Here, H0 has a trivial ground state (easy to prepare), and HT is the hamiltonian from which we want to obtain its ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Consider that the time-dependent function λ(t) runs from λ(t = 0) = 0 to λ(t = tf) = 1, where tf is the final time of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At t = 0 the state is prepared in the ground state of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' If ˙λ is sufficiently small compared to the instantaneous gap, by means of the adiabatic theorem the final state is the ground state of HT [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On the other hand, the adiabatic condition alerts us that as the gap closes, for example in continuous phase transitions, the execution time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' the circuit depth, scales with the inverse of this gap, thus also C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Importantly enough the adiabatic condition can be relaxed by introducing counter-diabatic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Generally speaking, instead of H(τ) (whose ground states are |ψ(tn)⟩) what is evolved is the “modified” Hamiltonian [56, 57]: H′(τ) = H(τ) + HCD(τ) (45) 4 In fact, in almost all algorithms the initial state seems to be |00 · · · 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 11 The last term ensures that the time evolution exactly matches the instantaneous ground state of H(τ) no matter how fast the evolution is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is known in the literature as shortcuts to adiabaticity and HCD is called counter-diabatic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' There are different ways of writing HCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In its original form we can write: HCD(τ) = i ˙λµ � ⟨m|∂µH|n⟩ En − Em |m⟩⟨n| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (46) with ∂µH ≡ ∂H/∂λµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We emphasize that at times 0 and t, |ψR⟩ and |ψT ⟩ are ground states of H(0) and H(t) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Explicitly |ψT ⟩ = T e− � t 0 H′(τ) dτ|ψR⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here τ means time, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' with Eq (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To connect this evolution with the previous sections, we note that the fidelity susceptibility can be written in terms of the HCD(τ) fluctuations [Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (9) and (13)]: χF = ⟨(H(τ) + HCD(τ))2⟩ − ⟨(H(τ) + HCD(τ))⟩2 = ⟨HCD(τ)2⟩ = ˙λµ ˙λν gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (47) In practice HCD is difficult to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, a systematic although approximate way of writing is convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Following [58] it can be rewritten as, HCD(τ) = ˙λµAµ (48) Here, A is the adiabatic gauge potential that can be approximated as: A(ℓ) µ = i ℓ � k=1 αk [H, [H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [H � �� � 2k−1 , ∂µH]]] (49) where (l) is the “degree of approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On top of that, the {αk} are found variationally by minimising the action [59]: Sℓ = Tr � G2 ℓ � , Gℓ = ˙λµ � ∂µH − i � H, A(ℓ) µ �� (50) In many cases of interest, in the adiabatic protocol, H(τ) is expected to be a local Hamiltonian, in particular a two body one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Notice that due to nested commutators, the higher the order (l), the longer the range of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Following the functional (6) three, four, or higher order body interactions will be highly penalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Thus, in what follows, we will restrict ourselves to l = 1 that introduces two body interactions at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This, in turn, provides a systematic way of preparing, via trotterization, quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity in adiabatic algorithms As has been previously discussed, in order to compute the complexity as defined by Nielsen [Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (I A 1)], we only need to express our unitary operation as the time evolution of some Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the present case, it is straightforward, with and without shortcuts, as the Hamiltonian (45) is given explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We study the Ising model in transverse field and the ZZXZ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For both, we use the function λ(t) = sin2 � π 2 sin2 � πt 2T �� to drive the evolution from the initial Hamiltonian, H0, to the target one, HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For the Ising model, we start from H0 = hx � i σx i , leaving the transverse field fixed and switching on the spin-spin interaction until we reach Hint = J � i σz i σz i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The counter-diabatic Hamiltonian follows from equations (46), (49) and (50) with l = 1 yielding HCD(t) = ˙λ(t)α(t) � i(σy i σz i+1 + σz i σy i+1), where α(t) is the variational parameter in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (49) and (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The full-time-dependent Hamiltonian reads H′(t) = H0 + λ(t)Hint + HCD(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (51) Since we need to implement our unitary evolution via Trotter decomposition, we have to split the total time T in T/δT steps, where δT is the time discretization employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The smaller δT, the more precise the implementation will be but more gates will be needed, increasing the complexity of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the simulations presented here δT = min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='1, T/30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, following the prescription given for the Nielsen complexity, CN, given in (6), it is straightforward to obtain5: CN = � T 0 dt � T δT �1/2 � N + (N − 1)λ(t)2J2 + 2(N − 1) ˙λ2(t)α2(t) �1/2 (52) 5 For the adiabatic evolution without shortcuts, the expression would be the same but with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 12 10 1 100 101 102 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 (a) |J| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 |J| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 |J| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 |J| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 |J| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 t/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 |E1 E0| (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 |J| 50 100 150 200 250 /L (c) L = 6 L = 8 L = 10 L = 12 L = 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity study for the Transverse Field Ising model using the adiabatic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing time lengths of the full algorithm and different target Js for L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At shorter times the shortcuts provide better results, being identical to the simple case (without shortcuts) for the longest times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (b) Evolution of the gap between the ground state and the first excited state during the algorithm for the same values of J as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The gap closes with an increasing value of |J|, explaining why longer times are needed for the larger |J| to obtain the same fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (c) Complexity per spin computed for different sizes with shortcuts (solid lines) and without shortcuts (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' As the gap closes, more gates are needed to achieve the fidelity threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 in this case) but we do not find relevant differences between applying shortcuts or not in the final result for the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Figure 3 summarises our results for the Ising model with adiabatic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The transverse field was fixed to hx = 1 and different values of J were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In Figure 3a we plot the fidelity between the final state obtained adiabatically and the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' As expected, the longer the time the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We also confirm that at lower times higher fidelities are achieved thanks to the counter-diabatic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Figure 3b shows the gap evolution within the adiabatic algorithm, giving insights about why as |J| is greater, it takes more time to achieve a high fidelity: the gap becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Finally, the last panel 3c shows the actual Nielsen complexity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Reflecting the fidelity behaviour, the complexity jumps around the transition as the gap is closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' As the adiabatic theorem states, crossing a QPT is hard for this kind of algorithms and complexity serves the purpose of quantifying such difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In order to check if this holds in other models, we also study the so-called antiferromagnetic ZZXZ model: HT = J � i σz i σz i+1 + hx � i σx i + hz � i σz i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (53) Due to the combination of longitudinal and transverse fields, this is a non-integrable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is ideal, then, to explore the phenomenology of complexity beyond the exactly solvable models considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4 we draw the phase diagram of the model at zero temperature as a function of the fields applied to the spins and the exchange constant [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The critical line separates paramagnetic and antiferromagnetic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For our particular purposes, keeping the same initial Hamiltonian, H0, we set the transverse field, hx = 1 and the target longitudinal field to hz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We thus study the quantum phase transition appearing when moving to different target values of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This path is shown as the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4, where the final point marks the maximum value simulated for the target J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, the transverse field is going to be fixed while we turn on both the longitudinal field and the magnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The counter-diabatic term can be computed in the same fashion as before, getting the same result as in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The time-dependent Hamiltonian reads H′(t) = H0 + λ(t) � i � Jσz i σz i+1 + hzσz i � + HCD(t) (54) and the complexity acquires the following expression CN = � T 0 dt � T δT �1/2 � N � 1 + h2 zλ2(t) � + (N − 1)λ(t)2J2 + ˙λ2(t) � Nα2(t) + 2(N − 1) � β2(t) + γ2(t) ���1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (55) 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 hx / J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 hz / J AFM PM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Phase diagram of the ZZXZ model for zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The black dotted line signals the critical region between phases for different ratios of the fields (hx, hz) to the magnitude of the exchange interaction (J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The coloured lines depict the path followed for the adiabatic algorithm (red) and the values computed in the VQE (blue) [Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' IV B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In figure 5 we show the results obtained for the different values of J and the chain sizes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The behaviour is equiva- lent to the previous model except that for sufficiently large values of J, the gap decreases sharply, closing completely (see figure 5b), causing the counter-diabatic terms to cause more error than the simple evolution itself, as we can see in panel (a) of the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is a consequence of the fact that our expression for the counter-diabatic term is not exact, but a first-order approximation of a general expression [cf Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (49)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The smaller the gap, the more careful we will have to be with the design of the CD term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Putting all together, we can conclude that, due to the gap closing at the QPT the CN with adiabatic algorithms diverges with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The inclusion of shortcuts does not provide any significant advantage in terms of complexity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is because we have constrained these shortcuts to be as local as possible, in our case l = 1 in (50), introducing two body interactions at much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' It is expected that by introducing long-range terms in (46) the complexity decreases as the system approaches to the QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This can be compared to the previous section II, where there was no restriction to local operations, so both CF and CN remained finite despite crossing the QPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Other paths investigated in this work are sent to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Circuit Complexity in VQEs VQEs, introduced in [34], use the fact that any quantum state can be written in terms of a unitary operation as |φ(⃗θ)⟩ = U(⃗θ)|0⟩ , (56) where U(⃗θ) is a parameterized unitary that transforms the initial state into the desired wave function |φ(⃗θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This unitary can be implemented in a quantum circuit as a set of quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The expectation value of the Hamiltonian where we encode our problem (H) results ⟨H⟩ = ⟨0|U †(⃗θ)HU(⃗θ)|0⟩ ≥ E0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (57) The optimization process consists on minimizing the average energy of the parameterized state: EVQE = min θ ⟨0|U(⃗θ)†HU(⃗θ)|0⟩ ≥ E0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (58) 14 10 1 100 101 102 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 (a) J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 t/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 |E1 E0| (b) 10 2 10 1 100 J 102 103 /L (c) L = 6 L = 8 L = 10 L = 12 L = 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Complexity study for the ZZXZ model using the adiabatic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The phenomenology is essentially the same as for the TFI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing time lengths of the full algorithm and L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In this case we see that, for sufficiently large values of J, no applying shortcuts works better than applying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is explained by the gap closing much more abruptly than in the TFI model, as can be seen in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (c) Complexity per qubit computed for different sizes with shortcuts (solid lines) and without shorcuts (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' As the gap closes, more gates are needed to achieve the fidelity threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The algorithm can be divided into three different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' First, we need to choose the trial wave function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='(56)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Choosing the unitary U(⃗θ) is equivalent to constructing the quantum circuit that transforms the initial state into the parameterized wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The circuit used to achieve |φ(⃗θ)⟩ is called the ansatz and can be represented as, q0 : U(⃗θ) q1 : q2 : q3 : q4 : (59) Choosing an appropriate ansatz is crucial for the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This choice depends completely on the model we are simulating and the set of gates available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We will dig into our choice of unitary below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The next step is constructing the Hamiltonian of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Since this Hamiltonian is going to be evaluated later, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (57), it must be written in terms of Pauli strings {I, σx, σy, σz}⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Pauli operators are related to spin observables, which are suitable for direct measurement in quantum devices [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' With the Hamiltonian and the wave function defined, we can measure the energy of the state, which is the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To compute this cost function, the expectation values of the Pauli observables are measured determining the value of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Since the technique uses quantum and classical processors, VQEs are cast as hybrid algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Our results are numerical and our Python code simply computes the product of the matrices U(⃗θ)†HU(⃗θ) previously defined and then projects onto the zero state obtaining ⟨0|U(⃗θ)†HU(⃗θ)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We will not discuss its measurement overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, we are interested in the circuit complexity for reaching the desired ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The final step is to minimize this cost function through the variation of the parameters θ in the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At the end of each iteration we obtain the value of the energy (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Then, a classical optimizer determines the best direction of variation of the parameter vector ⃗θ to minimize this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We use as many iterations as needed until we converge to a final solution for the coordinates of the parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Ideally, this solution is the absolute minimum in the space of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Still, obtaining this minimum is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The optimizer can get trapped in local minima which will imply serious limitations in the minimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This problem and others have been previously discussed in the literature [61, 62] and are out of scope for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 15 Summarizing, we assume a given ansatz, the set of available gates in U(⃗θ) in (56) and the hybrid algorithm finds the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' CN counts the number of gates, and once the VQE circuit is chosen, it can be done systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Local VQE ansatz We focus on a fixed geometry that is suitable for one-dimensional systems with single and two-qubit gates, besides the two-qubit gates act only on contiguous qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This ansatz can be interpreted as a Trotter approximation of continuous evolution by a local 1D Hamiltonian [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In this case, we can separate the terms of the Hamiltonian that act on even and odd links and obtain two sets, each made of mutually commuting gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In particular, the circuit is given by q0 : RY (θ[0]) RZ (−π/2) q1 : RY (θ[1]) RZ (π/2) RZ (−π/2) RY (θ[5]) RZ (−π/2) q2 : RY (θ[2]) RZ (−π/2) RY (θ[6]) RZ (π/2) RZ (−π/2) q3 : RY (θ[3]) RZ (π/2) RZ (−π/2) RY (θ[7]) RZ (−π/2) q4 : RY (θ[4]) RZ (π/2) RZ (−π/2) (60) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' it consists of fundamental blocks (or layers) (separated by dashed lines above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Each layer is made out of single-qubit rotations Ry(θ) and control-Z gates (CZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At the end of the circuit, we add a final column of rotations (Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For computing CN we rewrite the CZ gates in terms of Pauli operators, count the gates and use equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is a routine process that we send to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, we just give the final result: CN = d � j=1 � � � � 2(L−1) � i � θj i 2 �2 + 3(L − 1) �π 4 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (61) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' VQE complexity through QPTs As before, we focus on Ising and ZZXZ models, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (26) and (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In figure 6 we summarize our results for the Ising Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In panel a) we plot the complexity using the local VQE ansatz for obtaining the ground state at a given J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We see that CN grows when the ground state approaches the QPT, that in this case is given by Jc ∼= 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In fact, close enough to the transition the VQE cannot reach an acceptable ground state for a maximum depth of 8 (in our simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This can be checked in panel b) where the fidelity between the state obtained within the VQE algorithm and the exact ground state falls below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 in the gray region of panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, all indications are that, also with VQE, complexity increases as QPT is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' With what has been said so far this should not be a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Perhaps, the most remarkable thing here is that the complexity is only high near the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' When the target state is far from the critical point the complexity drops, even though the latter and the reference state may be in different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is due to the fact that, contrary to the adiabatic algorithm, the VQE does not necessarily need to visit states in the transition region to go from |ψR⟩ to |ψT ⟩, it can circumvent criticality and go directly from one phase to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is easy to understand in the Ising model, because in the paramagnetic phase the ground state is approximately given by |+, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', +⟩ (|+⟩ = 1/ √ 2(|0⟩ + |1⟩)), Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is easy to prepare: it can be obtained with single qubit rotations from the reference state |ψR⟩ = |0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Since we are dealing with finite simulations, deep in the ferromagnetic phase, the Z2 symmetry is not broken so the ground state manifold found by exact diagonalization is spanned by the states 1 2 (|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 0⟩ ± |1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 1⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The VQE 6 We say Jc ∼ = 1 since our simulations are done in finite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Jc = 1 in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 /L (a) 1 2 3 4 5 6 7 8 Layers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 (b) J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='90 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='92 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='94 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='96 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='98 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='02 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='04 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='06 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='08 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='10 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='30 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='50 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='70 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Transverse Field Ising model with bias, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='001 and size N = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Complexity per size as a function of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The grey zone indicates that the VQE does not converge for points inside that region in a reasonable number of layers to the fidelity threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (b) Fidelity obtained for different numbers of layers for points inside the grey box in (a) and in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For those points whose fidelity is above the threshold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9) it has only been plotted the best result for clarity’s sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 /L (a) 1 2 3 4 5 Layers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 (b) J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='90 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='92 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='94 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='96 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='98 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='02 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='04 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='06 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='08 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='10 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='30 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='50 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='70 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='90 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='10 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='30 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='50 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' ZZXZ Ising model for size N = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Complexity per size as a function of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The grey zone, as in the TFI model, indicates that the algorithm fails to achieve fidelity over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 for points within that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (b) The fidelity behaviour with the depth of the ansatz shows that, again, once the QPT is crossed the algorithm cannot reach fidelities over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In contrast to the TFI model, here we don’t recover high fidelity once we are fully in the antiferromagnetic phase, reaching a maximum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 for the highest values of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' reaches instead one of the fully polarized states, either |0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 0⟩ or |1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 1⟩, given that they are degenerate with the symmetric ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Our convergence criterion is based on reaching a fidelity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9 between the state generated by the VQE and the result of exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Because of the discrepancy in the ground states obtained by both methods, in the ferromagnetic phase the fidelity is capped at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 and the convergence criterion is never satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Driven by the physics of actual QPTs in the thermodynamic limit, where the symmetry is (spontaneously) broken, we decide to add a small bias, ϵ � σz i in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In doing so, the VQE should a priori be able to reach full convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is indeed the case as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Additionally, convergence is reached in very few layers, equivalently to what is observed in the PM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This low complexity can be explained by noticing that the symmetry broken ferromagnetic ground state is either the reference state or can be obtained from it by means of single-qubit rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We now consider the ZZXZ model, Hamiltonian (53)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, we are not going to explicitly break the symmetry 7 The parameters employed in the simulations are depicted as the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4, namely hx = 1, hz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 and J ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5] 17 0 1 2 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 Magnetization (a) Total Even sites Odd sites 0 1 2 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 (b) Single state Subspace 0 1 2 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='994 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 Energy accuracy (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' VQE state characterization in the ZZXZ model for N = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Magnetization of the spin chain as a function of J obtained from the states generated by VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The solid black line represents the total magnetization per site that the spin chain should have (obtained via exact diagonalization) whereas the dashed black line sets the magnetization per site in even/odd sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (b) Evolution of the best fidelity obtained as a function of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In blue it is computed the fidelity as the overlap between the state generated by the VQE and the exact ground state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' in red it is computed as the projection onto the subspace generated by the ground state and the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (c) Energy accuracy obtained for the same configurations displayed in the other panels computed as 1 − Erel, being Erel the relative error between the energy obtained from VQE and the exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' in order to discuss the scenario in which the symmetric ground state is sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the ZZXZ model, the QPT separates paramagnetic (PM) and antiferromagnetic (AFM) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the PM phase, the behavior is analogous to the Ising model, Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Deep in the AFM phase, the ground state manifold is spanned by the states |ψAFM⟩ ∼= 1 √ 2(|1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='⟩ ± |0, 1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Following the previous discussion, the VQE does not reach the symmetric ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Therefore, we see that CN grows as it approaches the phase transition (with our parameters Jc ∼= 1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 4) but does not decrease afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' At some point near criticality, the VQE cannot produce a ground state with a fidelity larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='9, see panel b), similar to the Ising model case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, however, the state remains difficult for the VQE after the near-transition region is surpassed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is further confirmed in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' There, we can see that although the total magnetization is well reproduced by the VQE (also the energy, in panel c), once we enter the antiferromagnetic phase the VQE generates either |1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='⟩ or |0, 1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='⟩, as can be seen by computing the magnetization per site, which should be close to 1/2 in the exact ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' However, the VQE gives 0 (1) for the even (odd) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To conclude our characterization, we see that all this is consistent with obtaining a F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5, as well as a F ∼= 1 if we compare the state generated by the VQE with the projection onto the subspace generated by the ground state and the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' DISCUSSION Knowing in advance how much a computation will cost, even if only approximately, is of great help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Unfortunately, this estimation can pose a great challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Computer science has traditionally categorized problems into different complexity classes, allowing one to know whether a given problem is tractable on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For a quantum computer, we can ask a similar question to know if the task we want to tackle is going to be feasible with the architecture we have at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' For this purpose, the concept of circuit complexity was invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Again, knowing the complexity of each task in any architecture seems too general to be able to give a concrete answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' On the other hand, we can shed some light on generic situations where some kind of general statement can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is the idea that motivated us to write this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We have studied the situation in which a critical region is crossed in the process of preparing a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Our work has shown that, regardless of the type of complexity one chooses, and for diverse models, it appears that complexity grows if the algorithm visits states near a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We have further proven that this is a characteristic trait of typical algorithms for state preparation such as VQE and adiabatic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The degree of divergence does depend on the definition of complexity used and on the allowed gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the case of local ans¨atze or evolutions, C tends to diverge as the system size grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Importantly, we have shown that VQEs, to the extent that they can go “directly” from the reference to the target state, can potentially avoid the divergence in complexity 18 even if the reference and target states lie in different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Whether this is possible depends on the model, as it is determined by the degree of entanglement of the target and reference states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In the case of adiabatic algorithms, keeping the complexity down seems to be a matter of allowing non-local gates in the evolution, to fully exploit shortcuts to adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This is supported analytically in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Here, the Ising critical point is traversed along a restricted path of states of the form (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Despite this restriction, these states are sufficiently non-local for CN to remain finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The impact of our work on the preparation of states in a quantum machine seems straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' What our results mean in the field of holography is another matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Unfortunately, we do not have the knowledge to anticipate anything, but it would be interesting to think in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Other ideas not discussed here would be the use of other types of complexity such as Krylov [22, 64–66] or mixed states and their behavior in thermal phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Note Added in Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='- While we were finishing writing this manuscript, the paper [67], which discusses the impor- tance of local and non-local gates in the computation of complexity, appeared in the arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Fernando Luis for his helpful comments and insights during the preparation of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), the Spanish Government Grants PID2020-115221GB-C41/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='13039/501100011033 and TED2021-131447B-C21 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='13039/501100011033 and the EU “NextGenerationEU”/PRTR, the Gobierno de Arag´on (Grant E09-17R Q-MAD) and the CSIC Quantum Technologies Platform PTI-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This work has been financially supported by the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' J R-R acknowledges support from the Ministry of Universities of the Spanish Government through the grant FPU2020- 07231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Appendix A: Complexity associated to the VQE To compute F we must express our ansatz as a unitary of the form U = T e−i � T 0 H(τ) dτ where H is written in terms of Pauli matrices {σx, σy, σz} and tensor products of these matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To do so, recall that the local VQE ansatz only contains one and two qubit gates (between nearest neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' To construct the effective Hamiltonian, notice that Ry(θi) = e−i θi 2 σy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (A1) Now, the C-Z gate, can be decomposed q0 : = q0 : RZ (−π/2) q1 : q1 : RZ (π/2) RZ (−π/2) Therefore C-Z = e−i π 4 (σ0 zσ1 z−σ0 z−σ1 z) = e−i π 4 σ0 zσ1 zei π 4 σ0 zei π 4 σ1 z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (A2) If we substitute in the representation of a layer of the ansatz, we find that each one of the building blocks marked with a dashed line in the main text is represented by a unitary of the form U = e−i � j θj 2 σj ye−i π 4 (σ0 zσ1 z−� j σj z) ≈ e−i(� j θj 2 σj y+ π 4 σ0 zσ1 z− π 4 � j σj z) , (A3) Finally, H = � j θj 2 σj y + π 4 σ0 zσ1 z − π 4 � j σj z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (A4) 19 More generally, each layer of the ansatz can be written as an operator of the type H = Heven + Hodd , (A5) where Heven = 1 t �� i θi 2 σi y − π 4 L−1 � i=0 σi z + π 4 � i=even σi zσi+1 z � , (A6) Hodd = 1 t �L−2 � i=0 θi+L 2 σi y − π 4 L � i=1 σi z + π 4 � i=odd σi zσi+1 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (A7) Now we use a Trotter decomposition to compute the complexity of this circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We have fixed the total evolution time to 1 and each layer is considered a Trotter step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This way, t = T/#steps = 1/d, where d is the number of layers of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Now, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (6) we find F(U) = � � � � 2(L−1) � i � dθi 2 �2 + 3(L − 1) � dπ 4 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (A8) Here, L − 1 corresponds to the number of C-Zs in the layer, with L is the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Now, the complexity is nothing but the integral of this functional across the number of layers in the circuit CN = � 1 0 F(U)dt ≈ d � j=1 F(U)1 d , (A9) which leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (61) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Appendix B: Other paths in the adiabatic algorithm In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' IV A we show an adiabatic evolution for the Transverse Field Ising model where we let the field fixed as we increase the interaction between the neighbouring spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' However, we could have let the interaction fixed and switched on the transverse field, going from a classical Ising model to the TFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 9 we show this possible adiabatic path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The behaviour of the gap between the ground state and the first excited state is qualitatively different, to the point of even closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' This results in a much worse performance for small values of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Similarly, the gap behaviour also causes a big impact in the ZZXZ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 10 we show that for odd number of spins in the chain we get a higher complexity as the gap presents a dip at intermediate times which makes necessary longer times to achieve the fidelity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Nielsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Dowling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Gu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Doherty, Science 311, 1133 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Nielsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Dowling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Gu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Doherty, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' A 73, 062323 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Heller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Marrochio, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Pastawski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' D 98, 126001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Hackl and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Myers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2018, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Susskind, Fortschr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 64, 24 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Roberts, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Susskind, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Swingle, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 116, 191301 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Huang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' D 103, 065002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Chapman and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Policastro, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' C 82, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Caceres, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Chapman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Couch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Hernandez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Myers, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Ruan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2020, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 20 10-1 100 101 102 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 F (a) hx = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 hx = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 hx = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 hx = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 hx = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 t/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 |E1 − E0| (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Adiabatic evolution for the TFI model by switching on the transverse field instead of the spin-spin interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' (a) Evolution of the fidelity obtained with shortcuts to adiabaticity (solid lines) and without them (dashed lines) for increasing time lengths of the full algorithm and L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' We see a clear difference with the plot in the main text, where the field is fixed and we vary the interaction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The gap closes much earlier for small field values (b), making the algorithm need much longer times to achieve high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 10-1 100 101 102 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 F (a) J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='25 J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='75 J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 J = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 t/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='5 |E1 − E0| (b) 10-2 10-1 100 |J| 102 103 104 105 C/L (c) L = 5 L = 7 L = 9 L = 11 L = 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Evolution in the ZZXZ model of the fidelity (a), the gap between the ground state and the first excited state (b) and the complexity (c) for spin chains with odd number of constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' The dip at intermediate times in the gap causes the complexity to increase compared to the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Di Giulio and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Tonni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 2020, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Ghodrati, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' D 98, 106011 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Xiong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Yao, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' B 101, 174305 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Venuti and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Zanardi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' 99, 095701 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} +page_content=' [46] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE3T4oBgHgl3EQfswvl/content/2301.04671v1.pdf'} 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gold nano-particle +enhanced radiotherapy with protons, +megavoltage photons and kilovoltage photons: A +Monte Carlo simulation” by Lin et al [Phys. Med. +Biol. 59 (2014) 7675–7689] +Hans Rabus 1 +1 Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany + +E-mail: hans.rabus@ptb.de + + +Abstract +In their article published in Phys. Med. Biol. 59 (2014) 7675–7689, Lin et al studied the dose +enhancement of gold nanoparticles (GNPs) for proton therapy, which they compared with the +case of photon irradiation. This comment points out two caveats to the methodlogy used by +Lin et al that may not be evident to readers and may contribute to confusion in the literature +about the dose enhancement by gold nanoparticles. + +Keywords: proton therapy, gold nanoparticles, Monte Carlo simulation, radiation therapy, microscopic dose enhancement + +1. Assessment of the dose enhancement by gold nanoparticles +In the paper of Lin et al (2014), dose enhancement by gold nanoparticles (GNPs) is assessed by Monte Carlo simulations of +radiation transport in a three-step procedure. First, simulations of radiation transport in water were performed to determine the +fluence of ionizing particles impinging on a GNP. Then, the fluence of electrons produced by the radiation impinging on a +spherical GNP with diameter of 50 nm or a water volume of the same size (a ‘water nanoparticle (WNP)’) was determined. +In the third step, the energy imparted by the electrons leaving the GNP or the WNP was scored for volumes at different +distances from the GNP surface. The ratio of the energy imparted to the mass of these volumes was then interpreted as the +“dose”, and the dose enhancement factor (DEF) was defined as the ratio between the “dose” obtained with the GNP and the +“dose” obtained with the “WNP”. +In reality, the quantity determined in step 3 of the simulations is the dose contribution of the electrons emitted from the +GNP or the “WNP” after the radiation interaction in the GNP or “WNP”. Therefore, defining the ratio of these two contributions +as the DEF is extremely misleading and gives values that do not correspond to dose enhancement. +Before detailing the arguments already presented in (Rabus et al. 2019, 2020, 2021), it may be useful to examine a simple +example. Consider a target volume of the same size as the GNP, with its center located at a radial distance of 250 nm from the +GNP center. Within 250 nm of the center of this target volume, there is a GNP of 50 nm diameter and about 399 "WNPs" of +50 nm diameter. If the GNP is not present, there are 400 "WNPs." Thus, considering only the dose contribution (DC) of +nanoparticles at 250 nm from the target volume, this dose contribution is that of one GNP and 399 "WNPs" when the GNP is + + + +2 + + +present, and that of 400 "WNPs" when the GNP is absent. The ratio of these two dose contributions, the dose contribution +enhancement factor (DCEF) of sources at 250 nm from the target volume is + +DCEF(250 nm) = 1+(DC(GNP)/DC(“WNP”)-1)/400 +(1) +Thus, even if the DC of a GNP for protons is a factor of 5 to 15 greater than the DC of a "WNP" (Fig. 3(c) in (Lin et al +2014)), the ratio DCEF(250 nm) is only about 1.025. And even if for photons the DC of a GNP is a factor of 1000 greater than +that of a "WNP", the DCEF(250 nm) drops to about 3.5. +However, in addition to the contribution from the GNP and “WNPs” at 250 nm, there are also contributions from “WNPs” +at smaller and larger distances from the target volume. In the absence of the GNP, these dose contributions simply add up to +the prescribed dose Dw. With the GNP present, the absorbed dose Dt,g in the target volume is approximately given by + +��,� = ������|�⃗�|� + +� +��� � ������|�⃗ − �⃗�|��� +�� += ������|�⃗�|� − ������|�⃗�|� + �� +(2) +where VNP is the nanoparticle volume, DCWNP is the dose contribution of a “WNP” located at �⃗, �⃗� is the target position, DCGNP +is the dose contribution from the GNP (located at the origin), and the integration domain V’ is defined by the range of the +secondary electrons and excludes the volume of the GNP. Therefore, the dose enhancement factor is given by + +����|�⃗�|� = 1 + ������,��|�⃗�|� − �����,��|�⃗�|� ��,� +! + +(3) +where �����,� and �����,� are the dose contributions from a GNP and a “WNP” per primary particle (shown in Fig. 3 and 4 +(a) and (b) of (Lin et al 2014)) and ��,� is the dose to water for a primary particle fluence "� of 1 per GNP cross-section, i.e., +"� = 5.1×1010 cm-2. ��,� can be estimated for protons from the stopping power (Berger et al 2005) and for photons from the +the mass-energy absorption coefficient (Hubbell and Seltzer 2004). For the monoenergetic 50 keV photon irradiation, this gives +0.016 Gy. From Fig. 4 (a), this is approximately the dose contribution from the GNP at about 20 nm from the GNP surface, +where then the DEF is 2 instead of 103. At 103 nm from the GNP surface, where the “DEF” increases in Fig. 4 (c) of Lin et al +(2014), the true DEF is about 1.0007 according to eq. 3. +For 10 MeV and 100 MeV protons, values of ��,� = 370 Gy and ��,� = 59 Gy, respectively, are obtained. The dose +contribution per incident proton at the GNP surface is about 25 Gy and 2.5 Gy at these two proton energies, and the resulting +DEF values at the surface of the GNP are 1.07 and 1.04 respectively. At 1000 nm from the GNP, where the “DEF” in Fig. 3(c) +of Lin et al (2014) saturates at about 15, the actual DEF for both energies is below 1.0001. (These values were obtained using +eq. 3 based on estimated dose values from the figures in Lin et al (2014) +2. Particle fluence estimations +In Sections 2.2.3 and 2.2.4 of their paper, Lin et al (2014) describe briefly how they proceeded to estimate the fluence of +particles interacting with the GNPs when the primary particles have an energy spectrum. Briefly, they performed radiation +transport simulations in a water phantom and scored all particles traversing a circular area with a radius of 25 mm to obtain a +distribution of phase-space coordinates. This distribution was then modified so that the lateral coordinates fell within a circle +with a radius of 25 nm, and the direction of motion was changed so that the particles moved along the direction of the incident +beam. A 1/cos correction was applied for the weight of the particles. Apart from this description of the procedure, no +justification for its correctness is given. +The background for the method used by Lin et al (2014) is the so-called common random number (CRN) approach. Let us +assume that the primary particles are emitted from a plane along the normal direction and let us refer to the collection of particles +and energy transfer points that occur during the simulation of a primary particle a “shower”. This shower is characterized by a +sequence of random numbers used in modeling of the various interactions that occur. If the emission point of the primary +particle is shifted in the source plane, and if the same sequence of random numbers is used in the simulation, then all energy +transfer points of the shower will be shifted by the same vector in a direction parallel to the source plane. +Thus, assume a particle resulting from a primary particle starting at a given position in the source plane traverses the plane +used for scoring at a location and along a direction of motion that does not transport the particle to the GNP. Applying a lateral +shift so that the trajectory of the particle from the shifted position on the scoring plane hits the GNP corresponds to a position +on the source plane shifted laterally by the same vector. If the source emits primary particles uniformly, all starting positions +are equivalent. Therefore, the lateral displacement simply “picks” a position on the source plane that is as likely as the one used +in the simulation but with the benefit that the trajectory intersects the GNP. The factor 1/cos accounts for the fact that the area +covered by possible starting points on the source plane is generally not a circle (like the cross-section of the GNP), but an +ellipse with one axis elongated by this factor 1/cos. + + + +3 + + +Estimation of the particle fluence at the GNP using this approach is justified when the possible lateral offsets from the +starting position of the primary particle is small compared to the lateral extent of the area used for the scoring. Under these +conditions, the occurrence of starting positions outside the part of the surface plane from which primary particles are emitted +is rare and negligible. For protons, this should be the case. For photons, on the other hand, using the CRN approach would +actually require an infinite lateral extension of the primary radiation field. Practically, this could be achieved by choosing the +size of the beam much larger than the area used for scoring. In addition, the radiation field for photons also depends on the +extent of the geometry in the direction of the primary particle beam. As shown by Rabus et al (2019) for a geometry with +infinite depth and a beam of infinite width, the contribution of scattered photons is by a factor of order 5 higher than the fluence +of primary beam for kilovoltage X-ray and by orders of magnitude for Co-60 radiation. +In addition, it should be noted that both Lin et al (2014) and Rabus et al (2019) estimated the fluence of particles at the GNP +using a simulation of radiation transport in water. This is appropriate when the case of a single GNP is considered. If the +concentration of gold atoms in the volume loaded with GNPs becomes significant, this will also change the particle fluence +spectra within this volume. In principle, this can be remedied by performing the simulation of the first step not for pure water, +but for a uniform mixture of gold and water. +Conclusions +From the above arguments, it appears that for photon fields whose fluence is determined by the procedure used by Lin et al +(2014), the fluence of particles impinging on the GNP may be underestimated, while the procedure is expected to be correct +for protons. With respect to determining the dose enhancement around a single GNP, the approach of Lin et al (2014) to use +the ratio of the dose contribution by electrons produced in the GNP to that of electrons produced in a water volume with the +same dimensions resulted in an overestimation of the DEF by orders of magnitude. +References +Berger M J, Coursey J S, Zucker M A and Chang J 2005 ESTAR, PSTAR, and ASTAR: Computer programs for calculating stopping- +power and range tables for electrons, protons, and helium ions (version 1.2.3) Available at: https://www.nist.gov/pml/stopping- +power-range-tables-electrons-protons-and-helium-ions (Gaithersburg, MD: National Institute of Standards and Technology) +Hubbell J H and Seltzer S M 2004 Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV +to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest (version 1.4). [Online] Available at: +https://physics.nist.gov/pml/x-ray-mass-attenuation-coefficients (Gaithersburg, MD: National Institute of Standards and +Technology) +Lin Y, McMahon S J, Scarpelli M, Paganetti H and Schuemann J 2014 Comparing gold nano-particle enhanced radiotherapy with protons, +megavoltage photons and kilovoltage photons: a Monte Carlo simulation Physics in Medicine and Biology 59 7675–89 +Rabus H, Gargioni E, Li W, Nettelbeck and H. C Villagrasa 2019 Determining dose enhancement factors of high-Z nanoparticles from +simulations where lateral secondary particle disequilibrium exists Phys. Med. Biol. 64 155016 (26 pp.) +Rabus H, Gargioni E, Li W B, Nettelbeck H and Villagrasa C 2020 Corrigendum: Determining dose enhancement factors of high-Z +nanoparticles from simulations where lateral secondary particle disequilibrium exists (2019 Phys. Med. Biol. 64 155016) Phys. +Med. Biol. 65 159501 +Rabus H, Li W B, Villagrasa C, Schuemann J, Hepperle P A, de la Fuente Rosales L, Beuve M, Maria S D, Klapproth A P, Li C Y, +Poignant F, Rudek B and Nettelbeck H 2021 Intercomparison of Monte Carlo calculated dose enhancement ratios for gold +nanoparticles irradiated by X-rays: Assessing the uncertainty and correct methodology for extended beams Phys. Medica 84 241– +53 + + diff --git a/GdA0T4oBgHgl3EQfBf_u/content/tmp_files/load_file.txt b/GdA0T4oBgHgl3EQfBf_u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c7d63cffbead8db204ce792381914f49bc8a6f1 --- /dev/null +++ b/GdA0T4oBgHgl3EQfBf_u/content/tmp_files/load_file.txt @@ -0,0 +1,110 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf,len=109 +page_content='1 Comment on “Comparing gold nano-particle enhanced radiotherapy with protons, megavoltage photons and kilovoltage photons: A Monte Carlo simulation” by Lin et al [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 59 (2014) 7675–7689] Hans Rabus 1 1 Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany E mail: hans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='rabus@ptb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='de Abstract In their article published in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 59 (2014) 7675–7689, Lin et al studied the dose enhancement of gold nanoparticles (GNPs) for proton therapy, which they compared with the case of photon irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' This comment points out two caveats to the methodlogy used by Lin et al that may not be evident to readers and may contribute to confusion in the literature about the dose enhancement by gold nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Keywords: proton therapy, gold nanoparticles, Monte Carlo simulation, radiation therapy, microscopic dose enhancement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Assessment of the dose enhancement by gold nanoparticles In the paper of Lin et al (2014), dose enhancement by gold nanoparticles (GNPs) is assessed by Monte Carlo simulations of radiation transport in a three-step procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' First, simulations of radiation transport in water were performed to determine the fluence of ionizing particles impinging on a GNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Then, the fluence of electrons produced by the radiation impinging on a spherical GNP with diameter of 50 nm or a water volume of the same size (a ‘water nanoparticle (WNP)’) was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In the third step, the energy imparted by the electrons leaving the GNP or the WNP was scored for volumes at different distances from the GNP surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' The ratio of the energy imparted to the mass of these volumes was then interpreted as the “dose”, and the dose enhancement factor (DEF) was defined as the ratio between the “dose” obtained with the GNP and the “dose” obtained with the “WNP”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In reality, the quantity determined in step 3 of the simulations is the dose contribution of the electrons emitted from the GNP or the “WNP” after the radiation interaction in the GNP or “WNP”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Therefore, defining the ratio of these two contributions as the DEF is extremely misleading and gives values that do not correspond to dose enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Before detailing the arguments already presented in (Rabus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 2019, 2020, 2021), it may be useful to examine a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Consider a target volume of the same size as the GNP, with its center located at a radial distance of 250 nm from the GNP center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Within 250 nm of the center of this target volume, there is a GNP of 50 nm diameter and about 399 "WNPs" of 50 nm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' If the GNP is not present, there are 400 "WNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='" Thus, considering only the dose contribution (DC) of nanoparticles at 250 nm from the target volume, this dose contribution is that of one GNP and 399 "WNPs" when the GNP is 2 present, and that of 400 "WNPs" when the GNP is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' The ratio of these two dose contributions, the dose contribution enhancement factor (DCEF) of sources at 250 nm from the target volume is DCEF(250 nm) = 1+(DC(GNP)/DC(“WNP”)-1)/400 (1) Thus, even if the DC of a GNP for protons is a factor of 5 to 15 greater than the DC of a "WNP" (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3(c) in (Lin et al 2014)), the ratio DCEF(250 nm) is only about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' And even if for photons the DC of a GNP is a factor of 1000 greater than that of a "WNP", the DCEF(250 nm) drops to about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' However, in addition to the contribution from the GNP and “WNPs” at 250 nm, there are also contributions from “WNPs” at smaller and larger distances from the target volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In the absence of the GNP, these dose contributions simply add up to the prescribed dose Dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' With the GNP present, the absorbed dose Dt,g in the target volume is approximately given by ��,� = ������|�⃗�|� + � ��� � ������|�⃗ − �⃗�|��� �� = ������|�⃗�|� − ������|�⃗�|� + �� (2) where VNP is the nanoparticle volume, DCWNP is the dose contribution of a “WNP” located at �⃗, �⃗� is the target position, DCGNP is the dose contribution from the GNP (located at the origin), and the integration domain V’ is defined by the range of the secondary electrons and excludes the volume of the GNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Therefore, the dose enhancement factor is given by ����|�⃗�|� = 1 + ������,��|�⃗�|� − �����,��|�⃗�|� ��,� !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' (3) where �����,� and �����,� are the dose contributions from a GNP and a “WNP” per primary particle (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3 and 4 (a) and (b) of (Lin et al 2014)) and ��,� is the dose to water for a primary particle fluence "� of 1 per GNP cross-section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=', "� = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='1×1010 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' ��,� can be estimated for protons from the stopping power (Berger et al 2005) and for photons from the the mass-energy absorption coefficient (Hubbell and Seltzer 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' For the monoenergetic 50 keV photon irradiation, this gives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='016 Gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 4 (a), this is approximately the dose contribution from the GNP at about 20 nm from the GNP surface, where then the DEF is 2 instead of 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' At 103 nm from the GNP surface, where the “DEF” increases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 4 (c) of Lin et al (2014), the true DEF is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='0007 according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' For 10 MeV and 100 MeV protons, values of ��,� = 370 Gy and ��,� = 59 Gy, respectively, are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' The dose contribution per incident proton at the GNP surface is about 25 Gy and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='5 Gy at these two proton energies, and the resulting DEF values at the surface of the GNP are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='07 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='04 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' At 1000 nm from the GNP, where the “DEF” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3(c) of Lin et al (2014) saturates at about 15, the actual DEF for both energies is below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' (These values were obtained using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3 based on estimated dose values from the figures in Lin et al (2014) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Particle fluence estimations In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='4 of their paper, Lin et al (2014) describe briefly how they proceeded to estimate the fluence of particles interacting with the GNPs when the primary particles have an energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Briefly, they performed radiation transport simulations in a water phantom and scored all particles traversing a circular area with a radius of 25 mm to obtain a distribution of phase-space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' This distribution was then modified so that the lateral coordinates fell within a circle with a radius of 25 nm, and the direction of motion was changed so that the particles moved along the direction of the incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' A 1/cos\uf071 correction was applied for the weight of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Apart from this description of the procedure, no justification for its correctness is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' The background for the method used by Lin et al (2014) is the so-called common random number (CRN) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Let us assume that the primary particles are emitted from a plane along the normal direction and let us refer to the collection of particles and energy transfer points that occur during the simulation of a primary particle a “shower”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' This shower is characterized by a sequence of random numbers used in modeling of the various interactions that occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' If the emission point of the primary particle is shifted in the source plane, and if the same sequence of random numbers is used in the simulation, then all energy transfer points of the shower will be shifted by the same vector in a direction parallel to the source plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Thus, assume a particle resulting from a primary particle starting at a given position in the source plane traverses the plane used for scoring at a location and along a direction of motion that does not transport the particle to the GNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Applying a lateral shift so that the trajectory of the particle from the shifted position on the scoring plane hits the GNP corresponds to a position on the source plane shifted laterally by the same vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' If the source emits primary particles uniformly, all starting positions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Therefore, the lateral displacement simply “picks” a position on the source plane that is as likely as the one used in the simulation but with the benefit that the trajectory intersects the GNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' The factor 1/cos\uf071 accounts for the fact that the area covered by possible starting points on the source plane is generally not a circle (like the cross-section of the GNP), but an ellipse with one axis elongated by this factor 1/cos\uf071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' 3 Estimation of the particle fluence at the GNP using this approach is justified when the possible lateral offsets from the starting position of the primary particle is small compared to the lateral extent of the area used for the scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Under these conditions, the occurrence of starting positions outside the part of the surface plane from which primary particles are emitted is rare and negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' For protons, this should be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' For photons, on the other hand, using the CRN approach would actually require an infinite lateral extension of the primary radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Practically, this could be achieved by choosing the size of the beam much larger than the area used for scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In addition, the radiation field for photons also depends on the extent of the geometry in the direction of the primary particle beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' As shown by Rabus et al (2019) for a geometry with infinite depth and a beam of infinite width, the contribution of scattered photons is by a factor of order 5 higher than the fluence of primary beam for kilovoltage X-ray and by orders of magnitude for Co-60 radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In addition, it should be noted that both Lin et al (2014) and Rabus et al (2019) estimated the fluence of particles at the GNP using a simulation of radiation transport in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' This is appropriate when the case of a single GNP is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' If the concentration of gold atoms in the volume loaded with GNPs becomes significant, this will also change the particle fluence spectra within this volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' In principle, this can be remedied by performing the simulation of the first step not for pure water, but for a uniform mixture of gold and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' Conclusions From the above arguments, it appears that for photon fields whose fluence is determined by the procedure used by Lin et al (2014), the fluence of particles impinging on the GNP may be underestimated, while the procedure is expected to be correct for protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' With respect to determining the dose enhancement around a single GNP, the approach of Lin et al (2014) to use the ratio of the dose contribution by electrons produced in the GNP to that of electrons produced in a water volume with the same dimensions resulted in an overestimation of the DEF by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content=' References Berger M J, Coursey J S, Zucker M A and Chang J 2005 ESTAR, PSTAR, and ASTAR: Computer programs for calculating stopping- power and range tables for electrons, protons, and helium ions (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='3) Available at: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdA0T4oBgHgl3EQfBf_u/content/2301.01978v1.pdf'} +page_content='gov/pml/stopping- 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a/JtE2T4oBgHgl3EQfAQZo/content/2301.03589v1.pdf b/JtE2T4oBgHgl3EQfAQZo/content/2301.03589v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..65c2a894f58db9e3682221fd15a409514d4a939e --- /dev/null +++ b/JtE2T4oBgHgl3EQfAQZo/content/2301.03589v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e302bffc6194cafa34178154b3ec8f11981a4b267bef0facdd51f4e072906384 +size 11506287 diff --git a/K9AyT4oBgHgl3EQfTveL/content/tmp_files/2301.00112v1.pdf.txt b/K9AyT4oBgHgl3EQfTveL/content/tmp_files/2301.00112v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37026f5e0dfd2a1a5d4d98acba62bd2f049f476d --- /dev/null +++ b/K9AyT4oBgHgl3EQfTveL/content/tmp_files/2301.00112v1.pdf.txt @@ -0,0 +1,586 @@ +Graphs with girth 2ℓ + 1 and without longer odd holes +are 3-colorable +Rong Chen +Center for Discrete Mathematics, Fuzhou University +Fuzhou, P. R. China +January 3, 2023 +1 +Abstract +For a number ℓ ≥ 2, let Gℓ denote the family of graphs which have girth +2ℓ + 1 and have no odd hole with length greater than 2ℓ + 1. Plummer and +Zha conjectured that every 3-connected and internally 4-connected graph in +G2 is 3-colorable. Wu, Xu, and Xu conjectured that every graph in � +ℓ≥2 Gℓ is +3-colorable. Chudnovsky et al. and Wu et al., respectively, proved that every +graph in G2 and G3 is 3-colorable. In this paper, we prove that every graph in +� +ℓ≥5 Gℓ is 3-colorable. +Key Words: chromatic number; odd holes +1 +Introduction +All graphs considered in this paper are finite, simple, and undirected. A proper +coloring of a graph G is an assignment of colors to the vertices of G such that no +two adjacent vertices receive the same color. A graph is k-colorable if it has a proper +coloring using at most k colors. The chromatic number of G, denoted by χ(G), is +the minimum number k such that G is k-colorable. +The girth of a graph G, denoted by g(G), is the minimum length of a cycle +in G. A hole in a graph is an induced cycle of length at least four. An odd hole +means a hole of odd length. For any integer ℓ ≥ 2, let Gℓ be the family of graphs +that have girth 2ℓ + 1 and have no odd holes of length at least 2ℓ + 3. Robertson +conjectured in [3] that the Petersen graph is the only graph in G2 that is 3-connected +and internally 4-connected. Plummer and Zha [4] disproved Robertson’s conjecture +1Mathematics Subject Classification: 05C15, 05C17, 05C69 +Email: rongchen@fzu.edu.cn (R. Chen). +1 +arXiv:2301.00112v1 [math.CO] 31 Dec 2022 + +and proposed the conjecture that all 3-connected and internally 4-connected graphs +in G2 have bounded chromatic number, and the strong conjecture that such graphs +are 3-colorable. The first was proved by Xu, Yu, and Zha [7], who proved that all +graphs in G2 is 4-colorable. Chudnovsky and Seymour [2] proved that every graph +in G2 is 3-colorable. In the same paper, Chudnovsky and Seymour also gave a short +and pretty proof of the result in [7]. Wu, Xu, and Xu [5] showed that graphs in +� +ℓ≥2 Gℓ are 4-colorable and proposed the following conjecture. +Conjecture 1.1. ([5], Conjecture 6.1.) Every graph in � +ℓ≥2 Gℓ is 3-colorable. +We say that a graph G has an odd K4-subdivision if some subgraph H of G is +isomorphic to a K4-subdivision and whose face cycles are all odd holes of G. The +subgraph H is also induced by ([1], Lemma 2.7). In the same paper, the author and +Zhou proved +Theorem 1.2. ([1], Theorem 1.1.) For each graph G in � +ℓ≥5 Gℓ, if G has an odd +K4-subdivision, then χ(G) = 3. +Based on Theorem 1.2, in this paper we prove that Conjecture 1.1 holds for +ℓ ≥ 5. That is, +Theorem 1.3. All graphs in � +ℓ≥5 Gℓ are 3-colorable. +We conjecture +Conjecture 1.4. For a graph G without triangles, if all odd holes of G have the +same length, then G is 3-colorable. +Conjecture 1.5. For an integer ℓ ≥ 2 and a graph G with g(G) = 2ℓ + 1, if the set +of lengths of odd holes of G have k members, then G is (k + 2)-colorable. +2 +Preliminary +A cycle is a connected 2-regular graph. Let G be a graph. For any subset U ⊆ V (G), +let G[U] be the induced subgraph of G defined on U. For subgraphs H and H′ of G, +set |H| := |E(H)| and H∆H′ := E(H)∆E(H′). Let H ∪ H′ denote the subgraph of +G with vertex set V (H) ∪ V (H′) and edge set E(H) ∪ E(H′). Let H ∩ H′ denote +the subgraph of G with edge set E(H) ∩ E(H′) and without isolated vertex. Let +N(H) be the set of vertices in V (G) − V (H) that have a neighbour in V (H). Set +N[H] := N(H) ∪ V (H). +Let P be an (x, y)-path and Q a (y, z)-path. +When P and Q are internally +disjoint paths, let PQ denote the (x, z)-path P ∪ Q. Evidently, PQ is a path when +x ̸= z, and PQ is a cycle when x = z. Let P ∗ denote the set of internal vertices +2 + +of P. When u, v ∈ V (P), let P(u, v) denote the subpath of P with ends u, v. For +simplicity, we will let P ∗(u, v) denote (P(u, v))∗. +For a number k ≥ 2, a graph G is k-vertex-critical if χ(G) = k and χ(G − v) ≤ +k − 1 for each vertex v of G. +Lemma 2.1. ([1], Lemma 2.2.) For any number k ≥ 4, each k-vertex-critical graph +has no 2-edge-cut. +For an i-vertex path P of a connected graph G, if G − V (P) is disconnected, +then we say that P is a Pi-cut. Usually, a P2-cut is also called a K2-cut. Evidently, +every k-vertex-critical graph has no K2-cut. Chudnovsky and Seymour in [2] proved +that every 4-vertex-critical graph G in G2 has no P3-cut. In fact, their methods can +also be used to prove the following result. +Lemma 2.2. ([2]) For any number ℓ ≥ 2, every 4-vertex-critical graph in Gℓ has +neither K2-cut nor P3-cut. +A theta graph is a graph that consists of a pair of distinct vertices joined by three +internally disjoint paths. Let C be a hole of a graph G. A path P of G is a chordal +path of C if C ∪ P is an induced theta-subgraph of G. Since g(G) = 2ℓ + 1 and +each odd hole of G has length 2ℓ + 1, by simple computation we have the following +result, which will be frequently used. +Lemma 2.3. ([1], Lemma 2.3.) Let C be an odd hole of a graph G ∈ Gℓ. Let P be +a chordal path of C, and P1, P2 be the internally disjoint paths of C that have the +same ends as P. Assume that |P| and |P1| have the same parity. Then +|P1| = 1 or ℓ ≥ |P2| < |P1| = |P| ≥ ℓ + 1. +In particular, when |P1| ≥ 2, all chordal paths of C with the same ends as P1 have +length |P1|. +Let C1, C2 be odd holes of a graph G ∈ Gℓ such that C1 ∪ C2 is a theta subgraph +G. When C1 ∪ C2 is not induced, since |C1∆C2| ≤ 4ℓ and g(G) = 2ℓ + 1, we have +that |C1∆C2| = 4ℓ and G[V (C1 ∪ C2)] is an odd K4-subdivision. For the similar +reason, if C is an even hole of length 4ℓ of G, then C has at most two chords, and +when C has two chords, G[V (C)] is an odd K4-subdivision. +3 +Graphs has a subgraph isomorphic to a K4-subdivision +Let H be a graph that is isomorphic to a K4-subdivision. For a path P of H whose +ends are degree-3 vertices of H, if P contains exactly two degree-3 vertices of H, then +we say it is an arris of H. Evidently, H has exactly six arrises. We say that a graph +3 + +u1 +u2 +u3 +u4 +C1 +C2 +C3 +C4 +P1 +Q2 +P2 +Q1 +L1 +L2 +H +Figure 1: u1, u2, u3, u4 are the degree-3 vertices of H. The face cycles C1, C2 are +odd holes, and C3, C4 are even holes. {P1, P2}, {Q1, Q2}, {L1, L2} are the pairs of +vertex disjoint arrises of H. +G contins a balanced K4-subdivision if G has an induced subgraph H isomorphic to +a K4-subdivision and exactly two face cycles of H are odd holes of G. +Lemma 3.1. If a graph G in Gℓ contains a balanced K4-subdivision H that is +pictured as the graph in Figure 1, then the following hold. +(1) |P1| ≤ ℓ and 1 ∈ {|Q1|, |Q2|, |L1|, |L2|}. +(2) |P2| ≥ ℓ. +(3) Assume that G has no odd K4-subdivision. If |Q1| = 1 and |L2| ≥ 2, then no +vertex v ∈ V (G) − V (H) has two neighbours in V (H). +Proof. Since C1, C2 are odd holes and g(G) = 2ℓ+1, we have |P1| ≤ ℓ. Assume that +1 /∈ {|Q1|, |Q2|, |L1|, |L2|}. Since P2Q2 is a chordal path of C1 and C4 is an even +hole, we have |P2|+|Q2| = |L1| by Lemma 2.3. Similarly, we have |P2|+|L1| = |Q2|, +which is a contradiction. So (1) holds. +Now we consider (2). +By (1) and symmetry we may assume that |Q1| = 1. +Since |P1| ≤ ℓ by (1), we have |L1| ≥ ℓ, so Q1P2 is a chordal path of C2. Then +|Q1P2| ≥ ℓ + 1 by Lemma 2.3. So |P2| ≥ ℓ. This proves (2). +Next, we prove that (3) is true. Assume not. Let v ∈ V (G)−V (H) have at least +two neighbours in V (H). Since |L2| > 1 and C3 is an even hole, C2∆C3 is an odd +hole and |L2| ≥ ℓ + 1 by Lemma 2.3. Moreover, since a vertex not in an odd hole +has at most one neighbour in the odd hole, each arris of H has at most one vertex +adjacent to v. +3.1.1. The vertex v has exactly one neighbour in V (C1 ∪C2) and exactly one neigh- +bour in V (P ∗ +2 ). +4 + +Subproof. Since v has at most one neighbour in V (P2), it suffices to show that v +has at most one neighbour in V (C1 ∪ C2). Assume not. Then v has exactly two +neighbours in V (C1 ∪ C2) with one in V (C1) − V (P1) and the other in V (C2) − +V (P2). Then G[V (C1 ∪ C2) ∪ {v}] is a balanced K4-subdivision as G has no odd +K4-subdivision, which is not possible by (2). +Note that C2∆C3 is an odd hole. Since v can not have two neighbours in an odd +hole of H, the vertex v has no neighbour in V (P1). For the same reason, if v has +a neighbour in V (Q1 ∪ Q2), the neighbours of v in V (H) must be in V (C1 ∪ C2), +which is not possible by 3.1.1. Hence, N(v) ∩ V (C1 ∪ C2) ⊆ V (L∗ +1 ∪ L∗ +2). Assume +that v has a neighbour in V (L∗ +1). +Since |C4| ≤ 4ℓ − 2 and g(G) = 2ℓ + 1, by +3.1.1, the two cycles in G[V (C4) ∪ {v}] containing v are odd holes. Moreover, since +|L2| ≥ ℓ+1, the subgraph G[V (C4)∪{v}]∪P1 ∪Q1 is an odd K4-subdivision, which +is not possible. Similarly we can show that v can not have a neighbour in V (L∗ +2). +This proves (3). +Let H be an induced subgraph of G pictured as the graph in Figure 1. When +|Q1| = 1, we say that H is a balanced K4-subdivision of type (1, 2) if |L2| > 1. Since +|Q1| = 1 implies that |L1| > 1, this definition is well defined. +Let H1, H2 be vertex disjoint induced subgraphs of a graph G. +An induced +(v1, v2)-path P is a direct connection linking H1 and H2 if v1 is the only vertex in +V (P) having a neighbour in H1 and v2 is the only vertex in V (P) having a neighbour +in H2. Evidently, the set of internal vertices of each shortest induced path joining +H1 and H2 induces a direct connection linking H1 and H2. +Theorem 3.2. Let ℓ ≥ 5 be an integer and G be a graph in Gℓ. If G is 4-vertex +critical, then G does not contain a balanced K4-subdivision of type (1, 2). +Proof. By Theorem 1.2, G does not have an odd K4-subdivision. Assume to the +contrary that G has a balanced K4-subdivision H of type (1, 2) that is pictured as +the graph in Figure 1. Without loss of generality we may assume that H is chosen +with |H| as small as possible.By Lemma 3.1 (1) and symmetry we may assume that +|Q1| = 1. By Lemmas 3.1 (2) and 2.3, we have +|Q1| = 1, |P2| ≥ ℓ, |L1|, |L2| ≥ ℓ + 1, and |P1| < ℓ. +(4.1) +Let e, f be the edges in P2 incident with u3, u4, respectively. Let P be a direct +connection in G\{e, f} linking P ∗ +2 and H − V (P ∗ +2 ). Lemma 2.1 implies that such +P exists. Let v1, v2 be the ends of P with v2 having a neighbour in V (P ∗ +2 ) and +v1 having a neighbour in V (H) − V (P ∗ +2 ). By Lemma 3.1 (3), both v1 and v2 have +a unique neighbour in V (H). +Let x be the neighbour of v1 in V (H) and y the +5 + +neighbour of v2 in V (H). Set P ′ := xv1Pv2y. Then H ∪ P ′ is an induced subgraph +of G. By Lemma 3.1 (3) again, we have |P ′| ≥ 3. +3.2.1. x ̸= u2. +Subproof. Assume that x = u2. Set C′ +3 := L2P2(u4, y)P ′. Since |L2| ≥ ℓ+1, we have +that C′ +3 is an even hole by Lemma 2.3. Since |P ′| ≥ 3 and C2∆C′ +3 is an odd hole, +(H ∪ P ′) − V (L∗ +2) is a balanced K4-subdivision of type (1, 2) whose edge number is +less than |H|, which is a contradiction to the choice of H. +3.2.2. x /∈ P1. +Subproof. Assume not. By 3.2.1, we have x ∈ V (P1)−{u2}. Set C′ +1 := P1(x, u2)Q1P2(u3, y)P ′. +Since |P1| < ℓ by (4.1), it follows from Lemma 2.3 that C′ +1 is an odd hole. Then +P1(x, u1)Q2P2(u4, y)P ′ is an even hole. Moreover, since |L2| ≥ ℓ+1, we have x = u1 +and |Q2| = 1 by Lemma 2.3. Since C1 and C′ +1 are odd holes, we have |L1| > |P ′|. +Moreover, since |P ′| ≥ 3, the subgraph C′ +1 ∪ C2 ∪ P2 is a balanced K4-subdivision +of type (1, 2) whose edge number is less than |H|, which is a contradiction. +3.2.3. When x = u3, we have |P2(u3, y)| = 1 or |P2(u3, y)| ≥ ℓ + 1. +Subproof. Assume that |P2(u3, y)| ̸= 1. Since Q1P2 and Q1P ′P2(y, u4) are chordal +paths of C2 that have the same ends as L2, they have length |L2| by Lemma 2.3. +So |P2(u3, y)P ′| = 2|P2(u3, y)|, implying that |P2(u3, y)| ≥ ℓ + 1. +3.2.4. If x ∈ V (L∗ +1), then xu3, yu3 ∈ E(G), and |Q2| = 1. +Subproof. Set C′ +3 := L1(x, u3)P2(u3, y)P ′. When C′ +3 is an odd hole, since C′ +3∆C4 +is an odd hole, C1 ∪ C′ +3 ∪ P2 ∪ Q2 is an odd K4-subdivision, which is not possible. +So C′ +3 is an even hole. Assume that xu3 /∈ E(G). Then |C′ +3| = 2|L1(x, u3)| by +Lemma 2.3. Moreover, since C1∆C′ +3 is an odd hole, (H ∪ P ′) − V (L∗ +1(x, u3)) is a +balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is +a contradiction to the choice of H. So xu3 ∈ E(G). +Assume that yu3 /∈ E(P2). Since Q1P2 and Q1L1(u3, x)P ′P2(y, u4) are chordal +paths of C2 that have the same ends as L2, they have length |L2| by Lemma 2.3. +Moreover, since xu3 ∈ E(G) and |L1| ≥ ℓ + 1 imply |L1(x, u1)| ≥ 2, the subgraph +(H ∪ P ′) − V (P ∗ +2 (y, u3)) is a balanced K4-subdivision of type (1, 2) whose edge +number is less than |H|, which is a contradiction. So yu3 ∈ E(P2), implying that +|P ′| ≥ 2ℓ. +Since C′ +3∆C3∆C1 is an odd cycle of length at least 2ℓ + 3, we have +|Q2| = 1. This proves 3.2.4. +3.2.5. x /∈ V (Q∗ +2). +6 + +Subproof. Assume not. Set C′ +3 := Q2(x, u4)P2(u4, y)P ′. Assume that C′ +3 is an odd +hole. Since C1 ∪ C2 ∪ (C3∆C′ +3) is an odd K4-subdivision when yu4 /∈ E(P2), we +have yu4 ∈ E(P2). Since |Q2| < ℓ as |L2| ≥ ℓ + 1, the graph C4∆C′ +3 is an odd +hole with length larger than 3ℓ by (4.1), which is not possible. So C′ +3 is an even +hole. Since |Q2| < ℓ, it follows from Lemma 2.3 that xu4 ∈ E(Q2). Moreover, +since P ′ is a chordal path of the odd hole C2∆C3 and C′ +3 is an even hole, ℓ + 1 ≤ +|P ′| = |P2(y, u4)| + 1 ≤ |P2| < |L1| by Lemma 2.3. +Hence, C2 ∪ C3 ∪ P ′ is a +balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is +a contradiction. +Let u′ +4 be the neighbour of u4 in L2. +3.2.6. If x ∈ V (L2), then x ∈ {u4, u′ +4}. +In particular, when x = u′ +4, we have +yu4 ∈ E(P2); and when x = u4, we have yu4 ∈ E(P2) or |P2(y, u4)| ≥ ℓ + 1. +Subproof. Set C′ +3 := L2(x, u4)P2(u4, y)P ′. Assume that x = u4 and yu4 /∈ E(P2). +Since Q1P2 and Q1P2(u3, y)P ′ are chordal paths of C2 with the same ends, |C′ +3| = +2|P2(y, u4)| by Lemma 2.3, so |P2(y, u4)| ≥ ℓ + 1. That is, when x = u4, 3.2.6 holds. +So we may assume that x ̸= u4. +By 3.2.2, we may assume that x ∈ V (L∗ +2). +When C′ +3 is an odd hole, since +x /∈ {u2, u4}, the graph C2 ∪ C3 ∪ P ′ is an odd K4-subdivision. So C′ +3 is an even +hole. When x ̸= u′ +4, by Lemma 2.3, we have that xu2 ∈ E(H), |C′ +3| = 2|L2(x, u4)| +and C3∆C′ +3 is an odd hole, so (H ∪ P ′) − V (L∗ +2(x, u4)) is a balanced K4-subdivision +of type (1, 2) whose edge number is less than |H|, which is not possible. Hence, +x = u′ +4. When yu4 /∈ E(P2), since Q1P2(u3, y)P ′ is a chordal path of C2, we have +|C′ +3| = 2|P2(y, u4)| by Lemma 2.3. So (H ∪ P ′) − V (P ∗ +2 (y, u4)) is a balanced K4- +subdivision of type (1, 2) whose edge number is less than |H|, which is not possible. +So yu4 ∈ E(G). +Let v be a vertex in V (P ∗ +2 ). When |P2| ≥ ℓ+1, let |P2(u3, v)| = ℓ−1, otherwise let +|P2(u3, v)| = ℓ − 2 ≥ 3. The vertex v is well defined as |P2| ≥ ℓ. Since |P2| ≤ 2ℓ − 2, +we have 2 ≤ |P2(u4, v)| ≤ ℓ − 1. Let u, w ∈ N(v) ∩ V (P ∗ +2 ) with u closer to u4 and w +closer to u3 along P2. Since {uv, vw} is not an edge cut of G by Lemma 2.1, there is +a shortest induced path Q in G\{uv, vw} linking v and H − v. By Lemma 3.1 (3), +H ∪ Q is an induced subgraph of G. Let v′ be the other end of Q that is not equal +to v. When v′ ∈ V (C1 ∪ C2), since Q∗ is a direct connection in G\{e, f} linking P ∗ +2 +and H −V (P ∗ +2 ), we get a contradiction to 3.2.2-3.2.6. So v′ ∈ V (P ∗ +2 )−{v}. Assume +that v′ ∈ V (P ∗ +2 (u3, v)). When v′ ̸= w, either Q2P1Q1P2(u3, v′)QP2(v, u4) is an odd +hole of length at least 2ℓ + 3 or QP2(v, v′) is a cycle of length 2|P2(v, v′)| that is at +most 2ℓ − 2 by the choice of v, which is not possible. So v′ = w. By symmetry, we +have v′ ∈ {u, w}. +7 + +Now we show that {v, v′} is a K2-cut of G, implying that Theorem 3.2 holds from +Lemma 2.2. Assume not. Let R′ be a direct connection in G−{v, v′} linking Q∗ and +H −{v, v′}. Let s1, s2 be the ends of R′ with s1 having a neighbour in V (H)−{v, v′} +and s2 having a neighbour in V (Q∗). By Lemma 3.1 (3), s1 has a unique neighbour, +say t1, in V (H) − {v, v′}. Since Q is chosen with |Q| as small as possible, s2 has a +unique neighbour, say t2, in V (Q∗). Set R := t1s1R′s2t2. Then H ∪ Q ∪ R is an +induced subgraph of G or some vertex in {v, v′} has a neighbour in V (R′). When +t1 ∈ V (P2), let P ′ +2 be an induced (u3, u4)-path in P2 ∪ Q ∪ R that is not equal to +P2. Since Q1P ′ +2 is a chordal path of C2 with length |L2| by Lemma 2.3, there is a +cycle in P2 ∪ P ′ +2 with length at most 2ℓ by the choice of v, which is not possible. So +t1 ∈ V (H) − V (P2). Then R ∪ Q∗ contains a direct connection in G\{e, f} linking +P ∗ +2 and H − V (P ∗ +2 ). Since |P2(u3, v)| ≥ ℓ − 2 ≥ 3 and 2 ≤ |P2(u4, v)| ≤ ℓ − 1, by +3.2.2-3.2.6, we have that v′ = u, t1 = u′ +4, and uu4 ∈ E(P2). Let P ′ +2 be an induced +(u3, u′ +4)-path in (P2 − {u4}) ∪ Q ∪ R. Since Q1P ′ +2 and Q1P2 are chordal paths of C2, +by Lemma 2.3, we have |P ′ +2(u′ +4, v)| = |P2(u4, v)| − 1, so u′ +4v ∈ E(G), which is not +possible. +4 +Jumps over an odd hole +Let C be an odd hole of a graph G and s, t ∈ V (C) nonadjacent. Let P be an +induced (s, t)-path. If V (C)∩V (P ∗) = ∅, we call P a jump or an (s, t)-jump over C. +Let Q1, Q2 be the internally disjoint (s, t)-paths of C. If some vertex in V (Q∗ +1) has a +neighbour in V (P ∗) and no vertex in V (Q∗ +2) has a neighbour in V (P ∗), we say that +P is a local jump over C across Q∗ +1. When there is no need to strengthen Q∗ +1, we +will also say that P is a local jump over C. In particular, when |V (Q∗ +1)| = 1, we say +that P is a local jump over C across one vertex. When no vertex in V (Q∗ +1 ∪ Q∗ +2) has +a neighbour in V (P ∗), we say that P is a short jump over C. Hence, short jumps +over C are chordal paths of C. But chordal paths over C maybe not short jumps +over C as the ends of a chordal path maybe adjacent. When P is a short jump over +C, if PQ1 is an odd hole, we say that PQ1 is a jump hole over C and P is a short +jump over C across Q∗ +1. Note that our definition of local jumps over C is a little +different from the definition given in [2]. By our definitions, no short jump over C +is local. +Lemma 4.1. Let C be an odd hole of a graph G ∈ Gℓ. Let P, Q1, Q2 be defined as +the last paragraph. If P is a local or short jump over C across Q∗ +1, then |P|, |Q2| +have the same parity, that is, PQ2 is an even hole and PQ1 is an odd cycle. +Proof. When P is short, the lemma holds from its definition. So we may assume that +P is local. Since some vertex in V (Q∗ +1) has a neighbour in V (P ∗) and g(G) = 2ℓ+1, +we have |P| ≥ 2ℓ. So PQ2 is an even hole. This proves Lemma 4.1. +8 + +Let C be a cycle of a graph G and e, f be chords of C. If no cycle in C ∪ e +containing e contains the two ends of f, we say e, f are crossing; otherwise they are +uncrossing. Note that by our definition, e, f are uncrossing when they share an end. +When C is an odd hole and Pi is an (ui, vi)-jump over C for each integer 1 ≤ i ≤ 2, +if u1v1 and u2v2 are uncrossing chords of C after we replace P1 by the edge u1v1 and +P1 by the edge u2v2, we say that P1, P2 are uncrossing; otherwise, they are crossing. +Lemma 4.2. Let C be an odd hole of a graph G ∈ Gℓ. If a jump P over C is not +local, then G[V (C ∪ P)] contains a short jump over C. +Proof. Without loss of generality we may assume that P is not a short jump over +C. Let s, t be the ends of P and Q1, Q2 be the internally disjoint (s, t)-paths of +C. Since P is neither local nor short, there are u1, u2 ∈ V (P ∗) such that u1 has a +neighbour in V (Q∗ +1) and u2 has a neighbour in V (Q∗ +2), and such that no vertex in +V (P ∗(u1, u2)) has a neighbour in V (C). Since no vertex outside an odd hole has +two neighbours in the odd hole, u1 ̸= u2 and ui has a unique neighbour, say wi, in +V (C) for each integer 1 ≤ i ≤ 2. Then w1u1P(u1, u2)u2w2 is a short jump over C. +This proves Lemma 4.2. +Lemma 4.3. Let C be an odd hole of a graph G ∈ Gℓ. If P is a local (v1, v2)-jump +over C, then G[V (C ∪P)] contains a local jump over C across one vertex or a short +jump over C. +Proof. Assume not. Among all local jumps over C not satisfying Lemma 4.3, let +P be chosen with |P| as small as possible. Let u1, u2 ∈ V (P ∗) have a neighbour +in V (C) − {v1, v2} that are closest to v1, v2, respectively. Let w1, w2 be the unique +neighbour of u1, u2 in V (C), respectively. Since w1u1P(u1, v1) is a short jump over C +when w1v1 /∈ E(G), we have w1v1 ∈ E(G). Similarly, we have w2v2 ∈ E(G). Hence, +when u1 = u2, w1 = w2 and w1 is adjacent to v1, v2, which is a contradiction. So +u1 ̸= u2. Since P is not a local jump over C across one vertex, we have w1 ̸= w2. +Let u3 be the neighbour of w1 in V (P ∗) closest to v2. Note that u3 maybe equal +to u1. By the choice of u2, we have u2 ∈ V (P(u3, v2)). Set P ′ := w1u3P(u3, v2). +Since P ′ is a local jump over C with |P ′| < |P|, by the choice of P, the subgraph +G[V (C ∪ P ′)] contains a local jump over C across one vertex or a short jump over +C, so is G[V (C ∪ P)], which is a contradiction. This proves Lemma 4.3. +Let C be a cycle and suppose that x1, x2, . . . , xn ∈ V (C) occur on C in this +cyclic order with n ≥ 3. +For any two distinct xi and xj, the cycle C contains +two (xi, xj)-paths. Let C(xi, xi+1, . . . , xj) denote the (xi, xj)-path in C containing +xi, xi+1, . . . , xj (and not containing xj+1 if i ̸= j + 1), where subscripts are modulo +n. Such path is uniquely determined as n ≥ 3. +9 + +Lemma 4.4. Let C be an odd hole of a graph G ∈ Gℓ, and Pi be a short (ui, vi)-jump +over C for each integer 1 ≤ i ≤ 2. If P1, P2 are crossing, then G contains an odd +K4-subdivision or a balanced K4-subdivision of type (1, 2). +Proof. Since P1, P2 are short jumps over C, either P1C(v1, u2)P2C(v2, u1) or P1C(u1, u2)P2C(v2, v1) +has length at most 4ℓ by Lemma 2.3. +By symmetry we may assume that the +first cycle is shorter than the last one. Set C′ := P1C(v1, u2)P2C(v2, u1). Since +g(G) = 2ℓ + 1, either C′ is a hole or |C′| = 4ℓ and C′ has exactly one or two +chords. When C′ is a hole, implying that |C(u1, u2)|, |C(v2, v1)| ≥ 2, the subgraph +C ∪ P1 ∪ P2 is induced and isomorphic to a balanced K4-subdivision of type (1, 2). +So we may assume that |C′| = 4ℓ and C′ has exactly one or two chords. Since +|C′| = 4ℓ and g(G) = 2ℓ+1, all cycles in G[V (C′)] using exactly one chord of C′ are +odd holes. Hence, when C′ has exactly two chords, the two chords of C′ are crossing +and G[V (C′)] is isomorphic to an odd K4-subdivision; and when C′ has exactly one +chord, G contains a balanced K4-subdivision of type (1, 2). +The proofs of Lemma 4.5 and Theorem 4.6 use some ideas in [2]. +Lemma 4.5. Let ℓ ≥ 4 be an integer and C be an odd hole of a graph G ∈ Gℓ. +For any integer 1 ≤ i ≤ 2, let Pi be a short or local (ui, vi)-jump over C such that +{u1, v1} ̸= {u2, v2} and u1, u2, v2, v1 appear on C in this order. Assume that Pi +is across C∗(ui, vi) and if Pi is local, we have |V (C∗(ui, vi))| = 1 for any integer +1 ≤ i ≤ 2. Then the following hold. +(1) When P1, P2 are short, G has an odd K4-subdivision. +(2) At most two vertices in V (C(u1, v1) ∪ C(u2, v2)) are not in a jump hole over +C. +Proof. Without loss of generality we may assume that P1, P2 are chosen with P ∗ +1 ∪P ∗ +2 +minimal. Set H := P1 ∪ P2 ∪ C(u1, u2) ∪ C(v1, v2). By symmetry we may assume +that |C(u1, u2)| ≤ |C(v1, v2)| ≥ 1. Note that u1 maybe equal to u2. +4.5.1. (1) is true. +Subproof. Since P1, P2 are short jumps over C, by Lemmas 2.3 and 4.1, |P1 ∪ P2| ≤ +4ℓ − 2 and |H| is odd and at most 6ℓ − 5. Then P1 ∪ P2 contains at most one cycle. +By the choice of P1 and P2, the subgraph P1 ∪ P2 contains no cycle. So H has at +most two cycles. When H has exactly two cycles, since |H| ≤ 6ℓ − 5, one cycle in H +is a hole, so we can find a new short jump over C to replace one of P1, P2 and get +a contradiction to the choice of P1, P2. Hence, H contains a unique cycle C′. Since +g(G) = 2ℓ + 1, the cycle C′ contains at most two chords, and the two chords are +uncrossing when C′ has two chords. No matter which case happens, G[V (P1 ∪ P2)] +10 + +contains two short jumps Q1, Q2 over C such that C ∪ Q1 ∪ Q2 is isomorphic to an +odd K4-subdivision. So (1) holds. +By 4.5.1, we may assume that some Pi is local. For any integer 1 ≤ i ≤ 2, let ai, bi +be the vertices in V (Pi) adjacent to ui, vi, respectively, and set Di := P ∗ +i − {ai, bi}. +Since ℓ ≥ 3, both D1 and D2 are nonempty by Lemma 2.3. Since v1 ̸= v2 and P1, P2 +are jumps over C across C∗(u1, v1) and C∗(u2, v2) respectively, we have that b1 ̸= b2 +and they are nonadjacent. +We say two vertex disjoint subgraphs of a graph are anticomplete if there are no +edges between them. +4.5.2. Either (2) is true or D1 ∪ {b1} is disjoint and anticomplete to D2 ∪ {b2}. +Subproof. Assume not. Since b1 ̸= b2 and they are nonadjacent, by symmetry we +may assume that G[V (D1 ∪ D2) ∪ {b1}] is connected. We claim that P2 is short. +Assume not. Let t be the vertex in V (C) adjacent to u2, v2. Since D2 ̸= ∅, there is a +minimal path Q linking V (C(u1, v1))−{u1} and t with interior in V (D1∪D2)∪{b1}. +Let s be the other end of Q. Since (2) holds when s ̸= v1, we have s = v1. When Q +is across V (C∗(v1, u1, u2, t)), (2) holds; when Q is across V (C∗(v1, v2, t)), replacing +P2 by Q, we get a contradiction to the choice of P1, P2. Hence, P2 is short, implying +that P1 is local and u1 ̸= u2, for otherwise (2) holds. +Let Q be a minimal (s, t)-path linking V (C(u1, v1)) − {u1} and {u2, v2} with +interior in V (D1 ∪ P ∗ +2 ) ∪ {b1}, with s ∈ V (C(u1, v1)) − {u1} and t ∈ {u2, v2}. Since +no vertex in V (C) − {s, t} has a neighbour in V (Q∗), either Q is a short (s, t)-jump +over C or st ∈ E(C). Assume that st ∈ E(C). Since 1 ≤ |C(u1, u2)| ≤ |C(v1, v2)|, +we have s = v1, t = v2, and |C(u1, u2)| = 1. Since P2 is short and P1 is across one +vertex, by Lemmas 2.3 and 4.1, we have |P2| = |C(u2, u1, v1, v2)| = 4 ≥ ℓ + 1, so +ℓ ≤ 3, which contradicts to the fact that ℓ ≥ 4. Hence, Q is a short (s, t)-jump over +C. +When t = u2, since P1 is local and 1 ≤ |C(u1, u2)| ≤ |C(v1, v2)|, the short jump +Q is across V (C∗(s, u1, u2)), so (2) holds as P1 is short. When t = v2, either (2) +holds or we can replace P1 by Q to get a contradiction to the choice of P1, P2. This +proves 4.5.2 as t ∈ {u2, v2}. +By 4.5.2, we may assume that D1∪{b1} is disjoint and anticomplete to D2∪{b2}. +By Lemma 4.1, |H| is odd and at least 2ℓ+3. By symmetry and 4.5.2, we may assume +that a1 has a neighbour in V (D2)∪{b2}. Let w be the vertex in N(a1)∩(V (D2)∪{b2}) +closest to v2 along P2. Set Q := u1a1wP2(w, v2). Then Q is a jump of C. Since +(2) holds when Q is short, we may assume that Q is not short, implying that P2 +is local and the vertex adjacent to v2, u2 has a neighbour in V (P2(w, v2)). Then +|Q| ≥ 3ℓ − 1, implying that P1(a1, v1)C(v1, v2)Q(v2, a1) is an even hole by 4.5.2. +11 + +Hence, by Lemma 4.1, C(u1, v1, v2)Q is an odd hole of length at least 3ℓ + 2, which +is not possible. +Theorem 4.6. Let ℓ ≥ 5 be an integer and C be an odd hole of a graph G ∈ Gℓ. +Then one of the following holds. +(1) G has an odd K4-subdivision. +(2) G contains a balanced K4-subdivision of type (1, 2). +(3) G has a P3-cut. +(4) G has a degree-2 vertex. +Proof. Assume that neither (1) nor (2) is true. Set C := v1v2 . . . v2ℓ+1v1. Let P be +a short jump over C with |P| as small as possible. By symmetry we may assume +that the ends of P are v2, vk with k ≤ ℓ + 2. By Lemmas 4.5 (1), 4.4 and 2.3, we +may assume that all short jumps over C have ends in {v2, v3, . . . , vk}. That is, (i) +no jump hole over C contains a vertex in V (C) − {v2, v3, . . . , vk}. +For each integer 1 ≤ i ≤ 2, let Pi be a local (si, ti)-jump over C across one vertex +and Qi be the (si, ti)-path on C of length three. When P1 and P2 are uncrossing, +by (i) and Lemma 4.5 (2), |V (Q1 ∪ Q2) − {v2, v3, . . . , vk}| ≤ 2. When P1 and P2 are +crossing, |Q1 ∪Q2| = 4. Since there is no short jump over C or at least one vertex in +{si, ti} is in {v2, v3, . . . , vk} by Lemma 4.5 (2), by possibly reordering we may assume +that all local jumps over C across one vertex have ends in {v1, v2, . . . , vk, vk+1} with +4 ≤ k ≤ ℓ + 2. For any integer 1 ≤ i ≤ k + 1, let Xi be the set of vertices adjacent +to vi that are in a local jump over C across one vertex with one end vi or a short +jump over C with one end vi. Set X := X1 ∪ X2 ∪ · · · ∪ Xk+1. Then X intersects +all short jumps over C and all local jumps over C across one vertex in at least two +vertices. +Since ℓ ≥ 5 and k ≤ ℓ + 2, we have k + 3 ≤ 2ℓ. Since no vertex in V (G) − V (C) +has two neighbours in V (C), the vertex vk+3 has no neighbour in X. Assume that +vk+3 has degree at least three, for otherwise (4) holds. There is a connected induced +subgraph D such that vk+3 has a neighbour in V (D) and V (D) ∩ (V (C) ∪ X) = ∅, +and D is maximal with these properties. Let N be the set of vertices in V (C) ∪ X +that have a neighbour in V (D). Evidently, vk+3 ∈ N. +4.6.1. N ∩ (X ∪ V (C)) ⊆ {vk+2, vk+3, vk+4}. +Subproof. Assume not. +When 1 ≤ i ≤ k + 1, set Wi := Xi ∪ {vi}; and when +k + 5 ≤ i ≤ 2ℓ + 1, set Wi := {vi}. Assume that N ∩ Wi ̸= ∅ for some integer +i /∈ {k + 2, k + 3, k + 4}. Let Q be a minimal (vi, vk+3)-path with interior in V (D)∪ +(Wi − {vi}). Then Q is a jump over C and |Q ∩ X| ≤ 1. Since X intersects each +12 + +local jumps over C across one vertex and each short jump over C in at least two +vertices, G[V (C ∪ Q)] does not contain a local jump over C across one vertex or a +short jump over C, which is not possible by Lemmas 4.2 and 4.3 +By 4.6.1, the set {vk+2, vk+3, vk+4} is a P3-cut of G. So (3) holds. +Now, we can prove Theorem 1.3, which is restated here for convenience. +Theorem 4.7. For any integer ℓ ≥ 5, each graph in Gℓ is 3-colorable. +Proof. Assume not. +Let G ∈ Gℓ be a minimal counterexample to Theorem 1.3. +Then G is 4-vertex-critical. So G has no degree-2 vertex. By Theorems 1.2 and 3.2, +G contains neither odd K4-subdivision nor balanced K4-subdivision of type (1, 2). +Hence, G has a P3-cut by Theorem 4.6, which is a contradiction to Lemma 2.2. +5 +Acknowledgments +This research was partially supported by grants from the National Natural Sciences +Foundation of China (No. 11971111). The author thanks Yidong Zhou for carefully +reading the paper and giving some suggestions. +References +[1] R. Chen, Y. Zhou, On coloring of graphs with girth 2ℓ + 1 and without longer +odd holes. odd K4-subdivisions, 2022, in arXiv:2210.12376v1. +[2] M. Chudnovsky, P. Seymour, Proof of a conjecture of Plummer and Zha, to +appear in J. Graph Theory, 2022. in arXiv:2201.11505v2. +[3] D. Nelson, M. Plummer, N. Robertson, X. Zha, On a conjecture concerning the +Petersen graph. Electron. J. Combin., 2011, 18: #P20. +[4] M. Plummer, X. Zha, On a conjecture concerning the Petersen Graph: Part II. +Electron. J. Combin., 2014, 21: #P1.34. +[5] D. Wu, B. Xu, Y. Xu, On coloring of graphs of girth 2ℓ + 1 without longer odd +holes (in Chinese), Sci Sin Math., 2022, 52: 1-18. in arXiv: 2204.06284v1. +[6] D. Wu, B. Xu, Y. Xu, The chromatic number of heptagraphs, 2022, in arXiv: +2206.01400v1. +[7] B. Xu, G. Yu, X. Zha, A note on chromatic number and induced odd cycles. +Electron. J. Combin., 2017, 24(4): #P4.32. +13 + diff --git a/K9AyT4oBgHgl3EQfTveL/content/tmp_files/load_file.txt b/K9AyT4oBgHgl3EQfTveL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdf2b50e41301a7e0951453df16ab88e789dcada --- /dev/null +++ b/K9AyT4oBgHgl3EQfTveL/content/tmp_files/load_file.txt @@ -0,0 +1,654 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf,len=653 +page_content='Graphs with girth 2ℓ + 1 and without longer odd holes are 3-colorable Rong Chen Center for Discrete Mathematics, Fuzhou University Fuzhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' China January 3, 2023 1 Abstract For a number ℓ ≥ 2, let Gℓ denote the family of graphs which have girth 2ℓ + 1 and have no odd hole with length greater than 2ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Plummer and Zha conjectured that every 3-connected and internally 4-connected graph in G2 is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Wu, Xu, and Xu conjectured that every graph in � ℓ≥2 Gℓ is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Chudnovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=', respectively, proved that every graph in G2 and G3 is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In this paper, we prove that every graph in � ℓ≥5 Gℓ is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Key Words: chromatic number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' odd holes 1 Introduction All graphs considered in this paper are finite, simple, and undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' A proper coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' A graph is k-colorable if it has a proper coloring using at most k colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The chromatic number of G, denoted by χ(G), is the minimum number k such that G is k-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The girth of a graph G, denoted by g(G), is the minimum length of a cycle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' A hole in a graph is an induced cycle of length at least four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' An odd hole means a hole of odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any integer ℓ ≥ 2, let Gℓ be the family of graphs that have girth 2ℓ + 1 and have no odd holes of length at least 2ℓ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Robertson conjectured in [3] that the Petersen graph is the only graph in G2 that is 3-connected and internally 4-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Plummer and Zha [4] disproved Robertson’s conjecture 1Mathematics Subject Classification: 05C15, 05C17, 05C69 Email: rongchen@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='cn (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Chen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='00112v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='CO] 31 Dec 2022 and proposed the conjecture that all 3-connected and internally 4-connected graphs in G2 have bounded chromatic number, and the strong conjecture that such graphs are 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The first was proved by Xu, Yu, and Zha [7], who proved that all graphs in G2 is 4-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Chudnovsky and Seymour [2] proved that every graph in G2 is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In the same paper, Chudnovsky and Seymour also gave a short and pretty proof of the result in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Wu, Xu, and Xu [5] showed that graphs in � ℓ≥2 Gℓ are 4-colorable and proposed the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' ([5], Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=') Every graph in � ℓ≥2 Gℓ is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' We say that a graph G has an odd K4-subdivision if some subgraph H of G is isomorphic to a K4-subdivision and whose face cycles are all odd holes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The subgraph H is also induced by ([1], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In the same paper, the author and Zhou proved Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' ([1], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=') For each graph G in � ℓ≥5 Gℓ, if G has an odd K4-subdivision, then χ(G) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Based on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, in this paper we prove that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 holds for ℓ ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' That is, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' All graphs in � ℓ≥5 Gℓ are 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' We conjecture Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For a graph G without triangles, if all odd holes of G have the same length, then G is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For an integer ℓ ≥ 2 and a graph G with g(G) = 2ℓ + 1, if the set of lengths of odd holes of G have k members, then G is (k + 2)-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 2 Preliminary A cycle is a connected 2-regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any subset U ⊆ V (G), let G[U] be the induced subgraph of G defined on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For subgraphs H and H′ of G, set |H| := |E(H)| and H∆H′ := E(H)∆E(H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let H ∪ H′ denote the subgraph of G with vertex set V (H) ∪ V (H′) and edge set E(H) ∪ E(H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let H ∩ H′ denote the subgraph of G with edge set E(H) ∩ E(H′) and without isolated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let N(H) be the set of vertices in V (G) − V (H) that have a neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set N[H] := N(H) ∪ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P be an (x, y)-path and Q a (y, z)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When P and Q are internally disjoint paths, let PQ denote the (x, z)-path P ∪ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Evidently, PQ is a path when x ̸= z, and PQ is a cycle when x = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P ∗ denote the set of internal vertices 2 of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When u, v ∈ V (P), let P(u, v) denote the subpath of P with ends u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For simplicity, we will let P ∗(u, v) denote (P(u, v))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For a number k ≥ 2, a graph G is k-vertex-critical if χ(G) = k and χ(G − v) ≤ k − 1 for each vertex v of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' ([1], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=') For any number k ≥ 4, each k-vertex-critical graph has no 2-edge-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For an i-vertex path P of a connected graph G, if G − V (P) is disconnected, then we say that P is a Pi-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Usually, a P2-cut is also called a K2-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Evidently, every k-vertex-critical graph has no K2-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Chudnovsky and Seymour in [2] proved that every 4-vertex-critical graph G in G2 has no P3-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In fact, their methods can also be used to prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' ([2]) For any number ℓ ≥ 2, every 4-vertex-critical graph in Gℓ has neither K2-cut nor P3-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' A theta graph is a graph that consists of a pair of distinct vertices joined by three internally disjoint paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be a hole of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' A path P of G is a chordal path of C if C ∪ P is an induced theta-subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since g(G) = 2ℓ + 1 and each odd hole of G has length 2ℓ + 1, by simple computation we have the following result, which will be frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' ([1], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=') Let C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P be a chordal path of C, and P1, P2 be the internally disjoint paths of C that have the same ends as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that |P| and |P1| have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then |P1| = 1 or ℓ ≥ |P2| < |P1| = |P| ≥ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In particular, when |P1| ≥ 2, all chordal paths of C with the same ends as P1 have length |P1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C1, C2 be odd holes of a graph G ∈ Gℓ such that C1 ∪ C2 is a theta subgraph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When C1 ∪ C2 is not induced, since |C1∆C2| ≤ 4ℓ and g(G) = 2ℓ + 1, we have that |C1∆C2| = 4ℓ and G[V (C1 ∪ C2)] is an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For the similar reason, if C is an even hole of length 4ℓ of G, then C has at most two chords, and when C has two chords, G[V (C)] is an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3 Graphs has a subgraph isomorphic to a K4-subdivision Let H be a graph that is isomorphic to a K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For a path P of H whose ends are degree-3 vertices of H, if P contains exactly two degree-3 vertices of H, then we say it is an arris of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Evidently, H has exactly six arrises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' We say that a graph 3 u1 u2 u3 u4 C1 C2 C3 C4 P1 Q2 P2 Q1 L1 L2 H Figure 1: u1, u2, u3, u4 are the degree-3 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The face cycles C1, C2 are odd holes, and C3, C4 are even holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' {P1, P2}, {Q1, Q2}, {L1, L2} are the pairs of vertex disjoint arrises of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' G contins a balanced K4-subdivision if G has an induced subgraph H isomorphic to a K4-subdivision and exactly two face cycles of H are odd holes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If a graph G in Gℓ contains a balanced K4-subdivision H that is pictured as the graph in Figure 1, then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (1) |P1| ≤ ℓ and 1 ∈ {|Q1|, |Q2|, |L1|, |L2|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (2) |P2| ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (3) Assume that G has no odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If |Q1| = 1 and |L2| ≥ 2, then no vertex v ∈ V (G) − V (H) has two neighbours in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since C1, C2 are odd holes and g(G) = 2ℓ+1, we have |P1| ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that 1 /∈ {|Q1|, |Q2|, |L1|, |L2|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P2Q2 is a chordal path of C1 and C4 is an even hole, we have |P2|+|Q2| = |L1| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Similarly, we have |P2|+|L1| = |Q2|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Now we consider (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By (1) and symmetry we may assume that |Q1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |P1| ≤ ℓ by (1), we have |L1| ≥ ℓ, so Q1P2 is a chordal path of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then |Q1P2| ≥ ℓ + 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So |P2| ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Next, we prove that (3) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let v ∈ V (G)−V (H) have at least two neighbours in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |L2| > 1 and C3 is an even hole, C2∆C3 is an odd hole and |L2| ≥ ℓ + 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since a vertex not in an odd hole has at most one neighbour in the odd hole, each arris of H has at most one vertex adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The vertex v has exactly one neighbour in V (C1 ∪C2) and exactly one neigh- bour in V (P ∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 4 Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since v has at most one neighbour in V (P2), it suffices to show that v has at most one neighbour in V (C1 ∪ C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then v has exactly two neighbours in V (C1 ∪ C2) with one in V (C1) − V (P1) and the other in V (C2) − V (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then G[V (C1 ∪ C2) ∪ {v}] is a balanced K4-subdivision as G has no odd K4-subdivision, which is not possible by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Note that C2∆C3 is an odd hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since v can not have two neighbours in an odd hole of H, the vertex v has no neighbour in V (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For the same reason, if v has a neighbour in V (Q1 ∪ Q2), the neighbours of v in V (H) must be in V (C1 ∪ C2), which is not possible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, N(v) ∩ V (C1 ∪ C2) ⊆ V (L∗ 1 ∪ L∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that v has a neighbour in V (L∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |C4| ≤ 4ℓ − 2 and g(G) = 2ℓ + 1, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, the two cycles in G[V (C4) ∪ {v}] containing v are odd holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since |L2| ≥ ℓ+1, the subgraph G[V (C4)∪{v}]∪P1 ∪Q1 is an odd K4-subdivision, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Similarly we can show that v can not have a neighbour in V (L∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let H be an induced subgraph of G pictured as the graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When |Q1| = 1, we say that H is a balanced K4-subdivision of type (1, 2) if |L2| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |Q1| = 1 implies that |L1| > 1, this definition is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let H1, H2 be vertex disjoint induced subgraphs of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' An induced (v1, v2)-path P is a direct connection linking H1 and H2 if v1 is the only vertex in V (P) having a neighbour in H1 and v2 is the only vertex in V (P) having a neighbour in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Evidently, the set of internal vertices of each shortest induced path joining H1 and H2 induces a direct connection linking H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let ℓ ≥ 5 be an integer and G be a graph in Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If G is 4-vertex critical, then G does not contain a balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, G does not have an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume to the contrary that G has a balanced K4-subdivision H of type (1, 2) that is pictured as the graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Without loss of generality we may assume that H is chosen with |H| as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (1) and symmetry we may assume that |Q1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (2) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, we have |Q1| = 1, |P2| ≥ ℓ, |L1|, |L2| ≥ ℓ + 1, and |P1| < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1) Let e, f be the edges in P2 incident with u3, u4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P be a direct connection in G\\{e, f} linking P ∗ 2 and H − V (P ∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 implies that such P exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let v1, v2 be the ends of P with v2 having a neighbour in V (P ∗ 2 ) and v1 having a neighbour in V (H) − V (P ∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (3), both v1 and v2 have a unique neighbour in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let x be the neighbour of v1 in V (H) and y the 5 neighbour of v2 in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set P ′ := xv1Pv2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then H ∪ P ′ is an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (3) again, we have |P ′| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' x ̸= u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that x = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ 3 := L2P2(u4, y)P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |L2| ≥ ℓ+1, we have that C′ 3 is an even hole by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |P ′| ≥ 3 and C2∆C′ 3 is an odd hole, (H ∪ P ′) − V (L∗ 2) is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is a contradiction to the choice of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' x /∈ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, we have x ∈ V (P1)−{u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ 1 := P1(x, u2)Q1P2(u3, y)P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |P1| < ℓ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1), it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3 that C′ 1 is an odd hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then P1(x, u1)Q2P2(u4, y)P ′ is an even hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since |L2| ≥ ℓ+1, we have x = u1 and |Q2| = 1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since C1 and C′ 1 are odd holes, we have |L1| > |P ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since |P ′| ≥ 3, the subgraph C′ 1 ∪ C2 ∪ P2 is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When x = u3, we have |P2(u3, y)| = 1 or |P2(u3, y)| ≥ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that |P2(u3, y)| ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q1P2 and Q1P ′P2(y, u4) are chordal paths of C2 that have the same ends as L2, they have length |L2| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So |P2(u3, y)P ′| = 2|P2(u3, y)|, implying that |P2(u3, y)| ≥ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If x ∈ V (L∗ 1), then xu3, yu3 ∈ E(G), and |Q2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ 3 := L1(x, u3)P2(u3, y)P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When C′ 3 is an odd hole, since C′ 3∆C4 is an odd hole, C1 ∪ C′ 3 ∪ P2 ∪ Q2 is an odd K4-subdivision, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So C′ 3 is an even hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that xu3 /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then |C′ 3| = 2|L1(x, u3)| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since C1∆C′ 3 is an odd hole, (H ∪ P ′) − V (L∗ 1(x, u3)) is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is a contradiction to the choice of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So xu3 ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that yu3 /∈ E(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q1P2 and Q1L1(u3, x)P ′P2(y, u4) are chordal paths of C2 that have the same ends as L2, they have length |L2| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since xu3 ∈ E(G) and |L1| ≥ ℓ + 1 imply |L1(x, u1)| ≥ 2, the subgraph (H ∪ P ′) − V (P ∗ 2 (y, u3)) is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So yu3 ∈ E(P2), implying that |P ′| ≥ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since C′ 3∆C3∆C1 is an odd cycle of length at least 2ℓ + 3, we have |Q2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' x /∈ V (Q∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 6 Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ 3 := Q2(x, u4)P2(u4, y)P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that C′ 3 is an odd hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since C1 ∪ C2 ∪ (C3∆C′ 3) is an odd K4-subdivision when yu4 /∈ E(P2), we have yu4 ∈ E(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |Q2| < ℓ as |L2| ≥ ℓ + 1, the graph C4∆C′ 3 is an odd hole with length larger than 3ℓ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1), which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So C′ 3 is an even hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |Q2| < ℓ, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3 that xu4 ∈ E(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Moreover, since P ′ is a chordal path of the odd hole C2∆C3 and C′ 3 is an even hole, ℓ + 1 ≤ |P ′| = |P2(y, u4)| + 1 ≤ |P2| < |L1| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, C2 ∪ C3 ∪ P ′ is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let u′ 4 be the neighbour of u4 in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If x ∈ V (L2), then x ∈ {u4, u′ 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In particular, when x = u′ 4, we have yu4 ∈ E(P2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' and when x = u4, we have yu4 ∈ E(P2) or |P2(y, u4)| ≥ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ 3 := L2(x, u4)P2(u4, y)P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that x = u4 and yu4 /∈ E(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q1P2 and Q1P2(u3, y)P ′ are chordal paths of C2 with the same ends, |C′ 3| = 2|P2(y, u4)| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, so |P2(y, u4)| ≥ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' That is, when x = u4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So we may assume that x ̸= u4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, we may assume that x ∈ V (L∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When C′ 3 is an odd hole, since x /∈ {u2, u4}, the graph C2 ∪ C3 ∪ P ′ is an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So C′ 3 is an even hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When x ̸= u′ 4, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, we have that xu2 ∈ E(H), |C′ 3| = 2|L2(x, u4)| and C3∆C′ 3 is an odd hole, so (H ∪ P ′) − V (L∗ 2(x, u4)) is a balanced K4-subdivision of type (1, 2) whose edge number is less than |H|, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, x = u′ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When yu4 /∈ E(P2), since Q1P2(u3, y)P ′ is a chordal path of C2, we have |C′ 3| = 2|P2(y, u4)| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So (H ∪ P ′) − V (P ∗ 2 (y, u4)) is a balanced K4- subdivision of type (1, 2) whose edge number is less than |H|, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So yu4 ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let v be a vertex in V (P ∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When |P2| ≥ ℓ+1, let |P2(u3, v)| = ℓ−1, otherwise let |P2(u3, v)| = ℓ − 2 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The vertex v is well defined as |P2| ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |P2| ≤ 2ℓ − 2, we have 2 ≤ |P2(u4, v)| ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let u, w ∈ N(v) ∩ V (P ∗ 2 ) with u closer to u4 and w closer to u3 along P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since {uv, vw} is not an edge cut of G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, there is a shortest induced path Q in G\\{uv, vw} linking v and H − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (3), H ∪ Q is an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let v′ be the other end of Q that is not equal to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When v′ ∈ V (C1 ∪ C2), since Q∗ is a direct connection in G\\{e, f} linking P ∗ 2 and H −V (P ∗ 2 ), we get a contradiction to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So v′ ∈ V (P ∗ 2 )−{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that v′ ∈ V (P ∗ 2 (u3, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When v′ ̸= w, either Q2P1Q1P2(u3, v′)QP2(v, u4) is an odd hole of length at least 2ℓ + 3 or QP2(v, v′) is a cycle of length 2|P2(v, v′)| that is at most 2ℓ − 2 by the choice of v, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So v′ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By symmetry, we have v′ ∈ {u, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 7 Now we show that {v, v′} is a K2-cut of G, implying that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2 holds from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let R′ be a direct connection in G−{v, v′} linking Q∗ and H −{v, v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let s1, s2 be the ends of R′ with s1 having a neighbour in V (H)−{v, v′} and s2 having a neighbour in V (Q∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1 (3), s1 has a unique neighbour, say t1, in V (H) − {v, v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q is chosen with |Q| as small as possible, s2 has a unique neighbour, say t2, in V (Q∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set R := t1s1R′s2t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then H ∪ Q ∪ R is an induced subgraph of G or some vertex in {v, v′} has a neighbour in V (R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When t1 ∈ V (P2), let P ′ 2 be an induced (u3, u4)-path in P2 ∪ Q ∪ R that is not equal to P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q1P ′ 2 is a chordal path of C2 with length |L2| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, there is a cycle in P2 ∪ P ′ 2 with length at most 2ℓ by the choice of v, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So t1 ∈ V (H) − V (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then R ∪ Q∗ contains a direct connection in G\\{e, f} linking P ∗ 2 and H − V (P ∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |P2(u3, v)| ≥ ℓ − 2 ≥ 3 and 2 ≤ |P2(u4, v)| ≤ ℓ − 1, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6, we have that v′ = u, t1 = u′ 4, and uu4 ∈ E(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P ′ 2 be an induced (u3, u′ 4)-path in (P2 − {u4}) ∪ Q ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since Q1P ′ 2 and Q1P2 are chordal paths of C2, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, we have |P ′ 2(u′ 4, v)| = |P2(u4, v)| − 1, so u′ 4v ∈ E(G), which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 4 Jumps over an odd hole Let C be an odd hole of a graph G and s, t ∈ V (C) nonadjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P be an induced (s, t)-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If V (C)∩V (P ∗) = ∅, we call P a jump or an (s, t)-jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let Q1, Q2 be the internally disjoint (s, t)-paths of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If some vertex in V (Q∗ 1) has a neighbour in V (P ∗) and no vertex in V (Q∗ 2) has a neighbour in V (P ∗), we say that P is a local jump over C across Q∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When there is no need to strengthen Q∗ 1, we will also say that P is a local jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' In particular, when |V (Q∗ 1)| = 1, we say that P is a local jump over C across one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When no vertex in V (Q∗ 1 ∪ Q∗ 2) has a neighbour in V (P ∗), we say that P is a short jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, short jumps over C are chordal paths of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' But chordal paths over C maybe not short jumps over C as the ends of a chordal path maybe adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When P is a short jump over C, if PQ1 is an odd hole, we say that PQ1 is a jump hole over C and P is a short jump over C across Q∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Note that our definition of local jumps over C is a little different from the definition given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By our definitions, no short jump over C is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P, Q1, Q2 be defined as the last paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If P is a local or short jump over C across Q∗ 1, then |P|, |Q2| have the same parity, that is, PQ2 is an even hole and PQ1 is an odd cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When P is short, the lemma holds from its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So we may assume that P is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since some vertex in V (Q∗ 1) has a neighbour in V (P ∗) and g(G) = 2ℓ+1, we have |P| ≥ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So PQ2 is an even hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 8 Let C be a cycle of a graph G and e, f be chords of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If no cycle in C ∪ e containing e contains the two ends of f, we say e, f are crossing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' otherwise they are uncrossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Note that by our definition, e, f are uncrossing when they share an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When C is an odd hole and Pi is an (ui, vi)-jump over C for each integer 1 ≤ i ≤ 2, if u1v1 and u2v2 are uncrossing chords of C after we replace P1 by the edge u1v1 and P1 by the edge u2v2, we say that P1, P2 are uncrossing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' otherwise, they are crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If a jump P over C is not local, then G[V (C ∪ P)] contains a short jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Without loss of generality we may assume that P is not a short jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let s, t be the ends of P and Q1, Q2 be the internally disjoint (s, t)-paths of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P is neither local nor short, there are u1, u2 ∈ V (P ∗) such that u1 has a neighbour in V (Q∗ 1) and u2 has a neighbour in V (Q∗ 2), and such that no vertex in V (P ∗(u1, u2)) has a neighbour in V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since no vertex outside an odd hole has two neighbours in the odd hole, u1 ̸= u2 and ui has a unique neighbour, say wi, in V (C) for each integer 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then w1u1P(u1, u2)u2w2 is a short jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If P is a local (v1, v2)-jump over C, then G[V (C ∪P)] contains a local jump over C across one vertex or a short jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Among all local jumps over C not satisfying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, let P be chosen with |P| as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let u1, u2 ∈ V (P ∗) have a neighbour in V (C) − {v1, v2} that are closest to v1, v2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let w1, w2 be the unique neighbour of u1, u2 in V (C), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since w1u1P(u1, v1) is a short jump over C when w1v1 /∈ E(G), we have w1v1 ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Similarly, we have w2v2 ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, when u1 = u2, w1 = w2 and w1 is adjacent to v1, v2, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So u1 ̸= u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P is not a local jump over C across one vertex, we have w1 ̸= w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let u3 be the neighbour of w1 in V (P ∗) closest to v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Note that u3 maybe equal to u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By the choice of u2, we have u2 ∈ V (P(u3, v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set P ′ := w1u3P(u3, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P ′ is a local jump over C with |P ′| < |P|, by the choice of P, the subgraph G[V (C ∪ P ′)] contains a local jump over C across one vertex or a short jump over C, so is G[V (C ∪ P)], which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be a cycle and suppose that x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , xn ∈ V (C) occur on C in this cyclic order with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any two distinct xi and xj, the cycle C contains two (xi, xj)-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C(xi, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , xj) denote the (xi, xj)-path in C containing xi, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , xj (and not containing xj+1 if i ̸= j + 1), where subscripts are modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Such path is uniquely determined as n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 9 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let C be an odd hole of a graph G ∈ Gℓ, and Pi be a short (ui, vi)-jump over C for each integer 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' If P1, P2 are crossing, then G contains an odd K4-subdivision or a balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P1, P2 are short jumps over C, either P1C(v1, u2)P2C(v2, u1) or P1C(u1, u2)P2C(v2, v1) has length at most 4ℓ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By symmetry we may assume that the first cycle is shorter than the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C′ := P1C(v1, u2)P2C(v2, u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since g(G) = 2ℓ + 1, either C′ is a hole or |C′| = 4ℓ and C′ has exactly one or two chords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When C′ is a hole, implying that |C(u1, u2)|, |C(v2, v1)| ≥ 2, the subgraph C ∪ P1 ∪ P2 is induced and isomorphic to a balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So we may assume that |C′| = 4ℓ and C′ has exactly one or two chords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since |C′| = 4ℓ and g(G) = 2ℓ+1, all cycles in G[V (C′)] using exactly one chord of C′ are odd holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, when C′ has exactly two chords, the two chords of C′ are crossing and G[V (C′)] is isomorphic to an odd K4-subdivision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' and when C′ has exactly one chord, G contains a balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The proofs of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6 use some ideas in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let ℓ ≥ 4 be an integer and C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any integer 1 ≤ i ≤ 2, let Pi be a short or local (ui, vi)-jump over C such that {u1, v1} ̸= {u2, v2} and u1, u2, v2, v1 appear on C in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that Pi is across C∗(ui, vi) and if Pi is local, we have |V (C∗(ui, vi))| = 1 for any integer 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (1) When P1, P2 are short, G has an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (2) At most two vertices in V (C(u1, v1) ∪ C(u2, v2)) are not in a jump hole over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Without loss of generality we may assume that P1, P2 are chosen with P ∗ 1 ∪P ∗ 2 minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set H := P1 ∪ P2 ∪ C(u1, u2) ∪ C(v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By symmetry we may assume that |C(u1, u2)| ≤ |C(v1, v2)| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Note that u1 maybe equal to u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (1) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P1, P2 are short jumps over C, by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, |P1 ∪ P2| ≤ 4ℓ − 2 and |H| is odd and at most 6ℓ − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then P1 ∪ P2 contains at most one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By the choice of P1 and P2, the subgraph P1 ∪ P2 contains no cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So H has at most two cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When H has exactly two cycles, since |H| ≤ 6ℓ − 5, one cycle in H is a hole, so we can find a new short jump over C to replace one of P1, P2 and get a contradiction to the choice of P1, P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, H contains a unique cycle C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since g(G) = 2ℓ + 1, the cycle C′ contains at most two chords, and the two chords are uncrossing when C′ has two chords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' No matter which case happens, G[V (P1 ∪ P2)] 10 contains two short jumps Q1, Q2 over C such that C ∪ Q1 ∪ Q2 is isomorphic to an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, we may assume that some Pi is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any integer 1 ≤ i ≤ 2, let ai, bi be the vertices in V (Pi) adjacent to ui, vi, respectively, and set Di := P ∗ i − {ai, bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since ℓ ≥ 3, both D1 and D2 are nonempty by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since v1 ̸= v2 and P1, P2 are jumps over C across C∗(u1, v1) and C∗(u2, v2) respectively, we have that b1 ̸= b2 and they are nonadjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' We say two vertex disjoint subgraphs of a graph are anticomplete if there are no edges between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Either (2) is true or D1 ∪ {b1} is disjoint and anticomplete to D2 ∪ {b2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since b1 ̸= b2 and they are nonadjacent, by symmetry we may assume that G[V (D1 ∪ D2) ∪ {b1}] is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' We claim that P2 is short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let t be the vertex in V (C) adjacent to u2, v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since D2 ̸= ∅, there is a minimal path Q linking V (C(u1, v1))−{u1} and t with interior in V (D1∪D2)∪{b1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let s be the other end of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since (2) holds when s ̸= v1, we have s = v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When Q is across V (C∗(v1, u1, u2, t)), (2) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' when Q is across V (C∗(v1, v2, t)), replacing P2 by Q, we get a contradiction to the choice of P1, P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, P2 is short, implying that P1 is local and u1 ̸= u2, for otherwise (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let Q be a minimal (s, t)-path linking V (C(u1, v1)) − {u1} and {u2, v2} with interior in V (D1 ∪ P ∗ 2 ) ∪ {b1}, with s ∈ V (C(u1, v1)) − {u1} and t ∈ {u2, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since no vertex in V (C) − {s, t} has a neighbour in V (Q∗), either Q is a short (s, t)-jump over C or st ∈ E(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that st ∈ E(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since 1 ≤ |C(u1, u2)| ≤ |C(v1, v2)|, we have s = v1, t = v2, and |C(u1, u2)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since P2 is short and P1 is across one vertex, by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, we have |P2| = |C(u2, u1, v1, v2)| = 4 ≥ ℓ + 1, so ℓ ≤ 3, which contradicts to the fact that ℓ ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, Q is a short (s, t)-jump over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When t = u2, since P1 is local and 1 ≤ |C(u1, u2)| ≤ |C(v1, v2)|, the short jump Q is across V (C∗(s, u1, u2)), so (2) holds as P1 is short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When t = v2, either (2) holds or we can replace P1 by Q to get a contradiction to the choice of P1, P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' This proves 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2 as t ∈ {u2, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, we may assume that D1∪{b1} is disjoint and anticomplete to D2∪{b2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, |H| is odd and at least 2ℓ+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By symmetry and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, we may assume that a1 has a neighbour in V (D2)∪{b2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let w be the vertex in N(a1)∩(V (D2)∪{b2}) closest to v2 along P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set Q := u1a1wP2(w, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then Q is a jump of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since (2) holds when Q is short, we may assume that Q is not short, implying that P2 is local and the vertex adjacent to v2, u2 has a neighbour in V (P2(w, v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then |Q| ≥ 3ℓ − 1, implying that P1(a1, v1)C(v1, v2)Q(v2, a1) is an even hole by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 11 Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, C(u1, v1, v2)Q is an odd hole of length at least 3ℓ + 2, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let ℓ ≥ 5 be an integer and C be an odd hole of a graph G ∈ Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (1) G has an odd K4-subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (2) G contains a balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (3) G has a P3-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' (4) G has a degree-2 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that neither (1) nor (2) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set C := v1v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' v2ℓ+1v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let P be a short jump over C with |P| as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By symmetry we may assume that the ends of P are v2, vk with k ≤ ℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5 (1), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, we may assume that all short jumps over C have ends in {v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' That is, (i) no jump hole over C contains a vertex in V (C) − {v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For each integer 1 ≤ i ≤ 2, let Pi be a local (si, ti)-jump over C across one vertex and Qi be the (si, ti)-path on C of length three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When P1 and P2 are uncrossing, by (i) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5 (2), |V (Q1 ∪ Q2) − {v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , vk}| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When P1 and P2 are crossing, |Q1 ∪Q2| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since there is no short jump over C or at least one vertex in {si, ti} is in {v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , vk} by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='5 (2), by possibly reordering we may assume that all local jumps over C across one vertex have ends in {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' , vk, vk+1} with 4 ≤ k ≤ ℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any integer 1 ≤ i ≤ k + 1, let Xi be the set of vertices adjacent to vi that are in a local jump over C across one vertex with one end vi or a short jump over C with one end vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Set X := X1 ∪ X2 ∪ · · · ∪ Xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then X intersects all short jumps over C and all local jumps over C across one vertex in at least two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since ℓ ≥ 5 and k ≤ ℓ + 2, we have k + 3 ≤ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since no vertex in V (G) − V (C) has two neighbours in V (C), the vertex vk+3 has no neighbour in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that vk+3 has degree at least three, for otherwise (4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' There is a connected induced subgraph D such that vk+3 has a neighbour in V (D) and V (D) ∩ (V (C) ∪ X) = ∅, and D is maximal with these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let N be the set of vertices in V (C) ∪ X that have a neighbour in V (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Evidently, vk+3 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' N ∩ (X ∪ V (C)) ⊆ {vk+2, vk+3, vk+4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Subproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' When 1 ≤ i ≤ k + 1, set Wi := Xi ∪ {vi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' and when k + 5 ≤ i ≤ 2ℓ + 1, set Wi := {vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume that N ∩ Wi ̸= ∅ for some integer i /∈ {k + 2, k + 3, k + 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let Q be a minimal (vi, vk+3)-path with interior in V (D)∪ (Wi − {vi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then Q is a jump over C and |Q ∩ X| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Since X intersects each 12 local jumps over C across one vertex and each short jump over C in at least two vertices, G[V (C ∪ Q)] does not contain a local jump over C across one vertex or a short jump over C, which is not possible by Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3 By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='1, the set {vk+2, vk+3, vk+4} is a P3-cut of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So (3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Now, we can prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3, which is restated here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' For any integer ℓ ≥ 5, each graph in Gℓ is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Assume not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Let G ∈ Gℓ be a minimal counterexample to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Then G is 4-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' So G has no degree-2 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' By Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2, G contains neither odd K4-subdivision nor balanced K4-subdivision of type (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' Hence, G has a P3-cut by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='6, which is a contradiction to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 5 Acknowledgments This research was partially supported by grants from the National Natural Sciences Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 11971111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' The 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfTveL/content/2301.00112v1.pdf'} diff --git a/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/2301.01288v1.pdf.txt b/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/2301.01288v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c93511ae92f08546a30aa52760a881ce9afef91 --- /dev/null +++ b/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/2301.01288v1.pdf.txt @@ -0,0 +1,2357 @@ +Astronomy & Astrophysics manuscript no. 45157corr +©ESO 2023 +January 4, 2023 +Constraints on the non-thermal desorption of methanol in the cold +core LDN 429-C⋆ +A. Taillard,1 V. Wakelam1, P. Gratier1, E. Dartois2, M. Chabot3, J. A. Noble4, J. V. Keane5, A. C. A. Boogert5, D. +Harsono6 +1 Laboratoire d’Astrophysique de Bordeaux (LAB), Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, +France e-mail: angele.taillard@u-bordeaux.fr +2 Institut des sciences Moléculaires d’Orsay, CNRS, Université Paris-Saclay, Bât 520, Rue André Rivière, 91405 Orsay, France +3 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France +4 Physique des Interactions Ioniques et Moléculaires, CNRS, Aix Marseille Univ., 13397 Marseille, France +5 Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822-1897, USA +6 Institute of Astronomy, Department of Physics, National Tsing Hua University, Hsinchu, Taiwan +January 3, 2023 +ABSTRACT +Context. Cold cores are one of the first steps of star formation, characterized by densities of a few 104 to 105 cm−3, low temperatures +(15 K and below), and very low external UV radiation. In these dense environments, a rich chemistry takes place on the surfaces of +dust grains. Understanding the physico-chemical processes at play in these environments is essential to tracing the origin of molecules +that are predominantly formed via reactions on dust grain surfaces. +Aims. We observed the cold core LDN 429-C (hereafter L429-C) with the NOEMA interferometer and the IRAM 30m single dish +telescope in order to obtain the gas-phase abundances of key species, including CO and CH3OH. Comparing the data for methanol +to the methanol ice abundance previously observed with Spitzer allows us to put quantitative constraints on the efficiency of the +non-thermal desorption of this species. +Methods. With physical parameters determined from available Herschel data, we computed abundance maps of 11 detected molecules +with a non-local thermal equilibrium (LTE) radiative transfer model. These observations allowed us to probe the molecular abundances +as a function of density (ranging from a few 103 to a few 106 cm−3) and visual extinction (ranging from 7 to over 75), with the variation +in temperature being restrained between 12 and 18 K. We then compared the observed abundances to the predictions of the Nautilus +astrochemical model. +Results. We find that all molecules have lower abundances at high densities and visual extinctions with respect to lower density +regions, except for methanol, whose abundance remains around 4.5 × 10−10 with respect to H2. The CO abundance spreads over a +factor of 10 (from an abundance of 10−4 with respect to H2 at low density to 1.8 × 10−5 at high density) while the CS, SO, and +H2S abundances vary by several orders of magnitude. No conclusion can be drawn for CCS, HC3N, and CN because of the lack of +detections at low densities. Comparing these observations with a grid of chemical models based on the local physical conditions, we +were able to reproduce these observations, allowing only the parameter time to vary. Higher density regions require shorter times +than lower density regions. This result can provide insights on the timescale of the dynamical evolution of this region. The increase +in density up to a few 104 cm−3 may have taken approximately 105 yr, while the increase to 106 cm−3 occurs over a much shorter +time span (104 yr). Comparing the observed gas-phase abundance of methanol with previous measurements of the methanol ice, +we estimate a non-thermal desorption efficiency between 0.002% and 0.09%, increasing with density. The apparent increase in the +desorption efficiency cannot be reproduced by our model unless the yield of cosmic-ray sputtering is altered due to the ice composition +varying as a function of density. +Key words. Astrochemistry, ISM: abundances, ISM: clouds, ISM: Individual objects: LDN 429-C, ISM: molecules +1. Introduction +In the last 20 years, our understanding of the overall process +of star formation has improved substantially (Jørgensen et al. +2020). The interplay of turbulence, magnetic field, and gravity +in the interstellar medium (ISM) leads to the formation of cold +cores (nH2 > 104 cm−2, T ∼ 10 K), which are the sites of rich +chemical processes. The dust grains contained in these cores pro- +vide a catalytic surface where complex molecules (COMs, as de- +⋆ This work is based on observations carried out under project num- +ber 079-20 and ID 111-21 with the IRAM NOEMA Interferometer and +30m telescope. IRAM is supported by INSU/CNRS (France), MPG +(Germany) and IGN (Spain). +fined by Herbst & van Dishoeck 2009) are formed, the simplest +of which is CH3OH. The icy mantles (on top of these grains) are +mostly made of H2O, CO, and CO2, with CH3OH (Boogert et al. +2015, and references therein). Among these molecules, methanol +is a key species as it is predominantly formed on dust grain sur- +faces, but commonly observed in the gas phase in cold cores +(Dartois et al. 1999; Dartois 2005; Pontoppidan et al. 2004). On +the grains, methanol can be efficiently formed by the hydrogena- +tion of CO (itself formed in the gas-phase and adsorbed on the +grains). The reaction barrier for two key hydrogenation steps +(H + CO and H + H2CO) is significant for CH3OH formation +(Fuchs et al. 2009). The presence of methanol in the gas-phase +of cold cores, even at low abundance, is a clear indicator that +Article number, page 1 of 24 +arXiv:2301.01288v1 [astro-ph.GA] 3 Jan 2023 + +A&A proofs: manuscript no. 45157corr +non-thermal desorption mechanisms are efficient in terms of re- +leasing molecules from the surface of the grains in regions where +simple thermal desorption are not (Garrod et al. 2007; Ioppolo +et al. 2011). +In cold cores, where the temperature is typically ∼ 10 K +in the absence of any heating source, thermal desorption of +methanol is not possible and different desorption mechanisms +must be considered. These mechanisms can include chemical +desorption (Dulieu et al. 2013; Minissale et al. 2016; Wake- +lam et al. 2017), UV-induced photodesorption and photolysis +(Öberg et al. 2007; Bertin et al. 2016; Cruz-Diaz et al. 2016), +and grain sputtering induced by cosmic-ray (CR) impacts (Dar- +tois et al. 2019, 2020). These mechanisms may be partly de- +structive, thereby releasing intact methanol along with frag- +ments of methanol, which may themselves participate in sub- +sequent chemical processes to reform methanol. The efficiency +of non-thermal desorption is not well known, but is regularly +investigated in laboratory experiments. Using an astrochemical +model, Wakelam et al. (2021) showed that the intrinsic efficiency +of these mechanisms depends on the local physical conditions: +moving inward into a cold core (with an increasing density, in- +creasing visual extinction, and decreasing temperature), the ini- +tially efficient photo-desorption will first be replaced by chem- +ical desorption, before cosmic-ray induced sputtering becomes +the majority process at the highest densities. With the increased +sensitivity of the new generation of telescopes and instruments, it +is now possible to efficiently detect methanol ice from the ground +(Chu et al. 2020; Goto et al. 2021) and from space (Dartois et al. +1999; Pontoppidan et al. 2003, 2004; Boogert et al. 2015; Shi- +monishi et al. 2016) using IR absorption spectroscopy along the +lines of sight toward background sources. Comparing these ice +observations to gas-phase methanol observations helps to assess +the efficiency of non-thermal desorption of this molecule. +In this article, we present observations of a number of +molecules in the cold core L429-C obtained with the IRAM +30m single dish telescope and the NOEMA interferometer. This +source is one of the few cores where CH3OH has been detected +in the solid phase and it is an obvious benchmark for gas-grain +models. We focus in particular on gas-phase methanol in order +to compare its abundance with ice abundances of methanol ob- +tained with Spitzer by Boogert et al. (2011) in the same region, +as well as to constrain the efficiency of non-thermal desorption +of this molecule from the grains. +The paper is organized as follows. Current knowledge of the +source properties and a description of our observations are given +in Sections 2 and 3. A description of the observed integrated in- +tensity maps and a kinematic analysis of the observations is pre- +sented in Section 4. From these observations, we compute abun- +dance maps for all detected molecules in Section 5 and compare +these results with the predictions of the Nautilus astrochemical +model in Section 6. Our conclusions are summarized in the final +section. +2. Observed source: LDN 429-C +LDN 429-C (hereafter L429-C) is a cold (T < 18 K) and dense +(column density NH ∼ 1022 cm−2) core (Stutz et al. 2009) lo- +cated in the Aquila Rift (∼ 200 pc away), with a visual extinc- +tion larger than 35 mag at the center (Crapsi et al. 2005; Caselli +et al. 2008). This core is characterized by high degrees of CO de- +pletion (15 to 20 times with respect to the canonical abundance +at the center, Bacmann et al. 2002; Caselli et al. 2008) and of +deuteration (Bacmann & Faure 2016; Crapsi et al. 2005; Caselli +et al. 2008). The isotopic fractionation of nitrogen presents a de- +pletion of 15N, as in other prestellar cores (Redaelli et al. 2018). +HCO, H2CO, and CH3OH have been detected by Bacmann et al. +(2002, 2003); Bacmann & Faure (2016) toward this source using +the IRAM 30m telescope, while CH3O (Bacmann & Faure 2016) +and O2 (Wirström et al. 2016) were searched for unsuccessfully. +The source has been proposed to be on the verge of collapsing +by Stutz et al. (2009), based on 70 µm observations with Spitzer. +Based on observed molecular line profiles, both Lee et al. (2004) +and Crapsi et al. (2005) classified this source under a possible +"infall" category, although the authors also discussed the pos- +sibility that the line asymmetry could be due to other types of +motion. +Herschel observations of this cold core are available from the +Herschel database1. We used the temperature and optical depth +maps at 353 GHz(τ353) derived by Sadavoy et al. (2018) from +SPIRE 250, 350 and 500 µm and PACS 160 µm, with a resolu- +tion of 36′′. Those authors fitted spectral energy distributions to +these maps in order to obtain temperature and τ maps on more +than 50 globules. To obtain the temperature maps, they averaged +the dust temperature along the line of sight. The dust tempera- +ture, within an extended map around L429-C, varies between 10 +and 18 K and the optical depth from 0.0001 to 0.001. We derived +a column density of H2 from the τ map following the method +described in Appendix A. The H2 column density map is shown +in Fig. 1 (left panel) together with the published Herschel dust +temperature. +Lastly, L429-C is one of the first cold cores where the signa- +ture of CH3OH ice was unambiguously detected with Spitzer. +In a survey of 16 isolated dense cores chosen from the c2d +legacy targets (Evans et al. 2009), Boogert et al. (2011) ob- +served a sample of 32 background stars in the 1-25 µm wave- +length range to determine the solid-phase molecular composi- +tion of dense cores. They identified and used four background +stars in the L429-C region. The authors were able to measure the +H2O, CO2, and CH3OH ice column densities, as well as a de- +tection attributed to NH+ +4. They found a H2O ice column density +up to 3.93 × 1018 cm−2 in the cloud with abundances of 43.12%, +6.13-9.08%, and 6.34-11.58% for CO2, NH+ +4, and CH3OH re- +spectively, with respect to water. Recently, two more methanol +ice detections in other cold cores have been reported using the +NASA Infrared Telescope Facility (IRTF). Chu et al. (2020) and +Goto et al. (2021) found CH3OH ice abundances relative to water +of around 14.2% and 10.6% towards L694 and L1544, respec- +tively. As L429-C is one of the few cores to have multiple clear +CH3OH detections in the solid phase, it is an obvious benchmark +for gas-grain modeling. +3. Observations +The NOEMA observations were conducted during summer 2020 +using the mosaic mode, with additional IRAM 30m short spac- +ing observations being made in winter 2020. The mosaic phase +center is R.A. =18h17m08s.00, DEC. =-8o14’00′′ (J2000). The +size of the mosaic is 300′′ ×300′′ with a synthesized beam of 7′′. +The short spacing maps are 360′′ × 360′′, slightly larger that the +NOEMA mosaic, and have a beam of ∼ 25′′. Velocity channels +were 0.2 km.s−1 and each cube contained 152 of them. The rms +sensitivity was, on average, between 0.10 and 0.20 K, depending +on the molecule at 7′′. +We observed three frequency bands with IRAM 30m: 94.5 - +102.2 GHz, 109.8 - 117.8 GHz, and 168 - 169.8 GHz. These se- +tups were made to focus on specific molecules such as methanol, +1 http://archives.esac.esa.int/hsa/whsa/ +Article number, page 2 of 24 + +Taillard et al.: Non-thermal desorption of methanol +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +12.75 +14.50 +16.25 +16.25 +1 +2 +3 +4 +5 +6 +7 +log(NH2 cm +2) +1e22 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +12.75 +14.50 +16.25 +16.25 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +log(nH2 cm +3) +Fig. 1: Physical parameter maps of L429-C. Left: H2 column density (NH2 in cm−2) in L429-C computed from the Herschel optical +depth map. The position of maximum continuum is represented by the dark blue cross. The other crosses represent different positions +where the spectra were studied (see Fig 2). Right: Computed molecular hydrogen density (nH2 in cm−3) map. On both maps, the +white contours are the dust temperatures. +CO isotopologues (12CO, 13CO, C17O, and C18O), and H2S. The +full list of molecules and detected lines is given in Table 1. +The data reduction was performed using the Gildas package +CLASS2. The line identification was done with the help of the +Cologne Database for Molecular Spectroscopy3 (CDMS, Müller +et al. 2001) and the Jet Propulsion Laboratory catalog4 (JPL, +Pickett et al. 1998), coupled with CLASS. The NOEMA data +reduction was performed using the Gildas package CLIC, the +standard pipeline provided by IRAM, to create the UV tables. +We then used MAPPING to produce the data cubes, using a nat- +ural weighting for the synthesized beam. Residuals of the clean- +ing residuals were systematically checked. The rms sensitivity +on these maps was on average between 0.15 and 0.25 K. +The NOEMA data cube does not show any signal at any fre- +quency. This indicates that there is a spatial filtering with no +molecular emission smaller than approximately 30′′. Merging +the two sets thus only ended up adding noise to the maps. We +decided to use the single dish observations only and all molecu- +lar emission maps have resolutions between 25′′ and 28′′. +4. Observational results +4.1. Molecular transitions and integrated intensity maps +Among the targeted molecules, we detected 19 lines (listed in +Table 1), corresponding to 11 different molecules, with a peak +intensity greater than three times the local rms . The integrated +intensity maps have been constructed from the data cubes with +the Spectral Cube python package (Ginsburg et al. 2019). In +Fig. B.1, we present the integrated intensity maps of all detected +molecules; when several lines were observed, we only show the +most intense ones. The molecules (and transitions) in the follow- +ing list were targeted but not detected: OCS (9-8), HNCO (4,0- +4,4), and c-C3H2 (4,3-1,0). These non-detections are discussed +in section E in the appendix. +2 http://www.iram.fr/IRAMFR/GILDAS +3 https://cdms.astro.uni-koeln.de/classic/ +4 https://spec.jpl.nasa.gov +Concerning the detected lines, most of the emissions are ex- +tended and not centered on the maximum continuum position: +C18O, SO, CS, and CH3OH exhibit "heart-shaped" emission, +namely, it is similar to the H2 column density (see Fig. 1). Their +integrated intensities peak in the upper-right part of the heart, +just below the dust maximum position (dark-blue cross). The +other molecules (CCS, HC3N, H2S, and CN) show a weak and +more localized emission close to the maximum dust emission. +The maps can be found in Appendix B. +4.2. Kinematic analysis +The line profiles of all detected molecules present a complex ve- +locity structure that varies with their spatial location. As an ex- +ample, in Fig. 2, we show the spectra of C18O, CS, and CH3OH +at four positions indicated by colored crosses in Fig. 1. Velocity +channels maps are shown in Appendix C. The C18O molecule +presents three velocity components, each of which can be fit- +ted with a Gaussian. The first component is between 5.95 and +6.25 km.s−1 (component 1), the second is between 6.25 and 7 +km.s−1 (component 2), while the higher velocity component is +between 7 and 7.6 km.s−1 (component 3, see Fig C.1). Compo- +nent 1 is the broadest feature with lowest peak intensity. Com- +ponent 2 corresponds to the vLS R of the cloud at 6.7 km.s−1 (see +also Spezzano et al. 2020). To better visualize the spatial distri- +bution of the emission, we integrated the signal over the three +ranges of velocities. In Fig. 3, we show the resulting contours +(together with the first-moment map, corresponding to the ve- +locity distribution of C18O). The lower (blue) and higher (red) +velocity components are only located in the left and right parts +of the map, respectively, while the central velocity component +is widely spread across the entire map and even superimposes +upon the red and blue emissions. +The other molecules also have multi-velocity components but +with less clear signatures. In particular, SO does not exhibit com- +ponent 1 but does have components 2 and 3 (with the strongest +emission coming from component 2, see also Fig C.2); CS has +a weak intensity component, possibly interpreted as component +Article number, page 3 of 24 + +A&A proofs: manuscript no. 45157corr +Table 1: List of detected lines and associated spectroscopic information. +Molecule +Frequency (MHz) +Transition +Eup (K) +gup +Aij (s−1) +CCS +93870.1 +(6-7) +19.9 +17 +3.8 × 10−5 +CH3OH +96739.3 +(2-1) +12.5 +5 +2.55 × 10−6 +CH3OH +96741.3 +(2,0)-(1,0) +7 +5 +3.40 × 10−6 +CH3OH +96744.5 +(2,0)-(1,0) +20.1 +5 +3.40 × 10−6 +CS +97980.9 +(2-1) +7.1 +5 +1.67 × 10−5 +SO +99299.8 +(1-0) +9.2 +7 +1.12 × 10−5 +HC3N +100076.5 +(11-10) +28.81 +69 +0.77 × 10−4 +C18O +109782.1 +(1-0) +5.27 +3 +6.26 × 10−8 +13CO +110201.3 +(1-0) +5.29 +3 +6.29 × 10−8 +C17O +112358.7 +(1-0) +5.39 +3 +6.69 × 10−8 +CN +113144.1 +(1.0,0.5,0.5)-(0.0,0.5,0.5) +5.43 +2 +0.10 × 10−4 +CN +113170.1 +(1.0,0.5,1.5)-(0.0,0.5,0.5) +5.43 +4 +0.51 × 10−5 +CN +113488.1 +(1.0,1.5,1.5)-(0.0,0.5,0.5) +5.43 +4 +0.67 × 10−5 +CN +113490.9 +(1.0,1.5,1.5)-(0.0,0.5,0.5) +5.44 +6 +0.11 × 10−4 +CN +113499.6 +(1.0,1.5,0.5)-(0.0,0.5,0.5) +5.44 +2 +0.10 × 10−4 +CN +113508.8 +(1.0,1.5,1.5)-(0.0,0.5,0.5) +5.44 +4 +0.51 × 10−5 +CN +113520.4 +(1.0,1.5,0.5)-(0.0,0.5,1.5) +5.44 +2 +0.13 × 10−5 +12CO +115271.2 +(1-0) +5.53 +3 +7.20 × 10−8 +H2S +168762.7 +(1,1,0)-(1,0,1) +27.9 +9 +2.65 × 10−5 +The most relevant non-detections are discussed in section E of the Appendix. +1, between 5.2 and 6.1 km.s−1. As for SO, component 2 is the +most intense (see also Fig C.3). Most of the methanol emission +can be fitted with one single component (component 2), but in +the right part of the map, a weak component 3 can be found (see +also Fig. C.4). The other molecules are weak and only present +emission at the velocity of component 2. This velocity structure +observed on the line profiles at large spatial scale is very likely +what Lee et al. (2004) and Crapsi et al. (2005) attributed to in- +fall, as they had only the spectra on the dust peak position for +Lee et al. and a very small map around it for Crapsi et al.. +We investigated the possibility that L429-C could be the re- +sult of a H i cloud - cloud collision. Looking strictly from a dy- +namical point of view, as discussed previously, we have multiple +velocity components in our spectra, possibly showing conver- +gent flows rather than turbulence. In the cloud-cloud collision +scenario, we would have two components, one moving towards +(blue) and one moving away (red) from us, converging to the +dust peak position and resulting in the formation of a denser re- +gion. In Bonne et al. (2020), the authors studied the formation +of the Musca filament and its origin from a cloud-cloud region +collision: a H i cloud colliding with a denser region. In the multi- +ple arguments in support of this hypothesis, they showed that the +result from such a collision would be a dense filament with the +apparition of CO from the resulting shocks. The CO emission +would therefore be blue-shifted in comparison to the HI emis- +sion. A key result would be that the matter would slide to the +core by the action of a curved magnetic field, thus becoming ac- +celerated. This would be manifested in the PV-diagram (shown +in Arzoumanian et al. (2018)) of CO, for example, and show a +"V-shaped" velocity diagram. Such a curved magnetic field is +observable with the Planck telescope: they observed magnetic +field lines arriving perpendicular to the filament and becoming +curved by going through the filament. +In the case of L429-C, from Planck Collaboration et al. +(2020a,b) released data, we can observe magnetic field lines ori- +ented perpendicularly to the cloud along the top-right axis. By +looking at the general direction of the field, we cannot say if the +lines are bent by the core’s presence. The Planck resolution at +353 GHz (Aghanim et al. 2020) is too large compared to the size +of cloud to observe any bending of these lines. Thus, we are not +able to confirm a similar effect as that observed in Musca. We +do observe CO in our cloud, with a very dense profile and an +asymmetry in our velocity profile. However, we do not observe +the "V-shaped" signature found by Arzoumanian et al. (2018) +in our PV diagram (see Fig. D.1) – rather, we simply have two +components converging toward the core position. Nor do we see +any temperature rise that could result from shocks. We conclude +that higher resolution magnetic field data would be needed to +conclude anything other than that the region is going through +dynamical behavior that more so appears to resemble conver- +gent flows. We cannot make other assumptions on a possible H i +cloud-cloud collision at this time. +5. Observed molecular abundances +5.1. Method +For all detected molecules, when the collisional rate coefficients +were available in the LAMDA database5 (Schöier et al. 2005), +we estimated the observed column density with an inversion +procedure and the RADEX radiative transfer code (van der Tak +et al. 2007). We first computed a theoretical grid of line inte- +grated intensities, using external constraints on the temperature, +line width, and H2 density. Then, comparing this grid of theoreti- +cal values with our observed ones through a χ2 minimization, we +constrained the molecular column densities at each pixel. We as- +sumed a gas temperature equal to that determined from Herschel +observations (see Fig 1). The resolution of the Herschel obser- +vations (36” resolution, from Sadavoy et al. 2018) is similar to +our 30m data (23 to 26.5”) The line widths were taken from each +spectrum from a Gaussian fitting (see Section 5.1.1), while the +5 https://home.strw.leidenuniv.nl/~moldata/ +Article number, page 4 of 24 + +Taillard et al.: Non-thermal desorption of methanol +5 +6 +7 +8 +9 +VLSR [km.s +1] +0 +1 +2 +3 +4 +Tmb [K] +CH3OH +CS +C18O +5 +6 +7 +8 +9 +VLSR [km.s +1] +0 +1 +2 +3 +4 +Tmb [K] +CH3OH +CS +C18O +5 +6 +7 +8 +9 +VLSR [km.s +1] +0 +1 +2 +3 +4 +Tmb [K] +CH3OH +CS +C18O +5 +6 +7 +8 +9 +VLSR [km.s +1] +0 +1 +2 +3 +4 +Tmb [K] +CH3OH +CS +C18O +Fig. 2: Spectra of CS (red line), CH3OH 96.741 GHz (blue line) and C18O (yellow line) for the four positions indicated as crosses +in Fig. 1. The colored crosses are shown in the upper left corner of each panel. +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +6.4 +6.6 +6.8 +7.0 +7.2 +7.4 +7.6 +7.8 +8.0 +Intensity-weighted velocity (km.s +1) +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +1 +2 +3 +4 +5 +6 +7 +log(NH2 cm +2) +1e22 +Fig. 3: Velocity distribution in the cloud. C18O integrated intensity contours over plotted on the NH2 Herschel map (gray color). +Contours are shown for the three velocity components detailed in the text (left). Cyan: 5.95 - 6.25 km.s−1 (with intensity levels as +follows: dotted is 0.4 K, dashed is 0.5 K and solid line is 1 K), pink: 6.25 - 7 km.s−1 (with dotted as 1 K, dashed is 1.5 K and solid +line is 2.4 K), red: 7 - 7.6 km.s−1 (with dotted as 1 K, dashed as 1.5 K and solid line as 2 K). Velocity (first-moment) map (right). +The dark blue cross shows the position of the continuum peak. +H2 density was derived from the Herschel data with a method +described below (Section 5.1.2). +5.1.1. Determining the line width and integrated intensities at +each pixel +The non-LTE RADEX code requires the line width at each spec- +trum in each pixel. The full width at half maximum (FWHM) +of the lines (dv) was obtained using the ROHSA method from +Marchal et al. (2019). The authors developed a Gaussian decom- +position algorithm using a nonlinear least-squares criterion to +perform a regression analysis. For each pixel, we computed the +Gaussian fits of the extracted spectra and computed the width +and the integral under the curve of this fit. We also compared the +difference between the intensity map obtained by ROHSA and +ours. It showed little variation (from 0.1 to 0.3 km.s−1 width) +between both models and so, we assumed that the dv value ob- +tained from the Gaussian fit is good enough to be used in our +Article number, page 5 of 24 + +A&A proofs: manuscript no. 45157corr +program. For the case of a molecule with multiple components, +we fit ROHSA with two Gaussian identifications and made sure +to sum the two widths for our final dv. It is also feasible as all +of our molecules present optically thin emission. It was more ac- +curate to use ROHSA’s new computed intensity as it gives us a +better signal-to-noise level and also takes into account the possi- +bility that some of the baselines are not completely centered at 0. +In the case of two velocity components, the integrated intensities +obtained with the two fitted Gaussian are also summed. +5.1.2. Determining the H2 density at each pixel +The number of detected lines per species was not enough to +determine the local volume density at each pixel from a radia- +tive transfer analysis. We therefore used the method described in +Bron et al. (2018), which estimates the H2 volume density from +the H2 column density (obtained from Herschel and described +in Appendix A). This method is particularly well adapted for +simple sources such as ours as it makes the assumption that the +medium is isotropic and that the density is smoothly increasing +from outer to inner regions of the cloud. It also assumes that +there is no preferential direction for the spatial density. It then +estimates the typical length scale l of the cloud, knowing the col- +umn density NH2. Finally, nH2 is given simply by dividing NH2 by +l. The values obtained for nH2 ranges from 3 × 103 to 106 cm−3 +and the obtained density map is shown in Fig. 1. +5.1.3. χ2 comparison with RADEX +We then used the non-LTE radiative transfer code RADEX +(van der Tak et al. 2007), with the LAMDA database to obtain +a grid of integrated intensities for one or multiple transitions per +molecule (for the latter, a grid was computed for each line). Col- +lision rates used are: SO from Lique et al. (2005), CS from Lique +et al. (2006), CN from Lique et al. (2010), CO from Yang et al. +(2010), H2S from Dubernet et al. (2009), CH3OH from Rabli & +Flower (2010), and HC3N from Faure et al. (2016). +We first made low-resolution grids using RADEX to estimate +the value ranges of the unknown parameters; the grids were wide +at first then narrowed by iteration. This process refines the grids +to avoid saturation on the extrema and smoothed the maps (for +example, starting with values for the column densities between +1011 and 1018 cm−2 before refining to values between 1012 and +1016 cm−2). The theoretical integrated intensity grid was com- +puted for H2 density values between 103 to 107 cm−3 (30 val- +ues in logarithmic space), temperatures between 11 and 18.5 K +(30 values in linear space), 5 values of dv (in linear space), be- +tween the minimum and maximum value, of each Gaussian file +of the molecules obtained by ROHSA. For the molecular column +density, we ran multiple tests to calibrate it for each molecule, +with a logarithm space of 60 values and the lowest input being +1012 cm−2 and the maximum 1018 cm−2. Once we ran the tests, +we adjusted the values to be the closest to the minimum and +maximum values obtained on the column densities maps (see +next steps). The final grid contains 270,000 values. +To circumvent the degeneracies between the RADEX input +parameters, we chose to fix most of the values (Tkin, nH2, dv) +to those we determined independently for each pixel from the +methods described in previous sections. This is done by interpo- +lating linearly on the intensity grid using the independently de- +rived parameters. The interpolated theoretical integrated intensi- +ties are then compared to the observed ones, through χ2 mini- +mization, to determine the best molecular column density. +Lastly, we created abundance maps by dividing the molecu- +lar column densities by the H2 column densities for each pixel. +These maps of the abundances with respect to H2 are shown +in Figs. 4 and 5. For CO, we computed the main isotopologue +abundance from the C18O abundance multiplied by 557 (Wilson +1999). We also computed it from the C17O abundance multiplied +by 2005 (Lodders 2003) and obtained the same abundance of the +main isotopologue. The maps of CCS and HC3N were computed +at LTE because no collisional coefficients are available. Similar +to CN, the lines are only detected in a small portion of the map. +For the non-detected molecules, we computed upper limits on +their column densities (given in Appendix E). +Finally, the optical depth for each detected molecule (that +has an entry in the LAMDA collisional database) was computed. +To do so, we used the four positions shown in Fig 1 and for +each associated pixel, we collected the kinetic temperature, the +H2 density, the line width, and the column density. We used +RADEX to compute τ for each position and each molecule. For +all molecules, we found τ values inferior to 1, indicating that the +emission averaged within the beam is optically thin. +5.2. Results +5.2.1. Abundance maps +The position of the dust peak is characterized by a decrease in the +abundance of most of the observed molecules (see Figs 4 and 5). +We obtained a CO abundance (with respect to H2) up to 10−4 in +the outer parts of the maps, whereas it is around 1.7×10−5 at the +dust peak. We computed a depletion factor of f = f(Xcan/X12CO), +with Xcan = 8.5 x 10−5 being the "canonical" abundance of 12CO +determined by (Frerking et al. 1982), from the 12CO/H2 abun- +dance map. At the position of the dust peak, we find a CO abun- +dance of 1.7 × 10−5, which gives us f +12CO = 4.91, showing an +underestimation of the abundance in comparison to the canoni- +cal one. This is almost three times smaller than Bacmann et al. +(2002), where the authors found a depletion factor of 15.5 in +L429-C. This discrepancy can be explained by a difference in +the adopted temperature used to determine the CO column den- +sity and in the adopted H2 column density. By using a lower +temperature for ∼ 7 K, the authors found f = 5 (and for ∼ 11 K, +f = 3), which is closer to our value. They also used a higher H2 +column density of 1.4 × 1023 cm−2 (we used NH2 = 7.2 x 1022 +cm−2), which produces a lower 12CO abundance compared to us. +We note that CS seems to be depleted over the entire "heart- +shaped" density structure. Its highest abundance is in the top of +the map (with a maximum value of 1.1 × 10−8), while the abun- +dance at the dust peak is 1.5 × 10−10, that is, almost 75 times +lower; SO presents a similar behavior, although its maximum +abundance is in the left part of the map, with a difference of 7.5 +between the maximum (2.5×10−9) and the dust peak (3.3×10−10) +abundances. The H2S intensity is weak so the values derived +here have to be considered with less robustness than for the other +species. The maximum abundance is obtained on the border of +the map, around 3.5 × 10−9, while it is 4.9 × 10−11 at the contin- +uum peak position. +Compared to the other species, the CH3OH abundance is +more homogeneous across the cloud. Its maximum abundance +is in the left part of the map (although not on the border of the +map) and is around 8.5 × 10−10, while its abundance on the dust +peak is about two times lower, being around 4.2×10−10. The CN +map was computed using several lines detected only in a small +area of the region. This results in a source-focused area with a +visible depletion on the continuum peak. The maximum abun- +Article number, page 6 of 24 + +Taillard et al.: Non-thermal desorption of methanol +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +CO / H2 Abundance map +5.0 +4.8 +4.6 +4.4 +4.2 +4.0 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +CH3OH / H2 Abundance map +10.0 +9.8 +9.6 +9.4 +9.2 +9.0 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +C18O / H2 Abundance map +7.5 +7.4 +7.3 +7.2 +7.1 +7.0 +6.9 +6.8 +6.7 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +H2S / H2 Abundance map +10.5 +10.0 +9.5 +9.0 +8.5 +8.0 +7.5 +7.0 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +CS / H2 Abundance map +10.5 +10.0 +9.5 +9.0 +8.5 +8.0 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +SO / H2 Abundance map +9.8 +9.6 +9.4 +9.2 +9.0 +8.8 +Fig. 4: On a logarithmic scale, observed gas-phase abundance maps with respect to H2. The green crosses on the methanol map +correspond to the positions of the methanol ice observations reported by Boogert et al. (2011). The dark blue cross is the continuum +peak. We note that the white cross on H2S / H2 represents a gap in the data. +dance is found to be just below the dust peak at 1.3 × 10−11 and +its minimum at 2.9 × 10−12 on the dust peak. CCS presents three +peaks, with its maximum in the left part of the map, where the +peak abundance is 6.1 × 10−11. The abundance on the dust peak +is around 4.5 × 10−11. In the wider region probed here, CCS is +constant and there is no evidence for depletion. HC3N shows lit- +tle to no variation in its abundance. The maximum abundance +(4.4 × 10−12) is found close to the dust peak position (with an +abundance of 2.3 × 10−12). +Article number, page 7 of 24 + +A&A proofs: manuscript no. 45157corr +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +CCS / H2 Abundance map +11.2 +11.0 +10.8 +10.6 +10.4 +10.2 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +HC3N / H2 Abundance map +12.50 +12.25 +12.00 +11.75 +11.50 +11.25 +11.00 +10.75 +10.50 +18h17m15s +10s +05s +00s +-8°12' +13' +14' +15' +16' +17' +Right Ascension +Declination +CN / H2 Abundance map +11.6 +11.5 +11.4 +11.3 +11.2 +11.1 +11.0 +10.9 +Fig. 5: On a logarithmic scale, observed gas-phase abundance +maps with respect to H2 for CCS, HC3N, and CN. The dark blue +cross is the continuum peak. +5.2.2. Abundances as a function of the physical parameters +In Figs. 6 and 7, we show the abundances (with respect to H2) +of each molecule for all pixels as a function of the three physi- +cal parameters (temperature, density, and visual extinction). The +CO, CS, SO, and H2S abundances decrease with density, as well +as with temperature and visual extinction, as these three param- +eters are linked. The lower molecular abundances of CO, CS, +SO, and H2S seen on the continuum peak position are indeed +explained by a higher density there. The slope of the decrease is +the strongest for H2S, which decreases by three orders of mag- +nitude when the density goes from 3 × 104 to 106 cm−3. Then, +CO has the smallest variation among these molecules with a less +than one order of magnitude decrease. The methanol abundance +is nearly flat. For the CCS, CN, and HC3N molecules, the abun- +dances also seem flat, but they are only observed at high density, +so they are not fully comparable with the other molecules. They +also present a larger spread of values at a specific density (or vi- +sual extinction), especially for lower densities (or visual extinc- +tions) and probably reflecting a larger uncertainty in their com- +puted abundances (due to lower line intensities). We note that +the CN abundance is varying over such a small range of values +that the distribution of the computed abundances (forming three +groups) reflects the sampling of the RADEX theoretical grid. +6. Chemical modeling of the region +To understand the trends in molecular abundances observed in +L429-C, we ran a suite of chemical models accounting for the +cloud’s physical conditions. +6.1. Model description +We used the chemical model Nautilus developed by Ruaud et al. +(2016). Nautilus is a three-phase gas-grain model that computes +the gas and ice abundances of molecules under ISM conditions. +All gas-phase chemical reactions are considered, based on up- +dates of the kida.uva.2014 chemical network (Wakelam et al. +2015), as listed in Wakelam et al. (2019). The gas and grain net- +work used for these simulations contain more than 14000 chem- +ical reactions (in the gas-phase, at the surface of the grains, in +the bulk, and at the interface between gas and grains, and sur- +face and bulk). Chemistry of the following elements is consid- +ered: H, He, C, N, O, Si, S, Fe, Na, Mg, Cl, P, and F. In the +model, species from the gas-phase can stick to interstellar grains +upon collision, through physisorption processes. They can then +diffuse and react. The thermal evaporation of adsorbed species +as well as a number of non-thermal desorption mechanisms are +included. Under the shielded and cold conditions of L429-C, +two non-thermal desorption processes are particularly impor- +tant: 1) chemical desorption, for which we adopted the formal- +ism of Minissale et al. (2016) for water ices, and 2) sputter- +ing by cosmic-rays (Dartois et al. 2018). As shown in Wakelam +et al. (2021), this latter process is the most efficient for releasing +icy molecules, in particular methanol, into the gas-phase under +dense conditions. Dartois et al. (2021) presented two yields for +this process, depending on the nature of the main constituent +of the ices: either water or CO2, with the latter being more ef- +ficient than the former. In this work, we tested both yields: a +"low sputtering yield" derived from data on pure H2O pure ices +and a "high sputtering yield" derived from data on pure CO2 +pure ices. In each case, we apply one yield to all species in the +model, which means that all species (both on the surface and in +the bulk) desorb with the same yield. We note that the fraction +of CO2 to H2O ice observed in L429-C by Boogert et al. (2011) +was as high as 43%, although this was only for one line of sight. +Our model also includes the non-thermal desorption of surface +species due to the heating of the entire grain by cosmic-rays (as +presented in Wakelam et al. 2021). This process was shown to be +important for some gas-phase species (such as CS, HC3N, and +Article number, page 8 of 24 + +Taillard et al.: Non-thermal desorption of methanol +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +4.75 +4.50 +4.25 +4.00 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +4.75 +4.50 +4.25 +4.00 +12 +14 +16 +18 +Temperature (K) +CO +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +9.6 +9.4 +9.2 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +9.6 +9.4 +9.2 +12 +14 +16 +Temperature (K) +CH3OH +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +10 +9 +8 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +10 +9 +8 +12 +14 +16 +18 +Temperature (K) +CS +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +9.5 +9.0 +8.5 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +9.5 +9.0 +8.5 +12 +14 +16 +Temperature (K) +SO +Fig. 6: Abundance as a function of of the logarithm of the volume density (right) and of the visual extinction, Av, (left). +HCO+) because of the efficient desorption of CH4 at high den- +sity > 2 × 104 cm−3). A more detailed description of the model +is given in Wakelam et al. (2021). +6.2. Model parameters +To compare our model results with our observed abundances, +we ran a grid of chemical models covering the observed phys- +ical conditions (temperature, density, and visual extinction). In +Appendix F, we show the observed relationship between the H2 +density, the visual extinction, and the temperature determined +from Herschel data. We note that the range of physical condi- +tions in that case is larger than the one probed by the methanol +lines because methanol was not detected throughout. We first +created a grid of eight H2 densities from 3×104 to 106 cm−3 in a +logarithm space. For each of the eight H2 densities (with associ- +ated derived dust and gas temperatures, and visual extinctions), +we considered two different CR sputtering yields and two dif- +Article number, page 9 of 24 + +A&A proofs: manuscript no. 45157corr +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +11.0 +10.5 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +11.0 +10.5 +12 +13 +14 +15 +Temperature (K) +CCS +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +12 +11 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +12 +11 +12.0 +12.5 +13.0 +13.5 +Temperature (K) +HC3N +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +10 +9 +8 +7 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +10 +9 +8 +7 +12 +14 +16 +Temperature (K) +H2S +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +log(nH2 cm +3) +11.6 +11.4 +11.2 +11.0 +log(Xobs) +0 +15 +30 +45 +60 +75 +Av +11.6 +11.4 +11.2 +11.0 +12.0 +12.5 +13.0 +Temperature (K) +CN +Fig. 7: Abundance as a function of of the logarithm of the volume density (right) and of the visual extinction, Av, (left) for CCS, +HC3N, H2S, and CN. +ferent values of the ionization rate ζ. Using the grid defined by +the methanol detection, associated visual extinctions vary from +15.2 to 77.7 while the temperatures vary from 11.8 to 15.7 K. We +note that we cover the values of densities where methanol was +detected and not the full range of some of the other molecules +such as CO. Since we were unable to determine the gas tem- +perature from our line analysis, for the model, we assume the +gas temperature as the measurement determined from Herschel +data. While an uncertainty of a few Kelvin in the gas tempera- +ture has little impact on the chemical modeling results, the ice +abundances can be strongly influenced by such a difference in +the dust temperature. The dust temperatures retrieved from Her- +schel observations are all above 11 K – even for the highest Av. +Dust temperatures derived for FIR emission tend to be overes- +timates of the true large grain temperature inside the cores be- +cause emission from warmer dust of the diffuse envelope can be +mixed in the observing beam (Marsh et al. 2015). For the dust +temperature, we therefore used the parametric expression for the +Article number, page 10 of 24 + +Taillard et al.: Non-thermal desorption of methanol +Table 2: Sets of chemical models. +Name +Sputtering yield +ζ (s−1) +Low yield and CR +H2O-rich ices +10−17 +Low yield and high CR +H2O-rich ices +3 × 10−17 +High yield and low CR +CO2-rich ices +10−17 +High yield and CR +CO2-rich ices +3 × 10−17 +4.6 +4.8 +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +log(nH2 cm +3) +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +log(Time yr) +Low sputtering yield - Low CR +Low sputtering yield - High CR +High sputtering yield - Low CR +High sputtering yield - High CR +Fig. 8: Best time obtained from the comparison between mod- +eled and observed abundances of CO, CS, H2S, and CH3OH as +a function of density. +dust temperature as a function of visual extinction from Hocuk +et al. (2017). This easy-to-handle parametrization was obtained +by semi-analytically solving the dust thermal balance for differ- +ent types of grains and comparing to a collection of observational +measurements. +In addition to the density, the dust and gas temperatures, and +the visual extinction, we considered two different values of the +ionization rate ζ: 10−17 s−1 (low ionization) and 3 × 10−17 s−1 +(high ionization). These two values cover the observational +range of ζ at high visual extinction (Av, e.g. Padovani et al. 2022, +, Fig.C1). We note that extending the first order function to de- +scribe the ionization attenuation with Av that is valid for translu- +cent clouds to moderate Av – as used in Wakelam et al. (2021) – +to such high Av would predict too low (< 10−18 s−1) ionization +rates. +In our simulations, we started from atoms (with abundance +values similar to those of Table 1 in Ruaud et al. 2016), with +the exception of hydrogen, which is assumed to be in molecular +form. In total, we have four sets of eight models as a function +of time. For the first two sets, we used the yield of sputtering for +water-rich ices (low sputtering yield) with two values of ζ (10−17 +and 3 × 10−17 s−1), while for the other two sets we use the yield +for CO2-rich ices (high sputtering yield) and the same two values +of ζ (Table 2). +6.3. Comparison between modeled and observed gas-phase +abundances +In these simulations, the abundances are computed as a function +of time. To quantify the agreement between model and observa- +tions, we computed the distance of disagreement, d, as described +in Wakelam et al. (2006): +d(t) = 1 +Ni +� +i +| log(Xmod,i(t)) − log(Xobs,i)|, +(1) +with t as the time, Ni the number of molecular species (four +in our case: CO, CS, H2S, and CH3OH) used in the compari- +son, Xmod,i the modeled abundance of species i at time, t, and +Xobs,i as the mean observed abundance of species i. A value of 1 +for d means that the mean difference between modeled and ob- +served abundance is a factor of 10. The smallest d value repre- +sents the best agreement and, thus, the best time. Figure 8 shows +the obtained best time as a function of density for the four sets +of models. Density is a fixed value during the time evolution of +the model. In Fig. H.1, we show d(t) as a function of time for all +eight models in each of the four model sets. +The best time, namely, the integration time used in the model +that best reproduces the observations – is similar for all sets of +models and decreases with density (see Fig. 8). The fact that +some of the models show a best agreement for exactly the same +time is a result of the sampling of the modeling time chosen to +get the model output and the small sensitivity of the agreement +for each model. For the models in Set 1, for instance (see Ta- +ble 2), the best time is 1.9 × 105 yr at a density of 3.2 × 104 +cm−3 and down to 1.0 × 104 yr at a density of 1.0 × 106 cm−3. +In other words, at a higher density, the observed abundances can +be achieved for a shorter integration time. The main constraint +on the time is given by the observed CO abundance. According +to the model, CO has a "simple" abundance curve with respect +to time. The molecule is progressively formed in the gas-phase +through gas-phase reactions. Its abundance reaches a peak at a +time that depends on density before decreasing as it is depleted +onto the grains and transformed into methanol and other species +(see also the discussion in Section 3 in Wakelam et al. 2021). +In our observations, the CO gas-phase abundance varies by less +than a factor of 10, while the density varies over several orders of +magnitude. As a result, the observed abundance at high density +cannot be achieved on the same timescale as that at lower den- +sities. Through our chemical modeling, we are able to evaluate +the dynamical evolution of this region. +We previously indicated that the number of molecular +species considered in the determination of the best evolution- +ary time was four, namely CO, CS, H2S, and CH3OH. We did +not use CCS, HC3N, and CN) because they were detected only +on a small fraction of the map. The SO molecule was detected +everywhere but was not reproduced by the model at a sufficient +level. +6.4. Goodness of fit +In Fig. 9, we plot the ratio between the modeled abundances +(at the best time for each condition) and the observed abun- +dances for the species used to determine the best times (CO, +CS, CH3OH, and H2S) to quantify the robustness of our models. +Overall, the abundances of these molecules are well reproduced +(i.e., within a factor of 10). CH3OH is not as well reproduced at +high density if low sputtering yield is assumed and at low density +if a higher sputtering yield is assumed. The ratio for the other +species (SO, HC3N, CN, and CCS) is shown in the appendix +(Fig.G.1). Specifically, SO, HC3N, and CN are overestimated by +the model at all densities; CCS is underestimated by the model, +with an agreement in excess of a factor of 10 at high density. +We cross-checked the upper limits derived for OCS, HNCO, and +c-C3H2 with our best models (see Appendix E). Upper limits on +Article number, page 11 of 24 + +A&A proofs: manuscript no. 45157corr +1.0 +0.5 +0.0 +0.5 +1.0 +Low sputtering yield +CO +CS +H2S +CH3OH +Low CR +High CR +4.6 +4.8 +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +log(nH2 cm +3) +1.0 +0.5 +0.0 +0.5 +1.0 +log(Xmod/Xobs) +High sputtering yield +CO +CS +H2S +CH3OH +Low CR +High CR +Fig. 9: Ratio between the modeled (Xmod) and observed gas-phase abundances (Xobs) of CO, CS, CH3OH, and H2S as a function of +the density for the best times of the four sets of models. +OCS and c-C3H2 are in agreement with our predictions, while +HNCO is overproduced by the model by at least a factor of 10 +at all densities. We also compared our model predictions to the +non-detections of O2 (with an abundance < 2× 10−6) and CH3O +(< 4.8× 10−12) reported at the continuum position by Wirström +et al. (2016) and Bacmann & Faure (2016), respectively. These +upper limits are in agreement with our model results. +7. Constraining the non-thermal desorption of +methanol +Combining our gas-phase abundances of methanol with the ice +observations of Boogert et al. (2011), we can offer constraints on +the efficiency of non-thermal desorption of methanol. Figure 10 +shows the ratio between the observed gas and ice column densi- +ties of methanol (black dots on the lower panel). The four points +represent the four positions reported by Boogert et al. (2011, +namely, in Table 6 of their paper) and shown in green crosses +in Fig. 4. The observed gas-phase column densities are the ones +derived in our study. On the same figure, we show our model +results (methanol gas-to-ice abundance ratios) obtained at the +best times for each density. As expected, the main reservoir of +methanol (empirically and theoretically) is in the ices. The gas- +phase abundance is several orders of magnitude lower than the +solid abundance (Drozdovskaya et al. 2016). From the observa- +tions, we computed the efficiency of non-thermal desorption of +CH3OH ices as Ngas +Nice × 100 and we obtained a desorption effi- +ciency between 0.002% (at low density ∼ 2.4 x 104 cm−3) and +0.09% (at high density ∼ 2.2 × 105 cm−3). Although we have +only four points, the observations seems to indicate that the ef- +ficiency of non-thermal desorption increases with density. The +gas-phase abundance at these positions does not vary much (4.0 +to 6.4 × 10−10), but the ice abundances (as shown in the upper +panel of Fig. 10) decrease by more than a factor of 2 with den- +sity. So at high density, to maintain the same gas-phase abun- +dance, the desorption needs to be more efficient by a factor of +45. Our models reproduce the observed ice column density of +methanol (see top Fig.10 ) within less than a factor of 2 for sets +2 and 4 (ζ = 3 × 10−17 s−1) and within a factor of 3 for sets +Article number, page 12 of 24 + +Taillard et al.: Non-thermal desorption of methanol +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +CH3OHice abundance +1e +5 +Low sputtering yield - Low CR +Low sputtering yield - High CR +High sputtering yield - Low CR +High sputtering yield - High CR +Spitzer observations +4.4 +4.6 +4.8 +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +log(nH2 cm +3) +10 +5 +10 +4 +CH3OHgas / CH3OHice +Fig. 10: Comparison between modeled and observed methanol. +Upper panel: Abundance of methanol ices obtained by our chem- +ical model as a function of density (solid lines; see text and Ta- +ble 2 for details on the model sets). The black points (with un- +certainties) are the observed values at the four positions probed +by Boogert et al. (2011). Red and orange curves overlap as well +as the green and blue curves. Lower panel: Gas-to-ice abundance +ratios of methanol as a function of density obtained by the dif- +ferent sets of models for the best times (solid lines) and the four +observation points (black dots). +1 and 3 (ζ = 10−17 s−1) at low density. At a higher density, all +models are in agreement with the CH3OH ice abundance. Both +the observations and the models seems to indicate a lower ice +abundance with increasing density. +Our different sets of models produce a gas-to-ice ratio that +decreases with density (contrary to the observations) but give +different values depending on the model parameters. The higher +cosmic ray sputtering yield produces a large ratio as does the +higher cosmic-ray ionization rate. Sets 1 and 2 (for water ices) +give a ratio closer to the observations at low density, while sets +3 and 4 (CO2 ices) give a ratio closer to the observations at high +density. None of the models seem to reproduce the lower density +point better than a factor of 3.5. Overall, and considering all the +uncertainties in the observations (both the ices and the gas), in +the density determination, and in the chemical model, we find +this agreement satisfactory. However, if the observed increase +in desorption efficiency of methanol with density is true, then +this cannot be explained by our model unless we change the ice +composition. If the ice composition changes from a water domi- +nated ice to a mixture where non-thermal desorption (such as the +cosmic-ray sputtering) is more efficient, then we can obtain the +same trend as the observations. Such change in the ice compo- +sition could occur during the catastrophic CO freeze out in cold +cores (Qasim et al. 2018). In fact, in our observations, the CO +abundance in the gas-phase is nearly decreased by a factor of 10 +from low to high density. The gas-to-ice CH3OH ratio that we +observe in L429-C may be an indication of a change in the ice +composition, as suggested by Navarro-Almaida et al. (2020), for +H2S in cold cores; although we note that paper’s focus was the +chemical desorption. +In comparison, Perotti et al. (2020) studied a dense star- +forming region, in the Serpens SVS 4 cluster, using the SubMil- +limeter Array, Atacama Pathfinder EXperiment and Very Large +Telescope observations. They estimated a CH3OH column den- +sity of approximately 1014 cm−2 in the gas-phase and 0.8× 1018 +cm−2 in the solid phase. They thus obtained a gas-to-ice ratio +varying between 1.4 × 10−4 and 3.7 × 10−3, which is higher than +in our findings. However, they do not provide information on the +densities within the region. Their gas-to-ice CH3OH ratio does +not show any trend with H2 column density. In addition, they +estimated the column densities of methanol in the gas-phase at +LTE, with a mean temperature of 15 K and using a high en- +ergy transition of methanol. It is possible that they are in the +sub-thermal excitation regime and would thus overestimate the +column densities, meaning they would actually have a lower gas- +to-ice ratio. +Among the other species studied here, H2S molecule is an +interesting case to highlight, as it is generally assumed that it +must be formed on the grains since there is no efficient gas- +phase pathway (Vidal et al. 2017). Similar to the role oxygen +atoms play in the formation of water, models predict that atomic +sulfur from the gas sticks onto the grains at low temperature and +is easily hydrogenated to form H2S. As such, large amounts of +H2S ice are predicted by chemical models but the molecule has +never been found in interstellar ices (Smith 1991; Boogert et al. +2015). This molecule is the dominant S specie sink in cometary +ices (Calmonte et al. 2016). Contrary to methanol, we found the +gas-phase abundance of H2S severely depleted at high density. +This means that the non-thermal desorption of H2S is much less +efficient at high density compared to methanol. One explanation +could be that the H2S formed on the grains at high density is sub- +sequently transformed into another product that still needs to be +identified. This could explain why H2S has not yet been detected +in ices. In our models, we were able to reproduce the observed +H2S because we had already adopted a depleted elemental abun- +dance of sulfur. +8. Conclusions +In this paper, we conducted observations of the cold core L429- +C with NOEMA and IRAM 30m telescopes (maps of 300′′ × +300′′). We detected 11 molecules, including methanol and iso- +topologues of CO. We determined the gas-phase abundances +of these species across the entire maps, constraining the col- +umn density with temperature determined from Herschel, den- +sity with the Bron et al. (2018) method, line widths with the +ROHSA method from Marchal et al. (2019). We interpolated +these three parameters with the theoretical integrated intensity +from RADEX. After a 3 σ cut, we computed the column den- +sity with a χ2 test. We divided the obtained column density with +the nH2 density to derive abundances. CCS, H2S, and HC3N +abundance maps were obtained from upper limits computation. +We compared our observations with the outputs of the Nautilus +chemical model. +We summarize our main findings below: +Article number, page 13 of 24 + +A&A proofs: manuscript no. 45157corr +– The short spacing of NOEMA does not show any signal, im- +plying that there is no molecular emission smaller than ap- +proximately 30′′. This also indicates that there is no protostar +formed yet, nor is the core at an advanced state of infall. +– We studied the cloud dynamics and showed that there were +multiple components (up to three) in the spectra. We did +not determine if the origin of the components was due to +turbulence or remnants of a cloud-cloud collision since ob- +servations of the magnetic field coupled with higher reso- +lution maps would be required. Considering that these ve- +locity components are seen at large spatial scales, this does +not seem to indicate any collapse at the maximum peak den- +sity, as was previously proposed based only on single-point +or spatially limited observations. +– The dust peak is characterized by a depletion in most of +our observed molecular species in the gas-phase, except +for methanol which has a fairly constant abundance along +the density range. We obtained a CO depletion factor f = +f(Xcan/X12CO) of 4.91 at the densest position. +– While comparing our observations with the Nautilus chemi- +cal model, we show that not all regions of the cloud can be +reproduced by the same cloud age. Higher density regions +seem to be younger by a factor of 10 compared to lower +density regions. The measured chemical abundances give an +indication of the dynamical evolution of the region. In other +words, the increase of density up to a few 104 cm−3 may have +taken approximately 105 yr while the increase to 106 cm−3 +happens over a much shorter time (104 yr). +– We observe that the methanol gas-to-ice ratio increases with +density, from 0.002% at 2.4 × 104 cm−3 to 0.09% at 2.2 × +105 cm−3. These values are reasonably well reproduced by +our models, although our model shows an overall trend of +decrease in the ratio with density. +– Our predicted methanol gas-to-ice ratio depends on both the +yield of cosmic-ray sputtering and the cosmic-ray ioniza- +tion, as the former process is the most efficient in releas- +ing methanol into the gas-phase in our model. The observed +slope of the gas-to-ice ratio could be an indication of an in- +crease in efficiency of cosmic-ray sputtering with density, +which may result from a change in the ice composition (from +water-dominated ices to a mixed composition). +– In our observations, we detected H2S in the gas-phase. Since +this molecule is also formed only at the surface of the grains, +its gas-phase abundance should be an indication of non- +thermal desorption from the grains. Contrary to CH3OH, its +abundance decreases by several orders of magnitude within +our observed range of densities. This result could indicate +that the non-thermal desorption process of H2S is differ- +ent from that of methanol and that its efficiency decreases +with density. Another possible explanation would be that the +reservoir of H2S on the grains decreases with density as it is +transformed in other chemical species. This last hypothesis +could also explain the non detection of H2S ices in interstel- +lar environments. +We expect the James Webb Space Telescope to provide ad- +ditional data on the interstellar ice composition thanks to its +unprecedented resolution and sensitivity. In particular, JWST +will increase in a statistical way our knowledge of the ice com- +position, probing a larger range of physical conditions. With +these data, we would be able to apply our methodology to many +other regions and better constrain the non-thermal desorption of +molecules formed at the surface of the grains. +Acknowledgements. AT, VW, PG, JN, ED, and MC acknowledge the CNRS pro- +gram "Physique et Chimie du Milieu Interstellaire" (PCMI) co-funded by the +Centre National d’Etudes Spatiales (CNES). 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W., Awad, Z., et al. 2007, The Astrophysical Journal, +662, L23 +Article number, page 15 of 24 + +A&A proofs: manuscript no. 45157corr +Appendix A: Computation of the NH2 density +We computed the H2 column density from the dust opacity map +τ350 obtained from the Herschel data at the frequency of ν = 350 +GHz (Sadavoy et al. 2018): +NH2 = Σgas +mH2 +, +(A.1) +where Σgas is the surface density of gas (in unit g cm−2) and +mH2 the mass of molecular hydrogen (3.34x10−24 g). The surface +density of gas can be computed by: +Σgas = dtg × Σdust, +(A.2) +where dtg is the dust to gas mass ratio (100 in our case) and Σdust +the surface density of dust. Σdust can be computed from the dust +opacity: +Σdust = τ350 +κ350 with κ350 = 0.4 × ( +ν +250GHz)2 = 0.8 cm2 g−1 +(Endrik Kruegel 2003; Siebenmorgen & Efstathiou 2001; +Kramer et al. 2010). +Article number, page 16 of 24 + +Taillard et al.: Non-thermal desorption of methanol +Appendix B: Integrated intensity maps +The integrated intensity maps of each molecule were obtained by +integrating the peak of emission across different velocity chan- +nels for each of them. It contains both red-shifted and blue- +shifted peaks for all. A 3σ noise cut has been applied. The con- +tour levels account for 90%, 70%, and 50% of the emission peak +value. The obtained map are shown in Fig. B.1. The maps shown +in the figure contains a sample of molecules with only the bright- +est transition when multiple ones were detected. +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +CCS - 93870 MHz +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +CH3OH - 96741 MHz +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +CS - 97980 MHz +0.0 +0.5 +1.0 +1.5 +2.0 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +SO - 99299 MHz +0.0 +0.5 +1.0 +1.5 +2.0 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +HC3N - 100076 MHz +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +C18O - 109782 MHz +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +CN - 113170 MHz +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Integrated intensity (K.km/s) +18h17m18s +12s +06s +00s +-8°12' +14' +16' +RA (J2000) +Dec (J2000) +H2S - 169782 MHz +0.0 +0.1 +0.2 +0.3 +0.4 +Integrated intensity (K.km/s) +Fig. B.1: Integrated intensity map for each detected molecule (brightest transition is shown when more than one was detected). +Article number, page 17 of 24 + +A&A proofs: manuscript no. 45157corr +Appendix C: Channel velocity maps +In Figs. C.1 to C.4, we show the velocity channel maps for C18O, +SO, CS, and CH3OH (at 96741 MHz). +Article number, page 18 of 24 + +Taillard et al.: Non-thermal desorption of methanol +Fig. C.1: C18O (109782 MHz) velocity channel map. The telescope beam is indicated on the lower side of each panel. Contours +show 25, 50, and 75% of the intensity (outer to inner contours). Color coding is in K. +Fig. C.2: SO (99299 MHz) velocity channel map. The telescope beam is indicated on the lower side of each panel. Contours show +25, 50, and 75% of the intensity (outer to inner contours). Color coding is in K. +Article number, page 19 of 24 + +A&A proofs: manuscript no. 45157corr +Fig. C.3: CS (97980 MHz) velocity channel map. The telescope beam is indicated on the lower side of each panel. Contours show +25, 50, and 75% of the intensity (outer to inner contours). Color coding is in K. +Fig. C.4: CH3OH (96741 MHz) velocity channel map. The telescope beam is indicated on the lower side of each panel. Contours +show 25, 50, and 75% of the intensity (outer to inner contours). Color coding is in K. +Article number, page 20 of 24 + +Taillard et al.: Non-thermal desorption of methanol +Appendix D: Position-velocity (PV) diagram +The PV diagram was obtained by integrating the velocity com- +ponents through the two vertical and horizontal axis. Here, C18O +shows multiple component on the horizontal axis of integration +and a simple gradient in the vertical axis. None of the PV dia- +grams shows the expected "V" shape found by Aghanim et al. +(2020). +Fig. D.1: PV diagram of C18O. Bottom-left: Integrated intensity +of C18O. Top-left: PV diagram obtained by integrating the ve- +locity along the horizontal axis. Bottom-right: PV diagram ob- +tained by integrating the velocity along the vertical axis. Top +right: Spectra associated with the crossing of the two axes. +Appendix E: Upper limits on the column densities +for non-detected molecules +To obtain upper limits on the abundance of non detected +molecules, we first computed the upper limits, Wupp, of the inte- +grated intensities : +Wupp < 1.064 × 3 × rms × dv, +where the rms (in K) is the noise level at the dust maximum +position. The values are around 0.06 to ∼ 0.2 K depending on +the molecule. We assumed a line width FWHM (dv) of 1 km.s−1. +The partition functions, provided by CDMS and JPL, are inter- +polated for the temperature of the cloud (10 K). For each species, +we computed the upper limit of the column density following +Mangum & Shirley (2015) and including the cosmological back- +ground radiation temperature : +Ni = 8πk × Qi × f2 +i × Wupp × 105 × +e +Eup,i +T +gup,i × h × c3 × Aul,i +, +where Qi is the partition function of species i, k is the Boltz- +mann constant, fi is the frequency of the transition (MHz), Eup +is the upper energy state of the transition, gup,i is the upper state +degeneracy, T is the temperature of the cloud, h is the Planck +constant, c is the speed of light, and Aul,i is the Einstein coeffi- +cient of the transition. The obtained upper limits are 2 × 1016, +2 × 1013, and 2 × 1013 cm−2 for c-C3H2 (95206.01 MHz), OCS +(97301.20 MHZ), and HNCO (109905.60 MHz), respectively. +Converted into abundances with the H2 column density at the +continuum peak, it gives 2.8×10−7, 2.8 × 10−10, and 2.8 × 10−10, +respectively. +Appendix F: Observed physical parameters +3.5 4.0 4.5 5.0 5.5 6.0 6.5 +log10(nH2) +10 +25 +40 +55 +70 +Av +12 +14 +16 +18 +Temperature (K) +Fig. F.1: Av as a function of nH2 (cm−3) and temperature (K) +observed in L429-C. All these parameters have been computed +from the Herschel observations (see text). +In this section, we compare the different physical parameters +observed in L429-C. Figure F.1 shows the visual extinction as a +function of nH2 (cm−3) and temperature (K) observed in L429-C. +The visual extinctions range from less than 10 to more than 80, +the H2 density from 5×103 to 3×106 cm−3, and the temperature +from approximately 12 up to 18 K. +Article number, page 21 of 24 + +A&A proofs: manuscript no. 45157corr +Appendix G: Goodness of fit for SO, CCS, HC3N, +and CN +Figure G.1 shows the ratio between the modeled and observed +gas-phase abundances of SO, CCS, HC3N, and CN as a function +of the density for the best times of the four sets of models (see +Section 6.4 and Table 2) . +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Low sputtering yield +SO +CCS +HC3N +CN +Low CR +High CR +4.6 +4.8 +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +log(nH2 cm +3) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +log(Xmod/Xobs) +High sputtering yield +SO +CCS +HC3N +CN +Low CR +High CR +Fig. G.1: Ratio between the modeled (Xmod) and observed gas- +phase abundances (Xobs) of SO, CCS, HC3N, and CN as a func- +tion of the density for the best times of the four sets of models. +Article number, page 22 of 24 + +Taillard et al.: Non-thermal desorption of methanol +Appendix H: Best time determination for each Av +The best time is determined by the lowest distance of disagree- +ment, d, defined in Section 6.3. Each figure represents the results +of one set of models. We can see that the higher the density, the +lower the time. The eight best times are illustrated in Fig. 8. +Article number, page 23 of 24 + +A&A proofs: manuscript no. 45157corr +100 +101 +102 +103 +104 +105 +106 +107 +Time (yr) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +d +Low sputtering yield - Low CR +Av=21.4, nH2 = 3.2e+04 cm +2, T = 14.9 K, +Av=27.2, nH2 = 5.2e+04 cm +2, T = 14.4 K, +Av=32.5, nH2 = 8.5e+04 cm +2, T = 14.0 K, +Av=39.7, nH2 = 1.4e+05 cm +2, T = 13.6 K, +Av=49.0, nH2 = 2.3e+05 cm +2, T = 13.1 K, +Av=58.5, nH2 = 3.7e+05 cm +2, T = 12.7 K, +Av=65.8, nH2 = 6.1e+05 cm +2, T = 12.3 K, +Av=73.4, nH2 = 1.0e+06 cm +2, T = 12.0 K, +100 +101 +102 +103 +104 +105 +106 +107 +Time (yr) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +d +Low sputtering yield - High CR +Av=21.4, nH2 = 3.2e+04 cm +2, T = 14.9 K, +Av=27.2, nH2 = 5.2e+04 cm +2, T = 14.4 K, +Av=32.5, nH2 = 8.5e+04 cm +2, T = 14.0 K, +Av=39.7, nH2 = 1.4e+05 cm +2, T = 13.6 K, +Av=49.0, nH2 = 2.3e+05 cm +2, T = 13.1 K, +Av=58.5, nH2 = 3.7e+05 cm +2, T = 12.7 K, +Av=65.8, nH2 = 6.1e+05 cm +2, T = 12.3 K, +Av=73.4, nH2 = 1.0e+06 cm +2, T = 12.0 K, +100 +101 +102 +103 +104 +105 +106 +107 +Time (yr) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +d +High sputtering yield - Low CR +Av=21.4, nH2 = 3.2e+04 cm +2, T = 14.9 K, +Av=27.2, nH2 = 5.2e+04 cm +2, T = 14.4 K, +Av=32.5, nH2 = 8.5e+04 cm +2, T = 14.0 K, +Av=39.7, nH2 = 1.4e+05 cm +2, T = 13.6 K, +Av=49.0, nH2 = 2.3e+05 cm +2, T = 13.1 K, +Av=58.5, nH2 = 3.7e+05 cm +2, T = 12.7 K, +Av=65.8, nH2 = 6.1e+05 cm +2, T = 12.3 K, +Av=73.4, nH2 = 1.0e+06 cm +2, T = 12.0 K, +100 +101 +102 +103 +104 +105 +106 +107 +Time (yr) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +d +High sputtering yield - High CR +Av=21.4, nH2 = 3.2e+04 cm +2, T = 14.9 K, +Av=27.2, nH2 = 5.2e+04 cm +2, T = 14.4 K, +Av=32.5, nH2 = 8.5e+04 cm +2, T = 14.0 K, +Av=39.7, nH2 = 1.4e+05 cm +2, T = 13.6 K, +Av=49.0, nH2 = 2.3e+05 cm +2, T = 13.1 K, +Av=58.5, nH2 = 3.7e+05 cm +2, T = 12.7 K, +Av=65.8, nH2 = 6.1e+05 cm +2, T = 12.3 K, +Av=73.4, nH2 = 1.0e+06 cm +2, T = 12.0 K, +Fig. H.1: Distance of disagreement d for all eight models as a function of time. Each figure represents the result of a set of model +as defined in Table 2. The legend gives each physical parameters associated for the model shown. Each vertical line represents the +lowest disagreement distance associated with each grid of parameters. +Article number, page 24 of 24 + diff --git a/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/load_file.txt b/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08a7aa84c144bd136d011ce0e0951ce46c9f2fe1 --- /dev/null +++ b/K9AzT4oBgHgl3EQfVvwt/content/tmp_files/load_file.txt @@ -0,0 +1,1813 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf,len=1812 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr ©ESO 2023 January 4, 2023 Constraints on the non-thermal desorption of methanol in the cold core LDN 429-C⋆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Taillard,1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Wakelam1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Gratier1, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Dartois2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Chabot3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Noble4, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Keane5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Boogert5, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Harsono6 1 Laboratoire d’Astrophysique de Bordeaux (LAB), Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France e-mail: angele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='taillard@u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='fr 2 Institut des sciences Moléculaires d’Orsay, CNRS, Université Paris-Saclay, Bât 520, Rue André Rivière, 91405 Orsay, France 3 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France 4 Physique des Interactions Ioniques et Moléculaires, CNRS, Aix Marseille Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=', 13397 Marseille, France 5 Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822-1897, USA 6 Institute of Astronomy, Department of Physics, National Tsing Hua University, Hsinchu, Taiwan January 3, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Cold cores are one of the first steps of star formation, characterized by densities of a few 104 to 105 cm−3, low temperatures (15 K and below), and very low external UV radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In these dense environments, a rich chemistry takes place on the surfaces of dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Understanding the physico-chemical processes at play in these environments is essential to tracing the origin of molecules that are predominantly formed via reactions on dust grain surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We observed the cold core LDN 429-C (hereafter L429-C) with the NOEMA interferometer and the IRAM 30m single dish telescope in order to obtain the gas-phase abundances of key species, including CO and CH3OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Comparing the data for methanol to the methanol ice abundance previously observed with Spitzer allows us to put quantitative constraints on the efficiency of the non-thermal desorption of this species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' With physical parameters determined from available Herschel data, we computed abundance maps of 11 detected molecules with a non-local thermal equilibrium (LTE) radiative transfer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These observations allowed us to probe the molecular abundances as a function of density (ranging from a few 103 to a few 106 cm−3) and visual extinction (ranging from 7 to over 75), with the variation in temperature being restrained between 12 and 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We then compared the observed abundances to the predictions of the Nautilus astrochemical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We find that all molecules have lower abundances at high densities and visual extinctions with respect to lower density regions, except for methanol, whose abundance remains around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 × 10−10 with respect to H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The CO abundance spreads over a factor of 10 (from an abundance of 10−4 with respect to H2 at low density to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 × 10−5 at high density) while the CS, SO, and H2S abundances vary by several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' No conclusion can be drawn for CCS, HC3N, and CN because of the lack of detections at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Comparing these observations with a grid of chemical models based on the local physical conditions, we were able to reproduce these observations, allowing only the parameter time to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Higher density regions require shorter times than lower density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This result can provide insights on the timescale of the dynamical evolution of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The increase in density up to a few 104 cm−3 may have taken approximately 105 yr, while the increase to 106 cm−3 occurs over a much shorter time span (104 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Comparing the observed gas-phase abundance of methanol with previous measurements of the methanol ice, we estimate a non-thermal desorption efficiency between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='002% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='09%, increasing with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The apparent increase in the desorption efficiency cannot be reproduced by our model unless the yield of cosmic-ray sputtering is altered due to the ice composition varying as a function of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Astrochemistry, ISM: abundances, ISM: clouds, ISM: Individual objects: LDN 429-C, ISM: molecules 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Introduction In the last 20 years, our understanding of the overall process of star formation has improved substantially (Jørgensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The interplay of turbulence, magnetic field, and gravity in the interstellar medium (ISM) leads to the formation of cold cores (nH2 > 104 cm−2, T ∼ 10 K), which are the sites of rich chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dust grains contained in these cores pro- vide a catalytic surface where complex molecules (COMs, as de- ⋆ This work is based on observations carried out under project num- ber 079-20 and ID 111-21 with the IRAM NOEMA Interferometer and 30m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' fined by Herbst & van Dishoeck 2009) are formed, the simplest of which is CH3OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The icy mantles (on top of these grains) are mostly made of H2O, CO, and CO2, with CH3OH (Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2015, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Among these molecules, methanol is a key species as it is predominantly formed on dust grain sur- faces, but commonly observed in the gas phase in cold cores (Dartois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Dartois 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' On the grains, methanol can be efficiently formed by the hydrogena- tion of CO (itself formed in the gas-phase and adsorbed on the grains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The reaction barrier for two key hydrogenation steps (H + CO and H + H2CO) is significant for CH3OH formation (Fuchs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The presence of methanol in the gas-phase of cold cores, even at low abundance, is a clear indicator that Article number, page 1 of 24 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='01288v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='GA] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr non-thermal desorption mechanisms are efficient in terms of re- leasing molecules from the surface of the grains in regions where simple thermal desorption are not (Garrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In cold cores, where the temperature is typically ∼ 10 K in the absence of any heating source, thermal desorption of methanol is not possible and different desorption mechanisms must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These mechanisms can include chemical desorption (Dulieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Minissale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Wake- lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2017), UV-induced photodesorption and photolysis (Öberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Cruz-Diaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016), and grain sputtering induced by cosmic-ray (CR) impacts (Dar- tois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These mechanisms may be partly de- structive, thereby releasing intact methanol along with frag- ments of methanol, which may themselves participate in sub- sequent chemical processes to reform methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The efficiency of non-thermal desorption is not well known, but is regularly investigated in laboratory experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Using an astrochemical model, Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021) showed that the intrinsic efficiency of these mechanisms depends on the local physical conditions: moving inward into a cold core (with an increasing density, in- creasing visual extinction, and decreasing temperature), the ini- tially efficient photo-desorption will first be replaced by chem- ical desorption, before cosmic-ray induced sputtering becomes the majority process at the highest densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' With the increased sensitivity of the new generation of telescopes and instruments, it is now possible to efficiently detect methanol ice from the ground (Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Goto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2021) and from space (Dartois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2003, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Shi- monishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016) using IR absorption spectroscopy along the lines of sight toward background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Comparing these ice observations to gas-phase methanol observations helps to assess the efficiency of non-thermal desorption of this molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In this article, we present observations of a number of molecules in the cold core L429-C obtained with the IRAM 30m single dish telescope and the NOEMA interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This source is one of the few cores where CH3OH has been detected in the solid phase and it is an obvious benchmark for gas-grain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We focus in particular on gas-phase methanol in order to compare its abundance with ice abundances of methanol ob- tained with Spitzer by Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011) in the same region, as well as to constrain the efficiency of non-thermal desorption of this molecule from the grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Current knowledge of the source properties and a description of our observations are given in Sections 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A description of the observed integrated in- tensity maps and a kinematic analysis of the observations is pre- sented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' From these observations, we compute abun- dance maps for all detected molecules in Section 5 and compare these results with the predictions of the Nautilus astrochemical model in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Our conclusions are summarized in the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Observed source: LDN 429-C LDN 429-C (hereafter L429-C) is a cold (T < 18 K) and dense (column density NH ∼ 1022 cm−2) core (Stutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2009) lo- cated in the Aquila Rift (∼ 200 pc away), with a visual extinc- tion larger than 35 mag at the center (Crapsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Caselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This core is characterized by high degrees of CO de- pletion (15 to 20 times with respect to the canonical abundance at the center, Bacmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Caselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2008) and of deuteration (Bacmann & Faure 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Crapsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Caselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The isotopic fractionation of nitrogen presents a de- pletion of 15N, as in other prestellar cores (Redaelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' HCO, H2CO, and CH3OH have been detected by Bacmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2002, 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Bacmann & Faure (2016) toward this source using the IRAM 30m telescope, while CH3O (Bacmann & Faure 2016) and O2 (Wirström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016) were searched for unsuccessfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The source has been proposed to be on the verge of collapsing by Stutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2009), based on 70 µm observations with Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Based on observed molecular line profiles, both Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2004) and Crapsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2005) classified this source under a possible "infall" category, although the authors also discussed the pos- sibility that the line asymmetry could be due to other types of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Herschel observations of this cold core are available from the Herschel database1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We used the temperature and optical depth maps at 353 GHz(τ353) derived by Sadavoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2018) from SPIRE 250, 350 and 500 µm and PACS 160 µm, with a resolu- tion of 36′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Those authors fitted spectral energy distributions to these maps in order to obtain temperature and τ maps on more than 50 globules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' To obtain the temperature maps, they averaged the dust temperature along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dust tempera- ture, within an extended map around L429-C, varies between 10 and 18 K and the optical depth from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0001 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We derived a column density of H2 from the τ map following the method described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The H2 column density map is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1 (left panel) together with the published Herschel dust temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Lastly, L429-C is one of the first cold cores where the signa- ture of CH3OH ice was unambiguously detected with Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In a survey of 16 isolated dense cores chosen from the c2d legacy targets (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2009), Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011) ob- served a sample of 32 background stars in the 1-25 µm wave- length range to determine the solid-phase molecular composi- tion of dense cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They identified and used four background stars in the L429-C region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The authors were able to measure the H2O, CO2, and CH3OH ice column densities, as well as a de- tection attributed to NH+ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They found a H2O ice column density up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='93 × 1018 cm−2 in the cloud with abundances of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='12%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='13-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='08%, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='34-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='58% for CO2, NH+ 4, and CH3OH re- spectively, with respect to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Recently, two more methanol ice detections in other cold cores have been reported using the NASA Infrared Telescope Facility (IRTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020) and Goto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021) found CH3OH ice abundances relative to water of around 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6% towards L694 and L1544, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As L429-C is one of the few cores to have multiple clear CH3OH detections in the solid phase, it is an obvious benchmark for gas-grain modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Observations The NOEMA observations were conducted during summer 2020 using the mosaic mode, with additional IRAM 30m short spac- ing observations being made in winter 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The mosaic phase center is R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' =18h17m08s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00, DEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' =-8o14’00′′ (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The size of the mosaic is 300′′ ×300′′ with a synthesized beam of 7′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The short spacing maps are 360′′ × 360′′, slightly larger that the NOEMA mosaic, and have a beam of ∼ 25′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Velocity channels were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 and each cube contained 152 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The rms sensitivity was, on average, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='20 K, depending on the molecule at 7′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We observed three frequency bands with IRAM 30m: 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 - 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 GHz, 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 - 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 GHz, and 168 - 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These se- tups were made to focus on specific molecules such as methanol, 1 http://archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='int/hsa/whsa/ Article number, page 2 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=" : Non-thermal desorption of methanol 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination 12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="25 1 2 3 4 5 6 7 log(NH2 cm 2) 1e22 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination 12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(nH2 cm 3) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1: Physical parameter maps of L429-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Left: H2 column density (NH2 in cm−2) in L429-C computed from the Herschel optical depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The position of maximum continuum is represented by the dark blue cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The other crosses represent different positions where the spectra were studied (see Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Right: Computed molecular hydrogen density (nH2 in cm−3) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' On both maps, the white contours are the dust temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CO isotopologues (12CO, 13CO, C17O, and C18O), and H2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The full list of molecules and detected lines is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The data reduction was performed using the Gildas package CLASS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The line identification was done with the help of the Cologne Database for Molecular Spectroscopy3 (CDMS, Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2001) and the Jet Propulsion Laboratory catalog4 (JPL, Pickett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1998), coupled with CLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The NOEMA data reduction was performed using the Gildas package CLIC, the standard pipeline provided by IRAM, to create the UV tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We then used MAPPING to produce the data cubes, using a nat- ural weighting for the synthesized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Residuals of the clean- ing residuals were systematically checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The rms sensitivity on these maps was on average between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The NOEMA data cube does not show any signal at any fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This indicates that there is a spatial filtering with no molecular emission smaller than approximately 30′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Merging the two sets thus only ended up adding noise to the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We decided to use the single dish observations only and all molecu- lar emission maps have resolutions between 25′′ and 28′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Observational results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Molecular transitions and integrated intensity maps Among the targeted molecules, we detected 19 lines (listed in Table 1), corresponding to 11 different molecules, with a peak intensity greater than three times the local rms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The integrated intensity maps have been constructed from the data cubes with the Spectral Cube python package (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1, we present the integrated intensity maps of all detected molecules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' when several lines were observed, we only show the most intense ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The molecules (and transitions) in the follow- ing list were targeted but not detected: OCS (9-8), HNCO (4,0- 4,4), and c-C3H2 (4,3-1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These non-detections are discussed in section E in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='iram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='fr/IRAMFR/GILDAS 3 https://cdms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='de/classic/ 4 https://spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='gov Concerning the detected lines, most of the emissions are ex- tended and not centered on the maximum continuum position: C18O, SO, CS, and CH3OH exhibit "heart-shaped" emission, namely, it is similar to the H2 column density (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Their integrated intensities peak in the upper-right part of the heart, just below the dust maximum position (dark-blue cross).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The other molecules (CCS, HC3N, H2S, and CN) show a weak and more localized emission close to the maximum dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maps can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Kinematic analysis The line profiles of all detected molecules present a complex ve- locity structure that varies with their spatial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As an ex- ample, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2, we show the spectra of C18O, CS, and CH3OH at four positions indicated by colored crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Velocity channels maps are shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The C18O molecule presents three velocity components, each of which can be fit- ted with a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The first component is between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='95 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (component 1), the second is between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 and 7 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (component 2), while the higher velocity component is between 7 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (component 3, see Fig C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Compo- nent 1 is the broadest feature with lowest peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Com- ponent 2 corresponds to the vLS R of the cloud at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (see also Spezzano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' To better visualize the spatial distri- bution of the emission, we integrated the signal over the three ranges of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 3, we show the resulting contours (together with the first-moment map, corresponding to the ve- locity distribution of C18O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The lower (blue) and higher (red) velocity components are only located in the left and right parts of the map, respectively, while the central velocity component is widely spread across the entire map and even superimposes upon the red and blue emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The other molecules also have multi-velocity components but with less clear signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In particular, SO does not exhibit com- ponent 1 but does have components 2 and 3 (with the strongest emission coming from component 2, see also Fig C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CS has a weak intensity component, possibly interpreted as component Article number, page 3 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr Table 1: List of detected lines and associated spectroscopic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Molecule Frequency (MHz) Transition Eup (K) gup Aij (s−1) CCS 93870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 (6-7) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 × 10−5 CH3OH 96739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 (2-1) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='55 × 10−6 CH3OH 96741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 (2,0)-(1,0) 7 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='40 × 10−6 CH3OH 96744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 (2,0)-(1,0) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='40 × 10−6 CS 97980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 (2-1) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='67 × 10−5 SO 99299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 (1-0) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='12 × 10−5 HC3N 100076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 (11-10) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='81 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='77 × 10−4 C18O 109782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 (1-0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='27 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='26 × 10−8 13CO 110201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 (1-0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='29 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='29 × 10−8 C17O 112358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 (1-0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='39 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='69 × 10−8 CN 113144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='43 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='10 × 10−4 CN 113170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='43 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='51 × 10−5 CN 113488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='43 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='67 × 10−5 CN 113490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='44 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='11 × 10−4 CN 113499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='44 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='10 × 10−4 CN 113508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='44 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='51 × 10−5 CN 113520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5)-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='44 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='13 × 10−5 12CO 115271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 (1-0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='53 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='20 × 10−8 H2S 168762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 (1,1,0)-(1,0,1) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='65 × 10−5 The most relevant non-detections are discussed in section E of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1, between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As for SO, component 2 is the most intense (see also Fig C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Most of the methanol emission can be fitted with one single component (component 2), but in the right part of the map, a weak component 3 can be found (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The other molecules are weak and only present emission at the velocity of component 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This velocity structure observed on the line profiles at large spatial scale is very likely what Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2004) and Crapsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2005) attributed to in- fall, as they had only the spectra on the dust peak position for Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' and a very small map around it for Crapsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='. We investigated the possibility that L429-C could be the re- sult of a H i cloud - cloud collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Looking strictly from a dy- namical point of view, as discussed previously, we have multiple velocity components in our spectra, possibly showing conver- gent flows rather than turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the cloud-cloud collision scenario, we would have two components, one moving towards (blue) and one moving away (red) from us, converging to the dust peak position and resulting in the formation of a denser re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In Bonne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020), the authors studied the formation of the Musca filament and its origin from a cloud-cloud region collision: a H i cloud colliding with a denser region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the multi- ple arguments in support of this hypothesis, they showed that the result from such a collision would be a dense filament with the apparition of CO from the resulting shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The CO emission would therefore be blue-shifted in comparison to the HI emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A key result would be that the matter would slide to the core by the action of a curved magnetic field, thus becoming ac- celerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This would be manifested in the PV-diagram (shown in Arzoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2018)) of CO, for example, and show a "V-shaped" velocity diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Such a curved magnetic field is observable with the Planck telescope: they observed magnetic field lines arriving perpendicular to the filament and becoming curved by going through the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the case of L429-C, from Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020a,b) released data, we can observe magnetic field lines ori- ented perpendicularly to the cloud along the top-right axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' By looking at the general direction of the field, we cannot say if the lines are bent by the core’s presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The Planck resolution at 353 GHz (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2020) is too large compared to the size of cloud to observe any bending of these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Thus, we are not able to confirm a similar effect as that observed in Musca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We do observe CO in our cloud, with a very dense profile and an asymmetry in our velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' However, we do not observe the "V-shaped" signature found by Arzoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2018) in our PV diagram (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1) – rather, we simply have two components converging toward the core position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Nor do we see any temperature rise that could result from shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We conclude that higher resolution magnetic field data would be needed to conclude anything other than that the region is going through dynamical behavior that more so appears to resemble conver- gent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We cannot make other assumptions on a possible H i cloud-cloud collision at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Observed molecular abundances 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Method For all detected molecules, when the collisional rate coefficients were available in the LAMDA database5 (Schöier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2005), we estimated the observed column density with an inversion procedure and the RADEX radiative transfer code (van der Tak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We first computed a theoretical grid of line inte- grated intensities, using external constraints on the temperature, line width, and H2 density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Then, comparing this grid of theoreti- cal values with our observed ones through a χ2 minimization, we constrained the molecular column densities at each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We as- sumed a gas temperature equal to that determined from Herschel observations (see Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The resolution of the Herschel obser- vations (36” resolution, from Sadavoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2018) is similar to our 30m data (23 to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5”) The line widths were taken from each spectrum from a Gaussian fitting (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1), while the 5 https://home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='strw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='leidenuniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='nl/~moldata/ Article number, page 4 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol 5 6 7 8 9 VLSR [km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s 1] 0 1 2 3 4 Tmb [K] CH3OH CS C18O 5 6 7 8 9 VLSR [km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s 1] 0 1 2 3 4 Tmb [K] CH3OH CS C18O 5 6 7 8 9 VLSR [km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s 1] 0 1 2 3 4 Tmb [K] CH3OH CS C18O 5 6 7 8 9 VLSR [km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s 1] 0 1 2 3 4 Tmb [K] CH3OH CS C18O Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2: Spectra of CS (red line), CH3OH 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='741 GHz (blue line) and C18O (yellow line) for the four positions indicated as crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The colored crosses are shown in the upper left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=" 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Intensity-weighted velocity (km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="s 1) 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination 1 2 3 4 5 6 7 log(NH2 cm 2) 1e22 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 3: Velocity distribution in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C18O integrated intensity contours over plotted on the NH2 Herschel map (gray color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contours are shown for the three velocity components detailed in the text (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Cyan: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='95 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (with intensity levels as follows: dotted is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K, dashed is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 K and solid line is 1 K), pink: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 - 7 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (with dotted as 1 K, dashed is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 K and solid line is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K), red: 7 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 (with dotted as 1 K, dashed as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 K and solid line as 2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Velocity (first-moment) map (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dark blue cross shows the position of the continuum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' H2 density was derived from the Herschel data with a method described below (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Determining the line width and integrated intensities at each pixel The non-LTE RADEX code requires the line width at each spec- trum in each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The full width at half maximum (FWHM) of the lines (dv) was obtained using the ROHSA method from Marchal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The authors developed a Gaussian decom- position algorithm using a nonlinear least-squares criterion to perform a regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For each pixel, we computed the Gaussian fits of the extracted spectra and computed the width and the integral under the curve of this fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We also compared the difference between the intensity map obtained by ROHSA and ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It showed little variation (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1 width) between both models and so, we assumed that the dv value ob- tained from the Gaussian fit is good enough to be used in our Article number, page 5 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the case of a molecule with multiple components, we fit ROHSA with two Gaussian identifications and made sure to sum the two widths for our final dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It is also feasible as all of our molecules present optically thin emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It was more ac- curate to use ROHSA’s new computed intensity as it gives us a better signal-to-noise level and also takes into account the possi- bility that some of the baselines are not completely centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the case of two velocity components, the integrated intensities obtained with the two fitted Gaussian are also summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Determining the H2 density at each pixel The number of detected lines per species was not enough to determine the local volume density at each pixel from a radia- tive transfer analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We therefore used the method described in Bron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2018), which estimates the H2 volume density from the H2 column density (obtained from Herschel and described in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This method is particularly well adapted for simple sources such as ours as it makes the assumption that the medium is isotropic and that the density is smoothly increasing from outer to inner regions of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It also assumes that there is no preferential direction for the spatial density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It then estimates the typical length scale l of the cloud, knowing the col- umn density NH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Finally, nH2 is given simply by dividing NH2 by l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The values obtained for nH2 ranges from 3 × 103 to 106 cm−3 and the obtained density map is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' χ2 comparison with RADEX We then used the non-LTE radiative transfer code RADEX (van der Tak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2007), with the LAMDA database to obtain a grid of integrated intensities for one or multiple transitions per molecule (for the latter, a grid was computed for each line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Col- lision rates used are: SO from Lique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2005), CS from Lique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2006), CN from Lique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2010), CO from Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2010), H2S from Dubernet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2009), CH3OH from Rabli & Flower (2010), and HC3N from Faure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We first made low-resolution grids using RADEX to estimate the value ranges of the unknown parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' the grids were wide at first then narrowed by iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This process refines the grids to avoid saturation on the extrema and smoothed the maps (for example, starting with values for the column densities between 1011 and 1018 cm−2 before refining to values between 1012 and 1016 cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The theoretical integrated intensity grid was com- puted for H2 density values between 103 to 107 cm−3 (30 val- ues in logarithmic space), temperatures between 11 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 K (30 values in linear space), 5 values of dv (in linear space), be- tween the minimum and maximum value, of each Gaussian file of the molecules obtained by ROHSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the molecular column density, we ran multiple tests to calibrate it for each molecule, with a logarithm space of 60 values and the lowest input being 1012 cm−2 and the maximum 1018 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Once we ran the tests, we adjusted the values to be the closest to the minimum and maximum values obtained on the column densities maps (see next steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The final grid contains 270,000 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' To circumvent the degeneracies between the RADEX input parameters, we chose to fix most of the values (Tkin, nH2, dv) to those we determined independently for each pixel from the methods described in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This is done by interpo- lating linearly on the intensity grid using the independently de- rived parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The interpolated theoretical integrated intensi- ties are then compared to the observed ones, through χ2 mini- mization, to determine the best molecular column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Lastly, we created abundance maps by dividing the molecu- lar column densities by the H2 column densities for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These maps of the abundances with respect to H2 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For CO, we computed the main isotopologue abundance from the C18O abundance multiplied by 557 (Wilson 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We also computed it from the C17O abundance multiplied by 2005 (Lodders 2003) and obtained the same abundance of the main isotopologue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maps of CCS and HC3N were computed at LTE because no collisional coefficients are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Similar to CN, the lines are only detected in a small portion of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the non-detected molecules, we computed upper limits on their column densities (given in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Finally, the optical depth for each detected molecule (that has an entry in the LAMDA collisional database) was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' To do so, we used the four positions shown in Fig 1 and for each associated pixel, we collected the kinetic temperature, the H2 density, the line width, and the column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We used RADEX to compute τ for each position and each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For all molecules, we found τ values inferior to 1, indicating that the emission averaged within the beam is optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Abundance maps The position of the dust peak is characterized by a decrease in the abundance of most of the observed molecules (see Figs 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We obtained a CO abundance (with respect to H2) up to 10−4 in the outer parts of the maps, whereas it is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7×10−5 at the dust peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We computed a depletion factor of f = f(Xcan/X12CO), with Xcan = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 x 10−5 being the "canonical" abundance of 12CO determined by (Frerking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1982), from the 12CO/H2 abun- dance map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' At the position of the dust peak, we find a CO abun- dance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 × 10−5, which gives us f 12CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='91, showing an underestimation of the abundance in comparison to the canoni- cal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This is almost three times smaller than Bacmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2002), where the authors found a depletion factor of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 in L429-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This discrepancy can be explained by a difference in the adopted temperature used to determine the CO column den- sity and in the adopted H2 column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' By using a lower temperature for ∼ 7 K, the authors found f = 5 (and for ∼ 11 K, f = 3), which is closer to our value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They also used a higher H2 column density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × 1023 cm−2 (we used NH2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 x 1022 cm−2), which produces a lower 12CO abundance compared to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that CS seems to be depleted over the entire "heart- shaped" density structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Its highest abundance is in the top of the map (with a maximum value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 × 10−8), while the abun- dance at the dust peak is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 × 10−10, that is, almost 75 times lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' SO presents a similar behavior, although its maximum abundance is in the left part of the map, with a difference of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 between the maximum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5×10−9) and the dust peak (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3×10−10) abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The H2S intensity is weak so the values derived here have to be considered with less robustness than for the other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maximum abundance is obtained on the border of the map, around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 × 10−9, while it is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 × 10−11 at the contin- uum peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Compared to the other species, the CH3OH abundance is more homogeneous across the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Its maximum abundance is in the left part of the map (although not on the border of the map) and is around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 × 10−10, while its abundance on the dust peak is about two times lower, being around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2×10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The CN map was computed using several lines detected only in a small area of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This results in a source-focused area with a visible depletion on the continuum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maximum abun- Article number, page 6 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=" : Non-thermal desorption of methanol 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination CO / H2 Abundance map 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="0 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination CH3OH / H2 Abundance map 10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="0 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination C18O / H2 Abundance map 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="7 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination H2S / H2 Abundance map 10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="0 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination CS / H2 Abundance map 10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="0 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination SO / H2 Abundance map 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 4: On a logarithmic scale, observed gas-phase abundance maps with respect to H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The green crosses on the methanol map correspond to the positions of the methanol ice observations reported by Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dark blue cross is the continuum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that the white cross on H2S / H2 represents a gap in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' dance is found to be just below the dust peak at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 × 10−11 and its minimum at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 × 10−12 on the dust peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CCS presents three peaks, with its maximum in the left part of the map, where the peak abundance is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 × 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The abundance on the dust peak is around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 × 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the wider region probed here, CCS is constant and there is no evidence for depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' HC3N shows lit- tle to no variation in its abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maximum abundance (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × 10−12) is found close to the dust peak position (with an abundance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 × 10−12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 7 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=" 45157corr 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination CCS / H2 Abundance map 11." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="2 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination HC3N / H2 Abundance map 12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="50 18h17m15s 10s 05s 00s 8°12' 13' 14' 15' 16' 17' Right Ascension Declination CN / H2 Abundance map 11." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5: On a logarithmic scale, observed gas-phase abundance maps with respect to H2 for CCS, HC3N, and CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dark blue cross is the continuum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Abundances as a function of the physical parameters In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6 and 7, we show the abundances (with respect to H2) of each molecule for all pixels as a function of the three physi- cal parameters (temperature, density, and visual extinction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The CO, CS, SO, and H2S abundances decrease with density, as well as with temperature and visual extinction, as these three param- eters are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The lower molecular abundances of CO, CS, SO, and H2S seen on the continuum peak position are indeed explained by a higher density there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The slope of the decrease is the strongest for H2S, which decreases by three orders of mag- nitude when the density goes from 3 × 104 to 106 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Then, CO has the smallest variation among these molecules with a less than one order of magnitude decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The methanol abundance is nearly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the CCS, CN, and HC3N molecules, the abun- dances also seem flat, but they are only observed at high density, so they are not fully comparable with the other molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They also present a larger spread of values at a specific density (or vi- sual extinction), especially for lower densities (or visual extinc- tions) and probably reflecting a larger uncertainty in their com- puted abundances (due to lower line intensities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that the CN abundance is varying over such a small range of values that the distribution of the computed abundances (forming three groups) reflects the sampling of the RADEX theoretical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Chemical modeling of the region To understand the trends in molecular abundances observed in L429-C, we ran a suite of chemical models accounting for the cloud’s physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Model description We used the chemical model Nautilus developed by Ruaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Nautilus is a three-phase gas-grain model that computes the gas and ice abundances of molecules under ISM conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' All gas-phase chemical reactions are considered, based on up- dates of the kida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2014 chemical network (Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2015), as listed in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The gas and grain net- work used for these simulations contain more than 14000 chem- ical reactions (in the gas-phase, at the surface of the grains, in the bulk, and at the interface between gas and grains, and sur- face and bulk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Chemistry of the following elements is consid- ered: H, He, C, N, O, Si, S, Fe, Na, Mg, Cl, P, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In the model, species from the gas-phase can stick to interstellar grains upon collision, through physisorption processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They can then diffuse and react.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The thermal evaporation of adsorbed species as well as a number of non-thermal desorption mechanisms are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Under the shielded and cold conditions of L429-C, two non-thermal desorption processes are particularly impor- tant: 1) chemical desorption, for which we adopted the formal- ism of Minissale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2016) for water ices, and 2) sputter- ing by cosmic-rays (Dartois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As shown in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021), this latter process is the most efficient for releasing icy molecules, in particular methanol, into the gas-phase under dense conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Dartois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021) presented two yields for this process, depending on the nature of the main constituent of the ices: either water or CO2, with the latter being more ef- ficient than the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In this work, we tested both yields: a "low sputtering yield" derived from data on pure H2O pure ices and a "high sputtering yield" derived from data on pure CO2 pure ices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In each case, we apply one yield to all species in the model, which means that all species (both on the surface and in the bulk) desorb with the same yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that the fraction of CO2 to H2O ice observed in L429-C by Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011) was as high as 43%, although this was only for one line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Our model also includes the non-thermal desorption of surface species due to the heating of the entire grain by cosmic-rays (as presented in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This process was shown to be important for some gas-phase species (such as CS, HC3N, and Article number, page 8 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 log(Xobs) 0 15 30 45 60 75 Av 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 12 14 16 18 Temperature (K) CO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 log(Xobs) 0 15 30 45 60 75 Av 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 12 14 16 Temperature (K) CH3OH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 10 9 8 log(Xobs) 0 15 30 45 60 75 Av 10 9 8 12 14 16 18 Temperature (K) CS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(Xobs) 0 15 30 45 60 75 Av 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 12 14 16 Temperature (K) SO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6: Abundance as a function of of the logarithm of the volume density (right) and of the visual extinction, Av, (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' HCO+) because of the efficient desorption of CH4 at high den- sity > 2 × 104 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A more detailed description of the model is given in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Model parameters To compare our model results with our observed abundances, we ran a grid of chemical models covering the observed phys- ical conditions (temperature, density, and visual extinction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In Appendix F, we show the observed relationship between the H2 density, the visual extinction, and the temperature determined from Herschel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that the range of physical condi- tions in that case is larger than the one probed by the methanol lines because methanol was not detected throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We first created a grid of eight H2 densities from 3×104 to 106 cm−3 in a logarithm space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For each of the eight H2 densities (with associ- ated derived dust and gas temperatures, and visual extinctions), we considered two different CR sputtering yields and two dif- Article number, page 9 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(Xobs) 0 15 30 45 60 75 Av 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 12 13 14 15 Temperature (K) CCS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 12 11 log(Xobs) 0 15 30 45 60 75 Av 12 11 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 Temperature (K) HC3N 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 10 9 8 7 log(Xobs) 0 15 30 45 60 75 Av 10 9 8 7 12 14 16 Temperature (K) H2S 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log(nH2 cm 3) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(Xobs) 0 15 30 45 60 75 Av 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Temperature (K) CN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 7: Abundance as a function of of the logarithm of the volume density (right) and of the visual extinction, Av, (left) for CCS, HC3N, H2S, and CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' ferent values of the ionization rate ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Using the grid defined by the methanol detection, associated visual extinctions vary from 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 while the temperatures vary from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that we cover the values of densities where methanol was detected and not the full range of some of the other molecules such as CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Since we were unable to determine the gas tem- perature from our line analysis, for the model, we assume the gas temperature as the measurement determined from Herschel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' While an uncertainty of a few Kelvin in the gas tempera- ture has little impact on the chemical modeling results, the ice abundances can be strongly influenced by such a difference in the dust temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The dust temperatures retrieved from Her- schel observations are all above 11 K – even for the highest Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Dust temperatures derived for FIR emission tend to be overes- timates of the true large grain temperature inside the cores be- cause emission from warmer dust of the diffuse envelope can be mixed in the observing beam (Marsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the dust temperature, we therefore used the parametric expression for the Article number, page 10 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol Table 2: Sets of chemical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Name Sputtering yield ζ (s−1) Low yield and CR H2O-rich ices 10−17 Low yield and high CR H2O-rich ices 3 × 10−17 High yield and low CR CO2-rich ices 10−17 High yield and CR CO2-rich ices 3 × 10−17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(nH2 cm 3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(Time yr) Low sputtering yield - Low CR Low sputtering yield - High CR High sputtering yield - Low CR High sputtering yield - High CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 8: Best time obtained from the comparison between mod- eled and observed abundances of CO, CS, H2S, and CH3OH as a function of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' dust temperature as a function of visual extinction from Hocuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This easy-to-handle parametrization was obtained by semi-analytically solving the dust thermal balance for differ- ent types of grains and comparing to a collection of observational measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In addition to the density, the dust and gas temperatures, and the visual extinction, we considered two different values of the ionization rate ζ: 10−17 s−1 (low ionization) and 3 × 10−17 s−1 (high ionization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These two values cover the observational range of ζ at high visual extinction (Av, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Padovani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2022, , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We note that extending the first order function to de- scribe the ionization attenuation with Av that is valid for translu- cent clouds to moderate Av – as used in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2021) – to such high Av would predict too low (< 10−18 s−1) ionization rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In our simulations, we started from atoms (with abundance values similar to those of Table 1 in Ruaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016), with the exception of hydrogen, which is assumed to be in molecular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In total, we have four sets of eight models as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the first two sets, we used the yield of sputtering for water-rich ices (low sputtering yield) with two values of ζ (10−17 and 3 × 10−17 s−1), while for the other two sets we use the yield for CO2-rich ices (high sputtering yield) and the same two values of ζ (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Comparison between modeled and observed gas-phase abundances In these simulations, the abundances are computed as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' To quantify the agreement between model and observa- tions, we computed the distance of disagreement, d, as described in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2006): d(t) = 1 Ni � i | log(Xmod,i(t)) − log(Xobs,i)|, (1) with t as the time, Ni the number of molecular species (four in our case: CO, CS, H2S, and CH3OH) used in the compari- son, Xmod,i the modeled abundance of species i at time, t, and Xobs,i as the mean observed abundance of species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A value of 1 for d means that the mean difference between modeled and ob- served abundance is a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The smallest d value repre- sents the best agreement and, thus, the best time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Figure 8 shows the obtained best time as a function of density for the four sets of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Density is a fixed value during the time evolution of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1, we show d(t) as a function of time for all eight models in each of the four model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The best time, namely, the integration time used in the model that best reproduces the observations – is similar for all sets of models and decreases with density (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The fact that some of the models show a best agreement for exactly the same time is a result of the sampling of the modeling time chosen to get the model output and the small sensitivity of the agreement for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For the models in Set 1, for instance (see Ta- ble 2), the best time is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 × 105 yr at a density of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 × 104 cm−3 and down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 × 104 yr at a density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 × 106 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In other words, at a higher density, the observed abundances can be achieved for a shorter integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The main constraint on the time is given by the observed CO abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' According to the model, CO has a "simple" abundance curve with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The molecule is progressively formed in the gas-phase through gas-phase reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Its abundance reaches a peak at a time that depends on density before decreasing as it is depleted onto the grains and transformed into methanol and other species (see also the discussion in Section 3 in Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In our observations, the CO gas-phase abundance varies by less than a factor of 10, while the density varies over several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As a result, the observed abundance at high density cannot be achieved on the same timescale as that at lower den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Through our chemical modeling, we are able to evaluate the dynamical evolution of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We previously indicated that the number of molecular species considered in the determination of the best evolution- ary time was four, namely CO, CS, H2S, and CH3OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We did not use CCS, HC3N, and CN) because they were detected only on a small fraction of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The SO molecule was detected everywhere but was not reproduced by the model at a sufficient level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Goodness of fit In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 9, we plot the ratio between the modeled abundances (at the best time for each condition) and the observed abun- dances for the species used to determine the best times (CO, CS, CH3OH, and H2S) to quantify the robustness of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Overall, the abundances of these molecules are well reproduced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=', within a factor of 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CH3OH is not as well reproduced at high density if low sputtering yield is assumed and at low density if a higher sputtering yield is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The ratio for the other species (SO, HC3N, CN, and CCS) is shown in the appendix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Specifically, SO, HC3N, and CN are overestimated by the model at all densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CCS is underestimated by the model, with an agreement in excess of a factor of 10 at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We cross-checked the upper limits derived for OCS, HNCO, and c-C3H2 with our best models (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Upper limits on Article number, page 11 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Low sputtering yield CO CS H2S CH3OH Low CR High CR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(nH2 cm 3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(Xmod/Xobs) High sputtering yield CO CS H2S CH3OH Low CR High CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 9: Ratio between the modeled (Xmod) and observed gas-phase abundances (Xobs) of CO, CS, CH3OH, and H2S as a function of the density for the best times of the four sets of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' OCS and c-C3H2 are in agreement with our predictions, while HNCO is overproduced by the model by at least a factor of 10 at all densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We also compared our model predictions to the non-detections of O2 (with an abundance < 2× 10−6) and CH3O (< 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8× 10−12) reported at the continuum position by Wirström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2016) and Bacmann & Faure (2016), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These upper limits are in agreement with our model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Constraining the non-thermal desorption of methanol Combining our gas-phase abundances of methanol with the ice observations of Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011), we can offer constraints on the efficiency of non-thermal desorption of methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Figure 10 shows the ratio between the observed gas and ice column densi- ties of methanol (black dots on the lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The four points represent the four positions reported by Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011, namely, in Table 6 of their paper) and shown in green crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The observed gas-phase column densities are the ones derived in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' On the same figure, we show our model results (methanol gas-to-ice abundance ratios) obtained at the best times for each density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As expected, the main reservoir of methanol (empirically and theoretically) is in the ices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The gas- phase abundance is several orders of magnitude lower than the solid abundance (Drozdovskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' From the observa- tions, we computed the efficiency of non-thermal desorption of CH3OH ices as Ngas Nice × 100 and we obtained a desorption effi- ciency between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='002% (at low density ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 x 104 cm−3) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='09% (at high density ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 × 105 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Although we have only four points, the observations seems to indicate that the ef- ficiency of non-thermal desorption increases with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The gas-phase abundance at these positions does not vary much (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × 10−10), but the ice abundances (as shown in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 10) decrease by more than a factor of 2 with den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' So at high density, to maintain the same gas-phase abun- dance, the desorption needs to be more efficient by a factor of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Our models reproduce the observed ice column density of methanol (see top Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='10 ) within less than a factor of 2 for sets 2 and 4 (ζ = 3 × 10−17 s−1) and within a factor of 3 for sets Article number, page 12 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='50 CH3OHice abundance 1e 5 Low sputtering yield - Low CR Low sputtering yield - High CR High sputtering yield - Low CR High sputtering yield - High CR Spitzer observations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(nH2 cm 3) 10 5 10 4 CH3OHgas / CH3OHice Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 10: Comparison between modeled and observed methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Upper panel: Abundance of methanol ices obtained by our chem- ical model as a function of density (solid lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' see text and Ta- ble 2 for details on the model sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The black points (with un- certainties) are the observed values at the four positions probed by Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Red and orange curves overlap as well as the green and blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Lower panel: Gas-to-ice abundance ratios of methanol as a function of density obtained by the dif- ferent sets of models for the best times (solid lines) and the four observation points (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1 and 3 (ζ = 10−17 s−1) at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' At a higher density, all models are in agreement with the CH3OH ice abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Both the observations and the models seems to indicate a lower ice abundance with increasing density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Our different sets of models produce a gas-to-ice ratio that decreases with density (contrary to the observations) but give different values depending on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The higher cosmic ray sputtering yield produces a large ratio as does the higher cosmic-ray ionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Sets 1 and 2 (for water ices) give a ratio closer to the observations at low density, while sets 3 and 4 (CO2 ices) give a ratio closer to the observations at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' None of the models seem to reproduce the lower density point better than a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Overall, and considering all the uncertainties in the observations (both the ices and the gas), in the density determination, and in the chemical model, we find this agreement satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' However, if the observed increase in desorption efficiency of methanol with density is true, then this cannot be explained by our model unless we change the ice composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' If the ice composition changes from a water domi- nated ice to a mixture where non-thermal desorption (such as the cosmic-ray sputtering) is more efficient, then we can obtain the same trend as the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Such change in the ice compo- sition could occur during the catastrophic CO freeze out in cold cores (Qasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In fact, in our observations, the CO abundance in the gas-phase is nearly decreased by a factor of 10 from low to high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The gas-to-ice CH3OH ratio that we observe in L429-C may be an indication of a change in the ice composition, as suggested by Navarro-Almaida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020), for H2S in cold cores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' although we note that paper’s focus was the chemical desorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In comparison, Perotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020) studied a dense star- forming region, in the Serpens SVS 4 cluster, using the SubMil- limeter Array, Atacama Pathfinder EXperiment and Very Large Telescope observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They estimated a CH3OH column den- sity of approximately 1014 cm−2 in the gas-phase and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8× 1018 cm−2 in the solid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' They thus obtained a gas-to-ice ratio varying between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × 10−4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 × 10−3, which is higher than in our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' However, they do not provide information on the densities within the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Their gas-to-ice CH3OH ratio does not show any trend with H2 column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In addition, they estimated the column densities of methanol in the gas-phase at LTE, with a mean temperature of 15 K and using a high en- ergy transition of methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It is possible that they are in the sub-thermal excitation regime and would thus overestimate the column densities, meaning they would actually have a lower gas- to-ice ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Among the other species studied here, H2S molecule is an interesting case to highlight, as it is generally assumed that it must be formed on the grains since there is no efficient gas- phase pathway (Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Similar to the role oxygen atoms play in the formation of water, models predict that atomic sulfur from the gas sticks onto the grains at low temperature and is easily hydrogenated to form H2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' As such, large amounts of H2S ice are predicted by chemical models but the molecule has never been found in interstellar ices (Smith 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This molecule is the dominant S specie sink in cometary ices (Calmonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contrary to methanol, we found the gas-phase abundance of H2S severely depleted at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This means that the non-thermal desorption of H2S is much less efficient at high density compared to methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' One explanation could be that the H2S formed on the grains at high density is sub- sequently transformed into another product that still needs to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This could explain why H2S has not yet been detected in ices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In our models, we were able to reproduce the observed H2S because we had already adopted a depleted elemental abun- dance of sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Conclusions In this paper, we conducted observations of the cold core L429- C with NOEMA and IRAM 30m telescopes (maps of 300′′ × 300′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We detected 11 molecules, including methanol and iso- topologues of CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We determined the gas-phase abundances of these species across the entire maps, constraining the col- umn density with temperature determined from Herschel, den- sity with the Bron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2018) method, line widths with the ROHSA method from Marchal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We interpolated these three parameters with the theoretical integrated intensity from RADEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' After a 3 σ cut, we computed the column den- sity with a χ2 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We divided the obtained column density with the nH2 density to derive abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' CCS, H2S, and HC3N abundance maps were obtained from upper limits computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We compared our observations with the outputs of the Nautilus chemical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We summarize our main findings below: Article number, page 13 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr – The short spacing of NOEMA does not show any signal, im- plying that there is no molecular emission smaller than ap- proximately 30′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This also indicates that there is no protostar formed yet, nor is the core at an advanced state of infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – We studied the cloud dynamics and showed that there were multiple components (up to three) in the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We did not determine if the origin of the components was due to turbulence or remnants of a cloud-cloud collision since ob- servations of the magnetic field coupled with higher reso- lution maps would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Considering that these ve- locity components are seen at large spatial scales, this does not seem to indicate any collapse at the maximum peak den- sity, as was previously proposed based only on single-point or spatially limited observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – The dust peak is characterized by a depletion in most of our observed molecular species in the gas-phase, except for methanol which has a fairly constant abundance along the density range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We obtained a CO depletion factor f = f(Xcan/X12CO) of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='91 at the densest position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – While comparing our observations with the Nautilus chemi- cal model, we show that not all regions of the cloud can be reproduced by the same cloud age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Higher density regions seem to be younger by a factor of 10 compared to lower density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The measured chemical abundances give an indication of the dynamical evolution of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In other words, the increase of density up to a few 104 cm−3 may have taken approximately 105 yr while the increase to 106 cm−3 happens over a much shorter time (104 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – We observe that the methanol gas-to-ice ratio increases with density, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='002% at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × 104 cm−3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='09% at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 × 105 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' These values are reasonably well reproduced by our models, although our model shows an overall trend of decrease in the ratio with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – Our predicted methanol gas-to-ice ratio depends on both the yield of cosmic-ray sputtering and the cosmic-ray ioniza- tion, as the former process is the most efficient in releas- ing methanol into the gas-phase in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The observed slope of the gas-to-ice ratio could be an indication of an in- crease in efficiency of cosmic-ray sputtering with density, which may result from a change in the ice composition (from water-dominated ices to a mixed composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' – In our observations, we detected H2S in the gas-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Since this molecule is also formed only at the surface of the grains, its gas-phase abundance should be an indication of non- thermal desorption from the grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contrary to CH3OH, its abundance decreases by several orders of magnitude within our observed range of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This result could indicate that the non-thermal desorption process of H2S is differ- ent from that of methanol and that its efficiency decreases with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Another possible explanation would be that the reservoir of H2S on the grains decreases with density as it is transformed in other chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' This last hypothesis could also explain the non detection of H2S ices in interstel- lar environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We expect the James Webb Space Telescope to provide ad- ditional data on the interstellar ice composition thanks to its unprecedented resolution and sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In particular, JWST will increase in a statistical way our knowledge of the ice com- position, probing a larger range of physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' With these data, we would be able to apply our methodology to many other regions and better constrain the non-thermal desorption of molecules formed at the surface of the grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' AT, VW, PG, JN, ED, and MC acknowledge the CNRS pro- gram "Physique et Chimie du Milieu Interstellaire" (PCMI) co-funded by the Centre National d’Etudes Spatiales (CNES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We would like to thank Lars Bonne and Sylvain Bontemps for their help on the dynamical study and for sharing with us 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Astrophys- ical Journal, 718, 1062, arXiv: 1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3923 Öberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=', Fuchs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=', Awad, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2007, The Astrophysical Journal, 662, L23 Article number, page 15 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr Appendix A: Computation of the NH2 density We computed the H2 column density from the dust opacity map τ350 obtained from the Herschel data at the frequency of ν = 350 GHz (Sadavoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2018): NH2 = Σgas mH2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1) where Σgas is the surface density of gas (in unit g cm−2) and mH2 the mass of molecular hydrogen (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='34x10−24 g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The surface density of gas can be computed by: Σgas = dtg × Σdust, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2) where dtg is the dust to gas mass ratio (100 in our case) and Σdust the surface density of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Σdust can be computed from the dust opacity: Σdust = τ350 κ350 with κ350 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 × ( ν 250GHz)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 cm2 g−1 (Endrik Kruegel 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Siebenmorgen & Efstathiou 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Kramer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 16 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol Appendix B: Integrated intensity maps The integrated intensity maps of each molecule were obtained by integrating the peak of emission across different velocity chan- nels for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' It contains both red-shifted and blue- shifted peaks for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' A 3σ noise cut has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The con- tour levels account for 90%, 70%, and 50% of the emission peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The obtained map are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The maps shown in the figure contains a sample of molecules with only the bright- est transition when multiple ones were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=" 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) CCS - 93870 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) CH3OH - 96741 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) CS - 97980 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) SO - 99299 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) HC3N - 100076 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) C18O - 109782 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) CN - 113170 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content="km/s) 18h17m18s 12s 06s 00s 8°12' 14' 16' RA (J2000) Dec (J2000) H2S - 169782 MHz 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 Integrated intensity (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='km/s) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: Integrated intensity map for each detected molecule (brightest transition is shown when more than one was detected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 17 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr Appendix C: Channel velocity maps In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, we show the velocity channel maps for C18O, SO, CS, and CH3OH (at 96741 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 18 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: C18O (109782 MHz) velocity channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The telescope beam is indicated on the lower side of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contours show 25, 50, and 75% of the intensity (outer to inner contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Color coding is in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2: SO (99299 MHz) velocity channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The telescope beam is indicated on the lower side of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contours show 25, 50, and 75% of the intensity (outer to inner contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Color coding is in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 19 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3: CS (97980 MHz) velocity channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The telescope beam is indicated on the lower side of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contours show 25, 50, and 75% of the intensity (outer to inner contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Color coding is in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4: CH3OH (96741 MHz) velocity channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The telescope beam is indicated on the lower side of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Contours show 25, 50, and 75% of the intensity (outer to inner contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Color coding is in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 20 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol Appendix D: Position-velocity (PV) diagram The PV diagram was obtained by integrating the velocity com- ponents through the two vertical and horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Here, C18O shows multiple component on the horizontal axis of integration and a simple gradient in the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' None of the PV dia- grams shows the expected "V" shape found by Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: PV diagram of C18O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Bottom-left: Integrated intensity of C18O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Top-left: PV diagram obtained by integrating the ve- locity along the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Bottom-right: PV diagram ob- tained by integrating the velocity along the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Top right: Spectra associated with the crossing of the two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Appendix E: Upper limits on the column densities for non-detected molecules To obtain upper limits on the abundance of non detected molecules, we first computed the upper limits, Wupp, of the inte- grated intensities : Wupp < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='064 × 3 × rms × dv, where the rms (in K) is the noise level at the dust maximum position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The values are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='06 to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 K depending on the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We assumed a line width FWHM (dv) of 1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The partition functions, provided by CDMS and JPL, are inter- polated for the temperature of the cloud (10 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' For each species,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' we computed the upper limit of the column density following Mangum & Shirley (2015) and including the cosmological back- ground radiation temperature : Ni = 8πk × Qi × f2 i × Wupp × 105 × e Eup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='i T gup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='i × h × c3 × Aul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' where Qi is the partition function of species i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' k is the Boltz- mann constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' fi is the frequency of the transition (MHz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Eup is the upper energy state of the transition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' gup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='i is the upper state degeneracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' T is the temperature of the cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' h is the Planck constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' c is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' and Aul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='i is the Einstein coeffi- cient of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The obtained upper limits are 2 × 1016, 2 × 1013, and 2 × 1013 cm−2 for c-C3H2 (95206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='01 MHz), OCS (97301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='20 MHZ), and HNCO (109905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='60 MHz), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Converted into abundances with the H2 column density at the continuum peak, it gives 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8×10−7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 × 10−10, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 × 10−10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Appendix F: Observed physical parameters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 log10(nH2) 10 25 40 55 70 Av 12 14 16 18 Temperature (K) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: Av as a function of nH2 (cm−3) and temperature (K) observed in L429-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' All these parameters have been computed from the Herschel observations (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' In this section, we compare the different physical parameters observed in L429-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 shows the visual extinction as a function of nH2 (cm−3) and temperature (K) observed in L429-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The visual extinctions range from less than 10 to more than 80, the H2 density from 5×103 to 3×106 cm−3, and the temperature from approximately 12 up to 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 21 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr Appendix G: Goodness of fit for SO, CCS, HC3N, and CN Figure G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 shows the ratio between the modeled and observed gas-phase abundances of SO, CCS, HC3N, and CN as a function of the density for the best times of the four sets of models (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 and Table 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 Low sputtering yield SO CCS HC3N CN Low CR High CR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(nH2 cm 3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 log(Xmod/Xobs) High sputtering yield SO CCS HC3N CN Low CR High CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: Ratio between the modeled (Xmod) and observed gas- phase abundances (Xobs) of SO, CCS, HC3N, and CN as a func- tion of the density for the best times of the four sets of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 22 of 24 Taillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' : Non-thermal desorption of methanol Appendix H: Best time determination for each Av The best time is determined by the lowest distance of disagree- ment, d, defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Each figure represents the results of one set of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' We can see that the higher the density, the lower the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The eight best times are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 23 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' 45157corr 100 101 102 103 104 105 106 107 Time (yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 d Low sputtering yield - Low CR Av=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 K, Av=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2, nH2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K, Av=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, Av=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 K, Av=49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0, nH2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 K, Av=58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 K, Av=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8, nH2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 K, Av=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0e+06 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, 100 101 102 103 104 105 106 107 Time (yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 d Low sputtering yield - High CR Av=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 K, Av=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2, nH2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K, Av=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, Av=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 K, Av=49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0, nH2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 K, Av=58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 K, Av=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8, nH2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 K, Av=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0e+06 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, 100 101 102 103 104 105 106 107 Time (yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 d High sputtering yield - Low CR Av=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 K, Av=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2, nH2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K, Av=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, Av=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 K, Av=49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0, nH2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 K, Av=58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 K, Av=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8, nH2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 K, Av=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0e+06 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, 100 101 102 103 104 105 106 107 Time (yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 d High sputtering yield - High CR Av=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='9 K, Av=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2, nH2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='2e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4 K, Av=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5e+04 cm 2, T = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, Av=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='6 K, Av=49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0, nH2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3e+05 cm 2, T = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1 K, Av=58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='5, nH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='7 K, Av=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='8, nH2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1e+05 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='3 K, Av=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='4, nH2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0e+06 cm 2, T = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='0 K, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content='1: Distance of disagreement d for all eight models as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Each figure represents the result of a set of model as defined in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' The legend gives each physical parameters associated for the model shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Each vertical line represents the lowest disagreement distance associated with each grid of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} +page_content=' Article number, page 24 of 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AzT4oBgHgl3EQfVvwt/content/2301.01288v1.pdf'} diff --git a/KE Research Center Intro/content/tmp_files/load_file.txt b/KE Research 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='硕士5名）。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 中心致力于综合知识库的构建和垂直领域的应用。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 吴信东 教授 之江实验室高级研究专家，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 俄罗斯工程院外籍院士、IEEE Fellow,' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' AAAS Fellow，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据挖掘和知识工程 领域专家，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='曾任美国佛蒙特大学计算 机系主任，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='路易斯安那大学计算机学 院院长，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='明略科技首席科学家 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='中心简介 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='成员背景 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='之江实验室 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ZHEJIANG LAB ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='RESEARCH INSTITUTE OF ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ARTIFICIAL INTELLIGENCE姓名?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 学位 职级 职称。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 专业 院校。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 孙泽懿 博士， 工程专家。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 正高。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 工业工程和运筹学 美国伊利诺伊大学芝加哥分校 黄飞 博士 工程专家。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 无 基础数学微分儿何 浙江大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 江金陵 博士 工程专家。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 无 计算机科学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 丹麦奥尔堡大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 邓祎。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 博士 高级研究专员 中级 电子工程。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 美国伦斯勒理工学院。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 张琛。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 硕士， 高级工程专员。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 无 计算机软件与理论 中国科学技术大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 吴凡 硕士 高级工程专员。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 无 电子工程。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 美国哥伦比亚大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 郭蒙浩 硕士， 工程专员。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 无 导航制导与控制 东北大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 孙山鑫 硕士 工程专员。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 计算机科学与技术。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 哈尔滨工业大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 战凯 硕士 工程专员。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 机械工程。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 大连理工大学。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='综合知识库 2 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='8亿条目，22.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='4亿关系，128个一层科 目，1139个二层科目 目 标 八项关键技术 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 多重关系联想与复合知识索引 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 自然人、数字人、机器人的全 息融合 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 开放环境下的持续学习 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 大规模知识图谱构建 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 多维信息感知、交互与融合 6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 跨媒体智能分析与理解 7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 数字人设计技术 8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 人机交互和双向双轮驱动 阶段性成果 主要研究方法 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' LLM+KG：专用知识的嵌入、大小 模型增强知识图谱、以及协同增强深 度融合符号主义（知识图谱）与联结 主义（神经网络深度学习）的结合 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 跨媒体知识的在线学习框架 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 基于深度学习和多模态知识图谱等 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='AI技术的知识计算 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='之江实验室 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ZHEJIANG ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='LAB ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='RESEARCH ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='INSTITUTE ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='OF ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ARTIFICIAL ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='INTELLIGENCEhace-KO：一片连通、综合、容纳、制衡、演化的知识海洋 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='笃志问题求解，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='贯通古今中外。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Chace-KO（a Connected，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Hybrid，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Accommodating Contained，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='and Evolving Knowledge-Ocean）含有全球最大的综合知识库，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='以跨语言、 多学科、可计算和增殖的知识计算为研究主题，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='以“大”“动”“亮”落地场景为应用特 色，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='以人机协的HAO智能为技术手，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='集成深度学习与图谱构建，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='开拓数据和知识双轮 驱动。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='吴信东 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='2022年10月19日连通、综合 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='容纳、制衡 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='演化的知识海 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='洋知识表示 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识获取 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识推理 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人机协同 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='复合关系结构 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='全息信息互连 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='开放持续学习 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识图谱构建 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='多维信息融合 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='跨媒体推理 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数字人设计 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='双向双轮驱动垂直领域 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='科学文献抽取大模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='于 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='K ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='B ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Q ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='A ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='和 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='L ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='L ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='M ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='资 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Q ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='A ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='图向量结构的新型数据库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作：天文、基因、制药、材料 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='通用大模型+领域知识库+Prompts ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='\uf070 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基于大模型的科学文献抽取调研评测报告 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='\uf070 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基于大模型的科学文献领域垂直数据集与评测平台 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='\uf070 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='国产科学文献抽取大模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='LLM+KG ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作团队：图计算研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='图结构+向量结构+连接缺失的复杂逻辑查询 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='3 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='星 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='地 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='计 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='算 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='多 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='源 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='识 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='融 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='智 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='能 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='航 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='线 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='规 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='划 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作团队：先进计算中心卫星团队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='目标识别+路径规划建模求解 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='之江实验室 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ZHEJIANG LAB ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='RESEARCH INSTITUTE OF ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ARTIFICIAL INTELLIGENCEjourn ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='autho ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='instit ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='acc ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='篇数 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='title ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='year ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='email ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='al ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='r ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ution ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基因 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='5 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='98% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='制药 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='2 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='85.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='7 天文 21 100% 100% 95.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='2% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='材料 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='8 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='95% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='100%遥感卫星 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='部著 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='遥感 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='部署 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='目标检测模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='图像 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='智能感知规划 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='星载计算机 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='冰山等 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='极地环境、 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='船白 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='障碍物 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='参数、航道规则 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='信息 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='等其他因素 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='输入 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='输入 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='电子海图 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='部著 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='输出 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='计划航线和 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='输入 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='航线规划模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='局部避障路径 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='地面计算机知识应用 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识搜索 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识问答 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识推荐 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='关联挖掘 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='智能决策 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识服务接口 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识服务 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='IAS、Bot Alpha、SmartKG等 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='生成式AI模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识服务模型（私有模型） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='企业知识图谱 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='互联网数据 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='（私有知识） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='（公开数据） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据类型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='文档 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='山表格 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='用 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='A ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='图片 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='（）音顿 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='视频admin ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资问答 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我的工资 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资图谱 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据分析 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我是Davis,' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='WendySue，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我2014年的工资是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 欢迎来到工资问答 预测ABAIEDJAMIEL在2023年的工资。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 假设2024年薪总包是288000000元，分配给Zimakas,Nilgun 你可以这样问我 T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的数额是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 请问职级为Professor的人，2015年平均工资是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 我是Davis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='WendvSue，我所在的职级年薪大干50000的人有多 2023 06 28 18:07:40 我是Davis,WendySue，我2014年的工资是多少？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:40 您好，Davis,WendySue的2014年的工资为153000。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:49 请问Davis,WendySue的2015年的工盗是否达到同部门的平均水平？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 202306 28 18:07:49 您好，能够达到！' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:57 请问职级为Professor的人，2015年平均工资是多少？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='询问我任何关于工资的问题 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='发送 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='京公网安备10000001000001号京ICP证010101号互联网新闻信息服务许可证11110110001网络文化经营许可证：京网文2023】1011 001号UserQuery ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='用户问题 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='融合图与向量结构的 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='通用知识库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='新型数据库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='领域知识库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='LLM ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='大语言 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='选代推理>知识融合 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Output ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ChatGPTDownstream tasks ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Query ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Result ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Neural Graph Database ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Neural Query ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Engine ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='QueryPlanner ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='QueryExecutor ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ApproximateGraphQuery ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='AnsweringFunction ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='EmbedQuery ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='GraphStore ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Neural Graph ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Retrieval ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Storage ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Encoder ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='FeatureStore ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Numbers ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Texts ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Images ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='EmbeddingStore技术框架与垂直领域 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='4 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='之江实验室 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ZHEJIANG LAB ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='RESEARCH INSTITUTE OF ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ARTIFICIAL INTELLIGENCEAIGC大模型时代的知识工程技术框架 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='理论创新 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='LLM+KG ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基于几何张量数学理论的 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识工程与生成式语言模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='牵引性创新任务 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='新型数据结构模型 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的互增强与融合路线图 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基于KBQA及语言大模 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='基于大语言模型的 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='多源知识融合的智能航 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='隐 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='型的工资问答系统 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='科学文献抽取与评测平台 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='线规划星地计算平台 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='私与安全保护机制 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识工程示范应用 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作：天文、基因、制药、材料 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作：先进计算中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='3 +1 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='垂直应用领域 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='与核心软件 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='融合向量与图结构的新型数据库 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知识枢纽数据基座 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='合作团队：图计算研究中心工资问答示范应用 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='系统界面 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='涨薪分配算法 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='三大技术亮点 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' KBQA：完整的知识图谱问答引擎 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 涨薪分配：多目标优化决策与机器学习预测综合应用 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 大语言模型：增强图谱查询及NL2Cypher的内容生成 https://ko.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='zhejianglab.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='com/salaryqa ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='5 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='之江实验室 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ZHEJIANG LAB ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='RESEARCH INSTITUTE OF ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='ARTIFICIAL INTELLIGENCEadmin ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资问答 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我的工资 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资图谱 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据分析 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我是Davis,' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='WendySue，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='我2014年的工资是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 欢迎来到工资问答 预测ABAIEDJAMIEL在2023年的工资。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 假设2024年薪总包是288000000元，分配给Zimakas,Nilgun 你可以这样问我 T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的数额是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 请问职级为Professor的人，2015年平均工资是多少?' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 我是Davis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='WendvSue，我所在的职级年薪大干50000的人有多 2023 06 28 18:07:40 我是Davis,WendySue，我2014年的工资是多少？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:40 您好，Davis,WendySue的2014年的工资为153000。' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:49 请问Davis,WendySue的2015年的工盗是否达到同部门的平均水平？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 202306 28 18:07:49 您好，能够达到！' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 06 28 18:07:57 请问职级为Professor的人，2015年平均工资是多少？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='询问我任何关于工资的问题 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='发送 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='京公网安备10000001000001号京ICP证010101号互联网新闻信息服务许可证11110110001网络文化经营许可证：京网文2023】1011 001号工资问答系统架构 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='业务层 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资问答 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资查询 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='图谱展示 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据分析 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='应用实现层 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='QA 问答引擎 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='Cypher查询推理 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='语言大模型内容生成 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='NLP Python工程 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据层 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工资文件(UMV) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='导入 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='数据库 Neo4J ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='萃取 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='知海 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工程底座 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='工程底座用户登录，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='权限，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='APP调度2023 07 0413:31;' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='45 假设2024年薪酬总包是288000000元，分配给Zimakas，NilgunT.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='的数额是多少？' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content=' 2023 07 04.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='13:31:45 ZIMAKAS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='NILGUNT当前职位为ASSISTANTPROFESSOR（COM），根据该职位平均薪酬变化趋势，以及 ZIMAKAS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='NILGUNT个人工资变化趋势，预测2024年在薪酬总包288000000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='0元中的分配数额为17694.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} +page_content='3 18,000 15,000 12,000 9,000 6,000 3,000' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\KE Research Center Intro\\content\\知识工程研究中心汇报20230710.pdf'} diff --git "a/KE Research Center Intro/content/tmp_files/\347\237\245\350\257\206\345\267\245\347\250\213\347\240\224\347\251\266\344\270\255\345\277\203\346\261\207\346\212\24520230710.pdf.txt" "b/KE Research Center Intro/content/tmp_files/\347\237\245\350\257\206\345\267\245\347\250\213\347\240\224\347\251\266\344\270\255\345\277\203\346\261\207\346\212\24520230710.pdf.txt" new file mode 100644 index 0000000000000000000000000000000000000000..5e20c1f2c09691436261e70e277482e3729e4f42 --- /dev/null +++ "b/KE Research Center Intro/content/tmp_files/\347\237\245\350\257\206\345\267\245\347\250\213\347\240\224\347\251\266\344\270\255\345\277\203\346\261\207\346\212\24520230710.pdf.txt" @@ -0,0 +1,401 @@ +人工智能研究院-知识工程研究中心 +2023年7月 + +之江实验室 +ZHEJIANGLAB中心简介 +1 +知识工程研究中心成立于2023年1月18日，现有 +全职员工9人，其中PI3人，工程/研究专员6人 +（博士4名，硕士5名）。 +中心致力于综合知识库的构建和垂直领域的应用。 +吴信东 教授 之江实验室高级研究专家， +俄罗斯工程院外籍院士、IEEE Fellow, +AAAS Fellow，数据挖掘和知识工程 +领域专家，曾任美国佛蒙特大学计算 +机系主任，路易斯安那大学计算机学 +院院长，明略科技首席科学家 +中心简介 +成员背景 + +之江实验室 +人工智能研究院 +ZHEJIANG LAB +RESEARCH INSTITUTE OF +ARTIFICIAL INTELLIGENCE姓名? +学位 +职级 +职称。 +专业 +院校。 +孙泽懿 +博士， +工程专家。 +正高。 +工业工程和运筹学 +美国伊利诺伊大学芝加哥分校 +黄飞 +博士 +工程专家。 +无 +基础数学微分儿何 +浙江大学。 +江金陵 +博士 +工程专家。 +无 +计算机科学。 +丹麦奥尔堡大学。 +邓祎。 +博士 +高级研究专员 +中级 +电子工程。 +美国伦斯勒理工学院。 +张琛。 +硕士， +高级工程专员。 +无 +计算机软件与理论 +中国科学技术大学。 +吴凡 +硕士 +高级工程专员。 +无 +电子工程。 +美国哥伦比亚大学。 +郭蒙浩 +硕士， +工程专员。 +无 +导航制导与控制 +东北大学。 +孙山鑫 +硕士 +工程专员。 +计算机科学与技术。 +哈尔滨工业大学。 +战凯 +硕士 +工程专员。 +机械工程。 +大连理工大学。综合知识库 +2 +4.8亿条目，22.4亿关系，128个一层科 +目，1139个二层科目 +目 标 +八项关键技术 +1. 多重关系联想与复合知识索引 +2. 自然人、数字人、机器人的全 +息融合 +3. 开放环境下的持续学习 +4. 大规模知识图谱构建 +5. 多维信息感知、交互与融合 +6. 跨媒体智能分析与理解 +7. 数字人设计技术 +8. 人机交互和双向双轮驱动 +阶段性成果 +主要研究方法 +1. LLM+KG：专用知识的嵌入、大小 +模型增强知识图谱、以及协同增强深 +度融合符号主义（知识图谱）与联结 +主义（神经网络深度学习）的结合 +2. 跨媒体知识的在线学习框架 +3. 基于深度学习和多模态知识图谱等 +AI技术的知识计算 + +之江实验室 +人工智能研究院 +ZHEJIANG LAB +RESEARCH INSTITUTE OF +ARTIFICIAL INTELLIGENCEhace-KO：一片连通、综合、容纳、制衡、演化的知识海洋 +笃志问题求解，贯通古今中外。Chace-KO（a Connected，Hybrid，Accommodating +Contained，and Evolving Knowledge-Ocean）含有全球最大的综合知识库，以跨语言、 +多学科、可计算和增殖的知识计算为研究主题，以“大”“动”“亮”落地场景为应用特 +色，以人机协的HAO智能为技术手，集成深度学习与图谱构建，开拓数据和知识双轮 +驱动。 +吴信东 2022年10月19日连通、综合 +容纳、制衡 +演化的知识海 +洋知识表示 +知识获取 +知识推理 +人机协同 +复合关系结构 +全息信息互连 +开放持续学习 +知识图谱构建 +多维信息融合 +跨媒体推理 +数字人设计 +双向双轮驱动垂直领域 +科学文献抽取大模型 +基 于 K B Q A 和 L L M 的 工 资 Q A +图向量结构的新型数据库 +合作：天文、基因、制药、材料 +通用大模型+领域知识库+Prompts + +基于大模型的科学文献抽取调研评测报告 + +基于大模型的科学文献领域垂直数据集与评测平台 + +国产科学文献抽取大模型 +LLM+KG +合作团队：图计算研究中心 +图结构+向量结构+连接缺失的复杂逻辑查询 +3 +星 地 计 算 多 源 知 识 融 合 的 智 能 航 +线 规 划 +合作团队：先进计算中心卫星团队 +目标识别+路径规划建模求解 + +之江实验室 +人工智能研究院 +ZHEJIANG LAB +RESEARCH INSTITUTE OF +ARTIFICIAL INTELLIGENCEjourn +autho +instit +acc +篇数 +title +year +email +al +r +ution +基因 +5 +100% +100% +100% +100% +98% +100% +制药 +2 +100% +100% +100% +100% +100% +100% +85.7 +天文 +21 +100% +100% +95.2% +100% +100% +% +材料 +8 +100% +100% +100% +100% +95% +100%遥感卫星 +部著 +遥感 +部署 +目标检测模型 +图像 +智能感知规划 +星载计算机 +冰山等 +极地环境、 +船白 +障碍物 +参数、航道规则 +信息 +等其他因素 +输入 +输入 +电子海图 +部著 +输出 +计划航线和 +输入 +航线规划模型 +局部避障路径 +地面计算机知识应用 +知识搜索 +知识问答 +知识推荐 +关联挖掘 +智能决策 +知识服务接口 +知识服务 +IAS、Bot-Alpha、SmartKG等 +生成式AI模型 +知识服务模型（私有模型） +企业知识图谱 +互联网数据 +（私有知识） +（公开数据） +数据类型 +文档 +山表格 +用 +数据库 +A +图片 +（）音顿 +视频admin +工资问答 +我的工资 +工资图谱 +数据分析 +我是Davis,WendySue，我2014年的工资是多少? +欢迎来到工资问答 +预测ABAIEDJAMIEL在2023年的工资。 +假设2024年薪总包是288000000元，分配给Zimakas,Nilgun +你可以这样问我 +T.的数额是多少? +请问职级为Professor的人，2015年平均工资是多少? +我是Davis.WendvSue，我所在的职级年薪大干50000的人有多 +2023-06-28 18:07:40 +我是Davis,WendySue，我2014年的工资是多少？ +2023-06-28 18:07:40 +您好，Davis,WendySue的2014年的工资为153000。 +2023-06-28 18:07:49 +请问Davis,WendySue的2015年的工盗是否达到同部门的平均水平？ +202306-28 18:07:49 +您好，能够达到！ +2023-06-28 18:07:57 +请问职级为Professor的人，2015年平均工资是多少？ +询问我任何关于工资的问题 +发送 +京公网安备10000001000001号京ICP证010101号互联网新闻信息服务许可证11110110001网络文化经营许可证：京网文2023】1011-001号UserQuery +用户问题 +知识库 +融合图与向量结构的 +通用知识库 +新型数据库 +领域知识库 +LLM +大语言 +模型 +4. +选代推理>知识融合 +Output +ChatGPTDownstream tasks +Query +Result +Neural Graph Database +Neural Query +Engine +QueryPlanner +QueryExecutor +ApproximateGraphQuery +AnsweringFunction +EmbedQuery +GraphStore +Neural Graph +Retrieval +Storage +Encoder +FeatureStore +Numbers +Texts +Images +EmbeddingStore技术框架与垂直领域 +4 + +之江实验室 +人工智能研究院 +ZHEJIANG LAB +RESEARCH INSTITUTE OF +ARTIFICIAL INTELLIGENCEAIGC大模型时代的知识工程技术框架 +理论创新 +LLM+KG +基于几何张量数学理论的 +知识工程与生成式语言模型 +牵引性创新任务 +新型数据结构模型 +的互增强与融合路线图 +基于KBQA及语言大模 +基于大语言模型的 +多源知识融合的智能航 +隐 +型的工资问答系统 +科学文献抽取与评测平台 +线规划星地计算平台 +私与安全保护机制 +知识工程示范应用 +合作：天文、基因、制药、材料 +合作：先进计算中心 +3 +1 +垂直应用领域 +与核心软件 +融合向量与图结构的新型数据库 +知识枢纽数据基座 +合作团队：图计算研究中心工资问答示范应用 +系统界面 +涨薪分配算法 +三大技术亮点 +1. KBQA：完整的知识图谱问答引擎 +2. 涨薪分配：多目标优化决策与机器学习预测综合应用 +3. 大语言模型：增强图谱查询及NL2Cypher的内容生成 +https://ko.zhejianglab.com/salaryqa +5 + +之江实验室 +人工智能研究院 +ZHEJIANG LAB +RESEARCH INSTITUTE OF +ARTIFICIAL INTELLIGENCEadmin +工资问答 +我的工资 +工资图谱 +数据分析 +我是Davis,WendySue，我2014年的工资是多少? +欢迎来到工资问答 +预测ABAIEDJAMIEL在2023年的工资。 +假设2024年薪总包是288000000元，分配给Zimakas,Nilgun +你可以这样问我 +T.的数额是多少? +请问职级为Professor的人，2015年平均工资是多少? +我是Davis.WendvSue，我所在的职级年薪大干50000的人有多 +2023-06-28 18:07:40 +我是Davis,WendySue，我2014年的工资是多少？ +2023-06-28 18:07:40 +您好，Davis,WendySue的2014年的工资为153000。 +2023-06-28 18:07:49 +请问Davis,WendySue的2015年的工盗是否达到同部门的平均水平？ +202306-28 18:07:49 +您好，能够达到！ +2023-06-28 18:07:57 +请问职级为Professor的人，2015年平均工资是多少？ +询问我任何关于工资的问题 +发送 +京公网安备10000001000001号京ICP证010101号互联网新闻信息服务许可证11110110001网络文化经营许可证：京网文2023】1011-001号工资问答系统架构 +业务层 +工资问答 +工资查询 +图谱展示 +数据分析 +应用实现层 +QA 问答引擎 +Cypher查询推理 +语言大模型内容生成 +NLP Python工程 +数据层 +工资文件(UMV) +导入 +数据库 Neo4J +萃取 +知海 +工程底座 +工程底座用户登录，权限，APP调度2023-07-0413:31;45 +假设2024年薪酬总包是288000000元，分配给Zimakas，NilgunT.的数额是多少？ +2023-07-04.13:31:45 +ZIMAKAS.NILGUNT当前职位为ASSISTANTPROFESSOR（COM），根据该职位平均薪酬变化趋势，以及 +ZIMAKAS.NILGUNT个人工资变化趋势，预测2024年在薪酬总包288000000.0元中的分配数额为17694.3 +18,000 +15,000 +12,000 +9,000 +6,000 +3,000 \ No newline at end of file diff --git a/KtFIT4oBgHgl3EQfaisW/content/2301.11257v1.pdf b/KtFIT4oBgHgl3EQfaisW/content/2301.11257v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0db43f6cc28d1888de606e800bb38ed138560ec7 --- /dev/null +++ b/KtFIT4oBgHgl3EQfaisW/content/2301.11257v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:70535ff6ca81a74100bfa45e0024049db797dd8d7b65544eddabaac994dac6c3 +size 1171535 diff --git a/KtFIT4oBgHgl3EQfaisW/vector_store/index.faiss b/KtFIT4oBgHgl3EQfaisW/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7deda391b9210165e7b57f7b84b103d90ff539f6 --- /dev/null +++ b/KtFIT4oBgHgl3EQfaisW/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f66408ca7f9fabce7f0167c219a17f21df1837749eda4510e4b66659cd61f5fd +size 6291501 diff --git a/KtFIT4oBgHgl3EQfaisW/vector_store/index.pkl b/KtFIT4oBgHgl3EQfaisW/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..02e9263ef194695c008f48adbc50f0c82269202e --- /dev/null +++ b/KtFIT4oBgHgl3EQfaisW/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f5cd30ecc12ad776fbe27e672b06cccd132717d070a22e07daf467e4c09988c8 +size 190588 diff --git a/LdAzT4oBgHgl3EQfkf0t/content/2301.01531v1.pdf b/LdAzT4oBgHgl3EQfkf0t/content/2301.01531v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9391296680a84d77da08d65b3345460c14cbb7c7 --- /dev/null +++ b/LdAzT4oBgHgl3EQfkf0t/content/2301.01531v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f72d45a09f1ddc007ce8c7ab9de92695904b81590bb41eba078029de5a4214b7 +size 3850678 diff --git a/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/2301.01537v1.pdf.txt b/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/2301.01537v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..030d901d21ece352517f38b4faef4cc351437736 --- /dev/null +++ b/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/2301.01537v1.pdf.txt @@ -0,0 +1,619 @@ +arXiv:2301.01537v1 [physics.flu-dyn] 4 Jan 2023 +Under consideration for publication in J. Fluid Mech. +1 +Banner appropriate to article type will appear here in typeset article +Intermittency in turbulent emulsions +M. Crialesi-Esposito1, G. Boffetta2, L. Brandt3,4, S. Chibbaro5 and S. Musacchio2 +1INFN, Sezione di Torino, via Pietro Giuria 1, 10125, Torino, Italy +2Dipartimento di Fisica and INFN, Università degli Studi di Torino, via P. Giuria 1, 10125 Torino, Italy. +3FLOW Centre, KTH Royal Institute of Technology, Stockholm, Sweden +4Department of Energy and Process Engineering, Norwegian University of Science and +Technology(NTNU), Trondheim, Norway +5Université Paris-Saclay, CNRS, LISN, 91400 Orsay, France +(Received xx; revised xx; accepted xx) +We investigate the statistics of turbulence in emulsions of two-immiscible fluids of same +density. We compute for the first time velocity increments between points conditioned to +be located in the same phase or in different phases and examine their probability density +functions (PDF) and the associated structure functions (SF). This enables us to demonstrate +that the the presence of the interface reduces the skewness of the PDF at scales below +the Kolmogorov-Hinze scale and therefore the magnitude of the energy flux towards the +dissipative scales, which is quantified by the third-order SF. The analysis of the higher order +SFs shows that multiphase turbulence is more intermittent than single-phase turbulence. In +particular, the local scaling exponents of the SFs display a saturation about the Kolmogorov- +Hinze scale and below, which indicates the presence of large velocity gradients across the +interface. Interestingly, the statistics approach of classic homogeneous isotropic turbulence +when significantly increasing the viscosity of the dispersed phase. +Key words: Emulsions, multiphase turbulence, intermittency, structure functions. +MSC Codes (Optional) Please enter your MSC Codes here +1. Introduction +Emulsions, i.e. mixtures composed of two immiscible (totally or partially) liquids with +similar densities, are extremely common in industrial applications environment such as +pharmaceuticals (Nielloud 2000; Spernath & Aserin 2006), food processing (McClements +2015) and oil production (Kokal & Others 2005; Mandal et al. 2010; Kilpatrick 2012). +Emulsions are also important in geophysical applications: as example, when oil or industrial +wastes spill into water streams (from rivers to oceans), the oil droplet distribution becomes +fundamental for quantifying the environmental damage (Li & Garrett 1998; French-McCay +2004; Gopalan & Katz 2010). + +2 +At very low-volume-fraction, turbulent emulsions are mainly characterized by the breakup +of droplets. The droplet size distribution is produced by the turbulent stresses and the feedback +of the dispersed phase on the carrier flow is small, and often neglected. The dynamics of +droplet breakup, for a dilute emulsion in a homogeneous and isotropic turbulent flow was +initially investigated by Kolmogorov (1949) and Hinze (1955), who derived an expression +for the maximum size of droplets resisting breakup as a function of the flow characteristics +and the fluid properties. This is usually referred to as the Kolmogorov-Hinze (KH) scale. +Recent numerical investigations of droplets, bubbles and emulsions confirmed the general +validity of the KH theory, both in isotropic and homogeneous turbulence (Perlekar et al. +2014; Mukherjee et al. 2019; Rivière et al. 2021; Crialesi-Esposito et al. 2022b; Girotto et al. +2022; Begemann et al. 2022), and in anisotropic flows (Soligo et al. 2019; Rosti et al. 2020; +Pandey et al. 2022), while some theoretical corrections were lately proposed to account for +scale-local nature of the process (Crialesi-Esposito et al. 2022a; Qi et al. 2022). +At finite volume fraction of the dispersed phase, the distribution of the droplet sizes results +from the interplay between breakup and coalescence. In this regime, the presence of droplets +also modulates the underlying turbulence, affecting the flow statistics both at large (Yi et al. +2021; Wang et al. 2022) and small scales (Mukherjee et al. 2019; Freund & Ferrante 2019; +Vela-Martín & Avila 2021; Crialesi-Esposito et al. 2022b). In particular, the presence of a +dispersed phase alters significantly the statistics at the small scales, producing large deviations +from the average values of dissipation and vorticity (Crialesi-Esposito et al. 2022b). +In this work, we address the effects of the dispersed phase on the velocity increments of +the turbulent flow at moderate (10%) and high (50%) volume fractions. We study the role +of the interface separating the two phases by examining the statistics of velocity increments +between two points which are conditioned to be either in the same phase or in different +phases. We find that the most important deviations from the statistics of single-phase flows, +quantified by the PDFs of the velocity increments, are concentrated in regions around the +interface, i.e. when the two points belongs to different phases. Moreover, we show that the +amplitude of the third-order structure function (SF) is reduced because of the contribution +of the points located around the interface. This is associated with a reduction of the flux of +kinetic energy in the turbulent cascade, which, in combination with the surface tension term, +alters significantly the energy transport across scales. Finally, we discuss the effects of the +droplets on the local scaling exponents of the high-order structure functions, which display +a striking saturation at small scales. +The remaining of this paper is organized as follows. In Section 2we introduce the numerical +method adopted for the simulations, Section 3 is devoted to the presentation of the results +and Section 4 summarises the main conclusions. +2. Methodology +We consider the velocity field 풖(풙, 푡) obeying the Navier-Stokes equations +휌 (휕푡풖 + 풖 · ∇푢) = −∇푝 + ∇ · +� +휇 +� +∇푢 + ∇푢푇 �� ++ 풇 휎 + 풇 +(2.1) +and the incompressibility condition ∇ · 풖 = 0. In Equation (2.1) 푝 is the pressure and 휌 +is the density and 휇(풙, 푡) is the local viscosity. The surface tension force is represented by +the term 풇 휎 = 휎휉훿푆풏 where 휎 is the surface tension coefficient, 휉 is the local interface +curvature, 풏 the surface normal unit vector and 훿푆 represents a delta function which ensures +that the surface force is applied at the interface only (Tryggvason et al. 2011). The last +term 풇 is a constant in time body force which sustains turbulence by injecting energy at +large scales. Here, we adopt the so-called ABC forcing (Mininni et al. 2006) which reads + +3 +풇 = (퐴 sin(푘 푓 푧) + 퐶 cos(푘 푓 푦), 퐵 sin(푘 푓 푥) + 퐴 cos(푘 푓 푧), 퐶 sin(푘 푓 푦) + 퐵 cos(푘 푓 푥)). The +forcing scale is given by 퐿 푓 = 2휋/푘 푓 . +We solve Equation (2.1) in a triply-periodic, cubic domanin of size 퐿 = 2휋, discertized +on a staggered uniform Cartesian grid. Spatial derivative are discretised with a second-order +centered finite difference scheme and time integration performed by means of a second-order +Adam-Bashford scheme. To reconstruct the interface, we use the algebraic Volume of Fluid +method, MTHINC, introduced by Ii et al. (2012). A constant-coefficient Poisson equation is +obtained using the pressure splitting method (Dodd & Ferrante 2014), which is solved using +a Fast Fourier Transform direct solver. All the simulations have been performed with the code +FluTAS, described in Crialesi-Esposito et al. (2023), where further details on the numerical +methods employed in this study can be found. +We consider four different cases, all using a fixed ABC forcing with 퐴 = 퐵 = 퐶 = 1 +and 푘 푓 = 2휋/퐿 푓 = 2. The reference single phase (SP) simulation assumes viscosity 휇 = +0.006, corresponding to a Taylor-scale Reynolds number 푅푒휆 = 15푘2/(휈휀)1/2 ≈ 137, with +휀 = 휈⟨(∇풖)2⟩ the energy dissipation rate, 푘 the turbulent kinetic energy and 휈 the kinematic +viscosity. We vary the volume fraction 훼 = 푉푑/푉, defined as the ratio between the volume of +the dispersed phase 푉푑 and the total volume 푉 = 퐿3, and the viscosity ratio 훾 = 휇푑/휇푐. As +regards different volume fractions, we analyze the two cases with 훼 = 0.1 (hereafter MP10) +and 훼 = 0.5 (hereafter MP50), while keeping 훾 = 1 for both cases. Finally, we study the case +훼 = 0.1 and 훾 = 100 (hereafter MPM). For all multiphase (MP) simulations the density ratio +among the two phases is kept equal to 1 and the Weber number 푊푒 = 휌퐿 푓 푢2 +푟푚푠/휎=42.6. +All the simulations are performed at a resolution 푁 = 512 which is sufficient to resolve all +the scales (see Crialesi-Esposito et al. 2022b); statistics are accumulated over several large +eddy turnover times 푇 = 퐿 푓 /푢푟푚푠 once statistical stationary conditions have been reached. +For further details on the simulation setup, we refer the reader to Crialesi-Esposito et al. +(2022b). +3. Statistics of the multiphase flow +In turbulent multiphase flows, part of the kinetic energy of the carrier phase is absorbed at +large scales by the deformation and breakup of the interface of the dispersed phase, while +the coalescence of small droplets, their surface oscillations and relaxation from high local +curvature re-inject energy in the carrier phase at scales smaller than the Kolmogorov-Hinze +scale (Crialesi-Esposito et al. 2022a). The consequences of this complex exchange of energy +between the two phases are evident in the kinetic energy spectrum shown in Figure 1. +Comparing the spectra of a multiphase flow with that of a single-phase flow sustained by the +same forcing, we observe a suppression of energy at low wavenumbers (i.e. large scales) and +an enhancement at high wavenumbers. This effect increases with the volume fraction 훼 of +the dispersed phase (Mukherjee et al. 2019; Crialesi-Esposito et al. 2022b). +Because of the injection of energy at small scales, due to the droplet dynamics, we expect +higher intermittency of the velocity fluctuations in the MP flow than in the SP flow at fixed +amplitude of the external forcing. In order to quantify this effect we compute the probability +density functions (PDF) of the longitudinal velocity increments 훿ℓ푢 = (풖(풙2)−풖(풙1))·(풙2− +풙1)/ℓ at distance ℓ = |풙2 − 풙1|. The comparison of the PDFs at two scales within the inertial +range, shown in Figure 2, confirms that the velocity increments have larger fluctuations in +the case of MP flows, in particular at smaller values of ℓ. This effect increases with the +concentration 훼 of the dispersed phase. We also observe that in the case 훾 = 100 (i.e. when +the dispersed phase is much more viscous than the carrier phase) the effect of the droplets on +the velocity increments vanishes due to the damping of fluctuations in the dispersed phase, +and we recover the statistics of the SP flow. + +4 +100 +101 +102 +κ +10−8 +10−6 +10−4 +10−2 +100 +E(κ) +MP10 +MP50 +SP +-5/3 +Figure 1: Kinetic energy spectra of SP flow (black continuous line), and MP flows at +훼 = 0.1 (red dashed line) and 훼 = 0.5 (blue dash-dotted line) +-20.0 +-10.0 +0.0 +10.0 +20.0 +∂uℓ/σuℓ,SP +10−9 +10−7 +10−5 +10−3 +10−1 +p.d.f. +ℓ/Lf =0.03 +(a) +-12.0 +-6.0 +0.0 +6.0 +12.0 +∂uℓ/σuℓ,SP +10−9 +10−7 +10−5 +10−3 +10−1 +p.d.f. +ℓ/Lf =0.12 +SP +MP10 +MP50 +MPM +(b) +Figure 2: PDF of velocity increments at distance ℓ = 0.03퐿 푓 (left panel) and ℓ = 0.12퐿 푓 +(right panel), normalized by the standard deviation of the SP case. The single-phase +Kolmogorov scale is at ℓ ≈ 0.008퐿 푓 +Note that the PDF shown in Figure 2 are computed over the full simulation domain, i.e. +the velocity increments are computed among points 풙1,2 which can belong to both phases +unconditionally. To understand the role of the interface in the turbulent statistics, we therefore +compute the PDF of the velocity increments conditioned to points belonging to the same +or to different phases. Hence, we introduce three different PDFs of the velocity increments +depending on which phase the two points 풙1 and 풙2 belong to. We denote by 푃푐푐, 푃푑푑 and +푃푐푑 the PDFs relative to points belonging only to the carrier phase 푐, only to the dilute phase +푑 and to both phases, respectively. We remark that, for small values of 훼, the statistics in the +two phases are different. For 훼 = 0.5 and 훾 = 1 the two phases are equivalent and therefore +푃푑푑 = 푃푐푐. +Figure 3 (panels a, b) shows the conditional PDF (normalized with the corresponding +variance of the SP case) pertaining the simulation with volume fraction 훼 = 0.1. First, we +note that the PDF of the carrier phase is not too far from that of the SP case (and this is the case + +5 +-8.0 +0.0 +8.0 +∂uℓ/σuℓ,SP +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +p.d.f. +ℓ/Lf =0.03 +(a) +-4.0 +0.0 +4.0 +∂uℓ/σuℓ,SP +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +p.d.f. +ℓ/Lf =0.12 +sp +cc +dd +cd +(b) +-8.0 +0.0 +8.0 +∂uℓ/σuℓ,SP +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +p.d.f. +ℓ/Lf =0.03 +(c) +-4.0 +0.0 +4.0 +∂uℓ/σuℓ,SP +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +p.d.f. +ℓ/Lf =0.12 +sp +cc +dd +cd +(d) +Figure 3: PDFs of velocity increments conditioned to the phases on which the two +velocities are measured: 푐푐 (both points in the carrier phase, red line), 푑푑 (both points in +the dispersed phase, orange line), 푐푑 (one point in each phase, blue line). Upper panel: +Simulation MP10 with 훼 = 0.1 and 훾 = 1. Lower panel: Simulation MP50 with 훼 = 0.5 +and 훾 = 1. Black line: PDF of velocity increments for a single-phase simulation with the +same parameters of the MP simulation. Dashed black line: Gaussian distribution. All the +PDFs are rescaled with the variance of the SP case. +also for the variance). In the dispersed phase, on the contrary, velocity increments develop +relatively larger tails at small separations. Remarkably, the PDF 푃푐푑 develops the largest +tails at small scale (panel a), a clear indication of the role of the interface for small-scale +intermittency in MP flows. +Similar observations can be made for the emulsion with 훼 = 0.5, shown in Figure 3 +(panels c, d). As expected 푃푐푐 = 푃푑푑, and also in this case the data show that the leading +contribution to the increased intermittency at small scales comes from velocity increments +across the interface, 푃푐푑. Note that, although the shapes of 푃푐푑 are similar for MP10 and +MP50, their contribution to the overall flow statistics is different because of the different +statistical weight (i.e. the different extension of the total interface). +A remarkable feature shown in Figure 3 is that the skewness of 푃푐푑 at small scales is +opposite (i.e. positive) to that of 푃푐푐. We remind that the sign of the skweness is linked to +the direction of the turbulent energy cascade via the third-order velocity structure function +(SF) defined as 푆3(ℓ) = ⟨(훿ℓ푢)3⟩. In the case of SP flows, under the assumption of statistical + +6 +10−2 +10−1 +ℓ/Lf +-0.80 +-0.60 +-0.40 +-0.20 +0.00 +S3 +cc +dd +dc +MP +SP +(a) +10−2 +10−1 +ℓ/Lf +-0.80 +-0.60 +-0.40 +-0.20 +0.00 +S3 +cc +dd +dc +MP +SP +(b) +Figure 4: Third-order stucture function 푆3(ℓ) averaged on the whole domain (violet line) +or conditioned on the two phases of the flows. Simulation with 훼 = 0.1 (left panel) +훼 = 0.5 (right panel) and 훾 = 1. The black line represents the SP case. +stationariety, homogeneity and isotropy, the Kolmogorov 4/5 law gives 푆3(ℓ) = −(4/5)휀ℓ, +where the viscous energy dissipation rate 휀 is equal to the flux of the turbulent cascade +(Frisch 1995). The negative skewness of the PDF of the longitudinal velocity increments is +therefore related to the direction of the energy transfer and the negative amplitude of 푆3(ℓ) +is proportional to the energy flux. +In MP flows, because of the opposite sign of the skewness of 푃푐푑 with respect to 푃푐푐, we +expect that the presence of the interface reduces the energy flux associated to the turbulent +cascade. This can be quantified by looking at the third-order velocity structure function +푆3(ℓ) = ⟨(훿ℓ푢)3⟩, whose average can be unconditioned, or conditioned to two points +belonging either to the carrier phase 푆푐푐 +3 (ℓ), or to the dispersed phase 푆푑푑 +3 (ℓ), or points +located on different sides of the interface 푆푑푐 +3 (ℓ). These are shown in Figure 4. +We see in the figure that the third-order SF of the MP turbulent flows is qualitatively +similar to the SP flow when averaged over the whole domain, yet with a smaller amplitude. +This is due the fact that part of the turbulence energy is used to break the interface and the +direct transfer of energy to small scales is reduced (Crialesi-Esposito et al. 2022b). If we +consider the same quantity averaged over one of the two phases only: 푆푐푐 +3 (ℓ) and 푆푑푑 +3 (ℓ), +which are equivalent in a binary flow, the magnitude of the flux increases and approaches the +SP limit, indicating that the turbulent cascade is not significantly affected when considering +flow structures living in one of the two phases. On the contrary, the flux across two points +belonging to different phases is strongly suppressed: the associated 푆푑푐 +3 (ℓ) is closer to zero +and even changes sign at intermediate scales for 훼 = 0.5 (consistently with what suggested +in Figure 3). The physical interpretation is that the interface “decouples” the velocity fields +in the two phases which become less correlated and therefore with a reduced energy flux, +signaled by the reduction of 푆3(ℓ). The precise behavior of 푆푑푐 +3 (ℓ) depends on the value of +훼, as shown by the comparison with the case 훼 = 0.1 (see Figure 4, left panel). Positive +values of 푆푑푐 +3 (ℓ), hint however to the possibility of scale-local backscatter, which becomes +more relevant at increasing volume fractions. Nevertheless, the reduction of the energy flux +at intermediate scales is a general feature, independent on 훼. +The effects of the presence of a dispersed phase on the statistics of the velocity fluctuations +affects also the scaling behavior of the structure functions of the absolute values of the +longitudinal velocity increments defined as 푆푎 +푝(ℓ) = ⟨|훿ℓ푢| 푝⟩. It is well known that in SP + +7 +10−2 +10−1 +ℓ/Lf +0.00 +0.60 +1.20 +1.80 +2.40 +3.00 +ζp +ℓ /ζ3 +ℓ +(a) +10−2 +10−1 +ℓ/Lf +0.00 +0.60 +1.20 +1.80 +2.40 +3.00 +ζp +ℓ /ζ3 +ℓ +(b) +10−2 +10−1 +ℓ/Lf +0.00 +0.60 +1.20 +1.80 +2.40 +3.00 +ζp +ℓ /ζ3 +ℓ +(c) +10−2 +10−1 +ℓ/Lf +0.00 +0.60 +1.20 +1.80 +2.40 +3.00 +ζp +ℓ /ζ3 +ℓ +(d) +Figure 5: Local scaling exponents 휁푝 of the structure functions, for the single phase +(Upper left panel), case MP10 at 훼 = 0.1 (Upper right panel), case MP50 at 훼 = 0.5 +(Lower left panel), case MPM at 훾 = 100 (Lower right panel). In each figure, the +exponents of different order 푝 assume increasing values. Vertical black dashed lines +represent the Kolmogorov-Hinze scale, computed in Crialesi-Esposito et al. (2022a). +Curves for 푝 = 3 is omitted. +flows the SFs display a power-law behavior 푆푎 +푝(ℓ) ∼ ℓ휁 푝 at scales ℓ in the inertial range +(Frisch 1995). In this context, intermittency manifests in the non-linear behavior of the scaling +exponents: 휁 푝 ≠ 푝/3. In the MP flows, because of the different physical processes which +occur at scales larger and smaller than the Kolmogorov-Hinze scale (dominated by brak- +up and coalescence respectively), we expect to observe a more complex scaling behavior. +To address this issue, we compute the local scaling exponents defined as the logarithmic +derivative of the SFs 휁 푝 +ℓ = d log(푆푎 +푝(ℓ))/d log(ℓ) and here applied to multiphase flows for +the first time. +The local scaling exponents 휁 푝 +ℓ are displayed for 푝 ⩽ 8 in figure 5, where they are divided +by the reference scaling exponent 휁3 +ℓ of the third order SF. In the SP case, panel a, we find +that the ratios 휁 푝 +ℓ /휁3 +ℓ are almost constant in the inertial range 0.09 ⩽ ℓ/퐿 푓 ⩽ 0.32. In the +MP flows, the value of the exponents are a little smaller but comparable to that of the SP +case only at large scales; we observe a dramatic decrease of the scaling exponents at scales ℓ +smaller than the KH scale 퐿퐾 퐻 ≈ 0.14퐿 푓 for the case MP10 (panel b) and 퐿퐾 퐻 ≈ 0.19퐿 푓 + +8 +for the case MP50 (panel b) (see vertical line in figure). In particular, we observe a striking +saturation of the scaling exponents of the high-order SF with 푝 ⩾ 5 at scales ℓ ≃ 0.02퐿 푓 for +the MP50 case and ℓ ≃ 0.04퐿 푓 for the MP10 case. The saturation of the scaling exponents +of the high order SFs reveals the presence of strong velocity differences across the interface +between the two phases, which originates from the pressure jump at the interface caused by +the surface tension forces. +Note also that the saturation of the exponents is not observed when the dispersed phase +presents higher viscosity, case MPM in panel d. This is consistent with the previous +observations in Figure 2, showing that when velocity gradients across the interface are +significantly reduced (e.g. by higher viscosity) no exponent saturation is observed. +4. Conclusions +We have discussed intermittency and scaling exponent obtained from direct numerical +simulations (DNS) of turbulent emulsions at moderate (10%) and high (50%) volume +fractions and two different values of the viscosity contrast. As observed in previous +works (Perlekar 2019; Pandey et al. 2020; Crialesi-Esposito et al. 2022b), the presence of +a deformable interface increases the intermittency in the flow and the energy content at small +scales, when the surface tension offers an alternative path for energy transport across scales. +By investigating the statistics of the velocity increments conditioned to points belonging +to a single phase or to different phases, we demonstrate that the increased intermittency is +mostly due to the presence of strong velocity differences across the interface between the +carrier and the dispersed phase. +We also show that the presence of the dispersed phase causes a decrease of the negative +skewness of the PDF of the longitudinal velocity increments. This is associated with a +reduction of the flux of the kinetic energy from the forcing scale to the viscous scales. In +other words, the presence of a deformable interface affects the vortex stretching and tilting +associated to the classic turbulent energy cascade of single-phase flows. +This effect becomes remarkable at the highest volume fraction considered here, when the +flux related to points lying on either side of the interface gives a positive contribution to +the distribution skewness. This suggests a not-negligible backscatter in multiphase flows, +expected in proximity of the interface separating the two fluids. We interpret this reduced +flux as due to the absorption and dissipation of part of the kinetic energy of the turbulent +flow by the deformation and break-up of drops of the dispersed phase. +Finally, to understand the local properties of turbulence, we have analysed the longitudinal +Structure Functions at higher orders. Interestingly, at scales larger than the Kolmogorov- +Hinze, the exponents are only slightly smaller than in the single-phase flow, which implies +increased intermittency, yet a similar anomalous scaling. More importantly, we report a neat +saturation of the exponents for structure functions higher than 3 at scales smaller than the +Kolmogorov-Hinze length. This is typically related to a strongly intermittent dynamics and +to the presence of jumps, here due to the pressure differences across the interface induced by +the surface tension. +A further demonstration that the interface is responsible of the increased intermittency +is given by the results for the flow at viscosity ratio 훾 = 100. In this case, small-scale +fluctuations are damped, especially in the more viscous dispersed phase, and the statistics +approach those of the single-phase turbulence with no exponent saturation. +These observations may prove fundamental for understanding small scale dynamics in +multiphase flows and for their future sub-grid modelling. Indeed, our results indicate that a +correct model would need to account for the reduction of the energy fluxes near an interface. +Moreover, we have shown that the turbulence statistics approach those of the single-phase + +9 +flow when the droplets consist of a highly viscous fluid. 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LB acknowledge the support from the Swedish Research Council via the multidisciplinary +research environment INTERFACE, Hybrid multiscalemodelling of transport phenomena for energy efficient +processes, Grant no. 2016-06119. +Declaration of interests. The authors report no conflict of interest. +Data availability statement. Data are available from the corresponding author upon reasonable request. + diff --git a/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/load_file.txt b/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29ad2dad448d58618311bce9d64c2995a328f60a --- /dev/null +++ b/M9AzT4oBgHgl3EQfkv2l/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf,len=433 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='01537v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 1 Banner appropriate to article type will appear here in typeset article Intermittency in turbulent emulsions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Crialesi-Esposito1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Boffetta2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Brandt3,4, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Chibbaro5 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Musacchio2 1INFN, Sezione di Torino, via Pietro Giuria 1, 10125, Torino, Italy 2Dipartimento di Fisica and INFN, Università degli Studi di Torino, via P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Giuria 1, 10125 Torino, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 3FLOW Centre, KTH Royal Institute of Technology, Stockholm, Sweden 4Department of Energy and Process Engineering, Norwegian University of Science and Technology(NTNU), Trondheim, Norway 5Université Paris-Saclay, CNRS, LISN, 91400 Orsay, France (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' accepted xx) We investigate the statistics of turbulence in emulsions of two-immiscible fluids of same density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We compute for the first time velocity increments between points conditioned to be located in the same phase or in different phases and examine their probability density functions (PDF) and the associated structure functions (SF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This enables us to demonstrate that the the presence of the interface reduces the skewness of the PDF at scales below the Kolmogorov-Hinze scale and therefore the magnitude of the energy flux towards the dissipative scales, which is quantified by the third-order SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The analysis of the higher order SFs shows that multiphase turbulence is more intermittent than single-phase turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In particular, the local scaling exponents of the SFs display a saturation about the Kolmogorov- Hinze scale and below, which indicates the presence of large velocity gradients across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Interestingly, the statistics approach of classic homogeneous isotropic turbulence when significantly increasing the viscosity of the dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Key words: Emulsions, multiphase turbulence, intermittency, structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' MSC Codes (Optional) Please enter your MSC Codes here 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Introduction Emulsions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' mixtures composed of two immiscible (totally or partially) liquids with similar densities, are extremely common in industrial applications environment such as pharmaceuticals (Nielloud 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Spernath & Aserin 2006), food processing (McClements 2015) and oil production (Kokal & Others 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Kilpatrick 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Emulsions are also important in geophysical applications: as example, when oil or industrial wastes spill into water streams (from rivers to oceans), the oil droplet distribution becomes fundamental for quantifying the environmental damage (Li & Garrett 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' French-McCay 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Gopalan & Katz 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2 At very low-volume-fraction, turbulent emulsions are mainly characterized by the breakup of droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The droplet size distribution is produced by the turbulent stresses and the feedback of the dispersed phase on the carrier flow is small, and often neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The dynamics of droplet breakup, for a dilute emulsion in a homogeneous and isotropic turbulent flow was initially investigated by Kolmogorov (1949) and Hinze (1955), who derived an expression for the maximum size of droplets resisting breakup as a function of the flow characteristics and the fluid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is usually referred to as the Kolmogorov-Hinze (KH) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Recent numerical investigations of droplets, bubbles and emulsions confirmed the general validity of the KH theory, both in isotropic and homogeneous turbulence (Perlekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Rivière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Girotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Begemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022), and in anisotropic flows (Soligo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Rosti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022), while some theoretical corrections were lately proposed to account for scale-local nature of the process (Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' At finite volume fraction of the dispersed phase, the distribution of the droplet sizes results from the interplay between breakup and coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In this regime, the presence of droplets also modulates the underlying turbulence, affecting the flow statistics both at large (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022) and small scales (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Freund & Ferrante 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Vela-Martín & Avila 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In particular, the presence of a dispersed phase alters significantly the statistics at the small scales, producing large deviations from the average values of dissipation and vorticity (Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In this work, we address the effects of the dispersed phase on the velocity increments of the turbulent flow at moderate (10%) and high (50%) volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We study the role of the interface separating the two phases by examining the statistics of velocity increments between two points which are conditioned to be either in the same phase or in different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We find that the most important deviations from the statistics of single-phase flows, quantified by the PDFs of the velocity increments, are concentrated in regions around the interface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' when the two points belongs to different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Moreover, we show that the amplitude of the third-order structure function (SF) is reduced because of the contribution of the points located around the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is associated with a reduction of the flux of kinetic energy in the turbulent cascade, which, in combination with the surface tension term, alters significantly the energy transport across scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Finally, we discuss the effects of the droplets on the local scaling exponents of the high-order structure functions, which display a striking saturation at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The remaining of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In Section 2we introduce the numerical method adopted for the simulations, Section 3 is devoted to the presentation of the results and Section 4 summarises the main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Methodology We consider the velocity field 풖(풙, 푡) obeying the Navier-Stokes equations 휌 (휕푡풖 + 풖 · ∇푢) = −∇푝 + ∇ · � 휇 � ∇푢 + ∇푢푇 �� + 풇 휎 + 풇 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1) and the incompressibility condition ∇ · 풖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1) 푝 is the pressure and 휌 is the density and 휇(풙, 푡) is the local viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The surface tension force is represented by the term 풇 휎 = 휎휉훿푆풏 where 휎 is the surface tension coefficient, 휉 is the local interface curvature, 풏 the surface normal unit vector and 훿푆 represents a delta function which ensures that the surface force is applied at the interface only (Tryggvason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The last term 풇 is a constant in time body force which sustains turbulence by injecting energy at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Here, we adopt the so-called ABC forcing (Mininni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2006) which reads 3 풇 = (퐴 sin(푘 푓 푧) + 퐶 cos(푘 푓 푦), 퐵 sin(푘 푓 푥) + 퐴 cos(푘 푓 푧), 퐶 sin(푘 푓 푦) + 퐵 cos(푘 푓 푥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The forcing scale is given by 퐿 푓 = 2휋/푘 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We solve Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1) in a triply-periodic, cubic domanin of size 퐿 = 2휋, discertized on a staggered uniform Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Spatial derivative are discretised with a second-order centered finite difference scheme and time integration performed by means of a second-order Adam-Bashford scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' To reconstruct the interface, we use the algebraic Volume of Fluid method, MTHINC, introduced by Ii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' A constant-coefficient Poisson equation is obtained using the pressure splitting method (Dodd & Ferrante 2014), which is solved using a Fast Fourier Transform direct solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' All the simulations have been performed with the code FluTAS, described in Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' (2023), where further details on the numerical methods employed in this study can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We consider four different cases, all using a fixed ABC forcing with 퐴 = 퐵 = 퐶 = 1 and 푘 푓 = 2휋/퐿 푓 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The reference single phase (SP) simulation assumes viscosity 휇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='006, corresponding to a Taylor-scale Reynolds number 푅푒휆 = 15푘2/(휈휀)1/2 ≈ 137, with 휀 = 휈⟨(∇풖)2⟩ the energy dissipation rate, 푘 the turbulent kinetic energy and 휈 the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We vary the volume fraction 훼 = 푉푑/푉, defined as the ratio between the volume of the dispersed phase 푉푑 and the total volume 푉 = 퐿3, and the viscosity ratio 훾 = 휇푑/휇푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' As regards different volume fractions, we analyze the two cases with 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 (hereafter MP10) and 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 (hereafter MP50), while keeping 훾 = 1 for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Finally, we study the case 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 and 훾 = 100 (hereafter MPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' For all multiphase (MP) simulations the density ratio among the two phases is kept equal to 1 and the Weber number 푊푒 = 휌퐿 푓 푢2 푟푚푠/휎=42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' All the simulations are performed at a resolution 푁 = 512 which is sufficient to resolve all the scales (see Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' statistics are accumulated over several large eddy turnover times 푇 = 퐿 푓 /푢푟푚푠 once statistical stationary conditions have been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' For further details on the simulation setup, we refer the reader to Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Statistics of the multiphase flow In turbulent multiphase flows, part of the kinetic energy of the carrier phase is absorbed at large scales by the deformation and breakup of the interface of the dispersed phase, while the coalescence of small droplets, their surface oscillations and relaxation from high local curvature re-inject energy in the carrier phase at scales smaller than the Kolmogorov-Hinze scale (Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The consequences of this complex exchange of energy between the two phases are evident in the kinetic energy spectrum shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Comparing the spectra of a multiphase flow with that of a single-phase flow sustained by the same forcing, we observe a suppression of energy at low wavenumbers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' large scales) and an enhancement at high wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This effect increases with the volume fraction 훼 of the dispersed phase (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Because of the injection of energy at small scales, due to the droplet dynamics, we expect higher intermittency of the velocity fluctuations in the MP flow than in the SP flow at fixed amplitude of the external forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In order to quantify this effect we compute the probability density functions (PDF) of the longitudinal velocity increments 훿ℓ푢 = (풖(풙2)−풖(풙1))·(풙2− 풙1)/ℓ at distance ℓ = |풙2 − 풙1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The comparison of the PDFs at two scales within the inertial range, shown in Figure 2, confirms that the velocity increments have larger fluctuations in the case of MP flows, in particular at smaller values of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This effect increases with the concentration 훼 of the dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We also observe that in the case 훾 = 100 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' when the dispersed phase is much more viscous than the carrier phase) the effect of the droplets on the velocity increments vanishes due to the damping of fluctuations in the dispersed phase, and we recover the statistics of the SP flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 4 100 101 102 κ 10−8 10−6 10−4 10−2 100 E(κ) MP10 MP50 SP 5/3 Figure 1: Kinetic energy spectra of SP flow (black continuous line), and MP flows at 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 (red dashed line) and 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 (blue dash-dotted line) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−9 10−7 10−5 10−3 10−1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='03 (a) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−9 10−7 10−5 10−3 10−1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='12 SP MP10 MP50 MPM (b) Figure 2: PDF of velocity increments at distance ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='03퐿 푓 (left panel) and ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='12퐿 푓 (right panel), normalized by the standard deviation of the SP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The single-phase Kolmogorov scale is at ℓ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='008퐿 푓 Note that the PDF shown in Figure 2 are computed over the full simulation domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' the velocity increments are computed among points 풙1,2 which can belong to both phases unconditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' To understand the role of the interface in the turbulent statistics, we therefore compute the PDF of the velocity increments conditioned to points belonging to the same or to different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Hence, we introduce three different PDFs of the velocity increments depending on which phase the two points 풙1 and 풙2 belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We denote by 푃푐푐, 푃푑푑 and 푃푐푑 the PDFs relative to points belonging only to the carrier phase 푐, only to the dilute phase 푑 and to both phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We remark that, for small values of 훼, the statistics in the two phases are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' For 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 and 훾 = 1 the two phases are equivalent and therefore 푃푑푑 = 푃푐푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Figure 3 (panels a, b) shows the conditional PDF (normalized with the corresponding variance of the SP case) pertaining the simulation with volume fraction 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' First, we note that the PDF of the carrier phase is not too far from that of the SP case (and this is the case 5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='03 (a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='12 sp cc dd cd (b) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='03 (c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='0 ∂uℓ/σuℓ,SP 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' ℓ/Lf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='12 sp cc dd cd (d) Figure 3: PDFs of velocity increments conditioned to the phases on which the two velocities are measured: 푐푐 (both points in the carrier phase, red line), 푑푑 (both points in the dispersed phase, orange line), 푐푑 (one point in each phase, blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Upper panel: Simulation MP10 with 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 and 훾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Lower panel: Simulation MP50 with 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 and 훾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Black line: PDF of velocity increments for a single-phase simulation with the same parameters of the MP simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Dashed black line: Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' All the PDFs are rescaled with the variance of the SP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' also for the variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In the dispersed phase, on the contrary, velocity increments develop relatively larger tails at small separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Remarkably, the PDF 푃푐푑 develops the largest tails at small scale (panel a), a clear indication of the role of the interface for small-scale intermittency in MP flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Similar observations can be made for the emulsion with 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5, shown in Figure 3 (panels c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' As expected 푃푐푐 = 푃푑푑, and also in this case the data show that the leading contribution to the increased intermittency at small scales comes from velocity increments across the interface, 푃푐푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Note that, although the shapes of 푃푐푑 are similar for MP10 and MP50, their contribution to the overall flow statistics is different because of the different statistical weight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' the different extension of the total interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' A remarkable feature shown in Figure 3 is that the skewness of 푃푐푑 at small scales is opposite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' positive) to that of 푃푐푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We remind that the sign of the skweness is linked to the direction of the turbulent energy cascade via the third-order velocity structure function (SF) defined as 푆3(ℓ) = ⟨(훿ℓ푢)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In the case of SP flows, under the assumption of statistical 6 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 S3 cc dd dc MP SP (a) 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 S3 cc dd dc MP SP (b) Figure 4: Third-order stucture function 푆3(ℓ) averaged on the whole domain (violet line) or conditioned on the two phases of the flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Simulation with 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 (left panel) 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 (right panel) and 훾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The black line represents the SP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' stationariety, homogeneity and isotropy, the Kolmogorov 4/5 law gives 푆3(ℓ) = −(4/5)휀ℓ, where the viscous energy dissipation rate 휀 is equal to the flux of the turbulent cascade (Frisch 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The negative skewness of the PDF of the longitudinal velocity increments is therefore related to the direction of the energy transfer and the negative amplitude of 푆3(ℓ) is proportional to the energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In MP flows, because of the opposite sign of the skewness of 푃푐푑 with respect to 푃푐푐, we expect that the presence of the interface reduces the energy flux associated to the turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This can be quantified by looking at the third-order velocity structure function 푆3(ℓ) = ⟨(훿ℓ푢)3⟩, whose average can be unconditioned, or conditioned to two points belonging either to the carrier phase 푆푐푐 3 (ℓ), or to the dispersed phase 푆푑푑 3 (ℓ), or points located on different sides of the interface 푆푑푐 3 (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' These are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We see in the figure that the third-order SF of the MP turbulent flows is qualitatively similar to the SP flow when averaged over the whole domain, yet with a smaller amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is due the fact that part of the turbulence energy is used to break the interface and the direct transfer of energy to small scales is reduced (Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' If we consider the same quantity averaged over one of the two phases only: 푆푐푐 3 (ℓ) and 푆푑푑 3 (ℓ), which are equivalent in a binary flow, the magnitude of the flux increases and approaches the SP limit, indicating that the turbulent cascade is not significantly affected when considering flow structures living in one of the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' On the contrary, the flux across two points belonging to different phases is strongly suppressed: the associated 푆푑푐 3 (ℓ) is closer to zero and even changes sign at intermediate scales for 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 (consistently with what suggested in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The physical interpretation is that the interface “decouples” the velocity fields in the two phases which become less correlated and therefore with a reduced energy flux, signaled by the reduction of 푆3(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The precise behavior of 푆푑푐 3 (ℓ) depends on the value of 훼, as shown by the comparison with the case 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 (see Figure 4, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Positive values of 푆푑푐 3 (ℓ), hint however to the possibility of scale-local backscatter, which becomes more relevant at increasing volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Nevertheless, the reduction of the energy flux at intermediate scales is a general feature, independent on 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The effects of the presence of a dispersed phase on the statistics of the velocity fluctuations affects also the scaling behavior of the structure functions of the absolute values of the longitudinal velocity increments defined as 푆푎 푝(ℓ) = ⟨|훿ℓ푢| 푝⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' It is well known that in SP 7 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 ζp ℓ /ζ3 ℓ (a) 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 ζp ℓ /ζ3 ℓ (b) 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 ζp ℓ /ζ3 ℓ (c) 10−2 10−1 ℓ/Lf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00 ζp ℓ /ζ3 ℓ (d) Figure 5: Local scaling exponents 휁푝 of the structure functions, for the single phase (Upper left panel), case MP10 at 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='1 (Upper right panel), case MP50 at 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='5 (Lower left panel), case MPM at 훾 = 100 (Lower right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In each figure, the exponents of different order 푝 assume increasing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Vertical black dashed lines represent the Kolmogorov-Hinze scale, computed in Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Curves for 푝 = 3 is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' flows the SFs display a power-law behavior 푆푎 푝(ℓ) ∼ ℓ휁 푝 at scales ℓ in the inertial range (Frisch 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In this context, intermittency manifests in the non-linear behavior of the scaling exponents: 휁 푝 ≠ 푝/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In the MP flows, because of the different physical processes which occur at scales larger and smaller than the Kolmogorov-Hinze scale (dominated by brak- up and coalescence respectively), we expect to observe a more complex scaling behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' To address this issue, we compute the local scaling exponents defined as the logarithmic derivative of the SFs 휁 푝 ℓ = d log(푆푎 푝(ℓ))/d log(ℓ) and here applied to multiphase flows for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The local scaling exponents 휁 푝 ℓ are displayed for 푝 ⩽ 8 in figure 5, where they are divided by the reference scaling exponent 휁3 ℓ of the third order SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In the SP case, panel a, we find that the ratios 휁 푝 ℓ /휁3 ℓ are almost constant in the inertial range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='09 ⩽ ℓ/퐿 푓 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In the MP flows, the value of the exponents are a little smaller but comparable to that of the SP case only at large scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' we observe a dramatic decrease of the scaling exponents at scales ℓ smaller than the KH scale 퐿퐾 퐻 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='14퐿 푓 for the case MP10 (panel b) and 퐿퐾 퐻 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='19퐿 푓 8 for the case MP50 (panel b) (see vertical line in figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In particular, we observe a striking saturation of the scaling exponents of the high-order SF with 푝 ⩾ 5 at scales ℓ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='02퐿 푓 for the MP50 case and ℓ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='04퐿 푓 for the MP10 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The saturation of the scaling exponents of the high order SFs reveals the presence of strong velocity differences across the interface between the two phases, which originates from the pressure jump at the interface caused by the surface tension forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Note also that the saturation of the exponents is not observed when the dispersed phase presents higher viscosity, case MPM in panel d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is consistent with the previous observations in Figure 2, showing that when velocity gradients across the interface are significantly reduced (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' by higher viscosity) no exponent saturation is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Conclusions We have discussed intermittency and scaling exponent obtained from direct numerical simulations (DNS) of turbulent emulsions at moderate (10%) and high (50%) volume fractions and two different values of the viscosity contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' As observed in previous works (Perlekar 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Crialesi-Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022b), the presence of a deformable interface increases the intermittency in the flow and the energy content at small scales, when the surface tension offers an alternative path for energy transport across scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' By investigating the statistics of the velocity increments conditioned to points belonging to a single phase or to different phases, we demonstrate that the increased intermittency is mostly due to the presence of strong velocity differences across the interface between the carrier and the dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We also show that the presence of the dispersed phase causes a decrease of the negative skewness of the PDF of the longitudinal velocity increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is associated with a reduction of the flux of the kinetic energy from the forcing scale to the viscous scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In other words, the presence of a deformable interface affects the vortex stretching and tilting associated to the classic turbulent energy cascade of single-phase flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This effect becomes remarkable at the highest volume fraction considered here, when the flux related to points lying on either side of the interface gives a positive contribution to the distribution skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This suggests a not-negligible backscatter in multiphase flows, expected in proximity of the interface separating the two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' We interpret this reduced flux as due to the absorption and dissipation of part of the kinetic energy of the turbulent flow by the deformation and break-up of drops of the dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Finally, to understand the local properties of turbulence, we have analysed the longitudinal Structure Functions at higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Interestingly, at scales larger than the Kolmogorov- Hinze, the exponents are only slightly smaller than in the single-phase flow, which implies increased intermittency, yet a similar anomalous scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' More importantly, we report a neat saturation of the exponents for structure functions higher than 3 at scales smaller than the Kolmogorov-Hinze length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This is typically related to a strongly intermittent dynamics and to the presence of jumps, here due to the pressure differences across the interface induced by the surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' A further demonstration that the interface is responsible of the increased intermittency is given by the results for the flow at viscosity ratio 훾 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' In this case, small-scale fluctuations are damped, especially in the more viscous dispersed phase, and the statistics approach those of the single-phase turbulence with no exponent saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' These observations may prove fundamental for understanding small scale dynamics in multiphase flows and for their future sub-grid modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Indeed, our results indicate that a correct model would need to account for the reduction of the energy fluxes near an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Moreover, we have shown that the turbulence statistics approach those of the single-phase 9 flow when the droplets consist of a highly viscous fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' This suggests that, despite several global measures seem to indicate a similar dynamics (Olivieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Yousefi 2022), the turbulence modulation is significantly different in the case of rigid particles and deformable intrusions.' metadata={'source': 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Federico & Sun, Chao 2021 Global and local statistics in turbulent emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Journal of Fluid Mechanics 912, A13, arXiv: 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content='00963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Yousefi, Ali 2022 Transport and mixing by finite-size particles in turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' PhD thesis, KTH Royal Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' MCE, GB and SM acknowledge the support from the Departments of Excellence grant (MIUR) and INFN22-FieldTurb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The authors acknowledge computer time provided by the National Infrastructure for High Performance Computing and Data Storage in Norway (Sigma2, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' NN9561K) and by SNIC (Swedish National Infrastructure for Computing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' LB acknowledge the support from the Swedish Research Council via the multidisciplinary research environment INTERFACE, Hybrid multiscalemodelling of transport phenomena for energy efficient processes, Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' 2016-06119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Declaration of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Data availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AzT4oBgHgl3EQfkv2l/content/2301.01537v1.pdf'} +page_content=' Data are 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sha256:9cbb12ad313d7a6aeab386c2b4d2aefe86d986b2cc6f549c4ade3a5f59455572 +size 98152 diff --git a/PdE5T4oBgHgl3EQfYg_i/content/tmp_files/2301.05575v1.pdf.txt b/PdE5T4oBgHgl3EQfYg_i/content/tmp_files/2301.05575v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c50f31684882c012a38c1f0fac96b04ec1b80e4 --- /dev/null +++ b/PdE5T4oBgHgl3EQfYg_i/content/tmp_files/2301.05575v1.pdf.txt @@ -0,0 +1,2286 @@ +Deep learning-based approaches for human motion decoding in smart walkers +for rehabilitation +Carolina Gonçalvesa,b, João M. Lopesa,b, Sara Mocciac, Daniele Berardinid, Lucia Migliorellid, Cristina P. +Santosa,b,e +aCenter for Microelectromechanical Systems (CMEMS), University of Minho, Guimarães, Portugal +bLABBELS - Associate Laboratory, Braga/Guimarães, Portugal +cThe BioRobotics Institute and Department of Excellence in Robotics and AI, Scuola Superiore Sant’Anna, Pisa, Italy +dDepartment of Information Engineering, Università Politecnica delle Marche, Ancona, Italy +eClinical Academic Center of Braga (2CA-Braga), Braga, Portugal +Abstract +Gait disabilities are among the most frequent impairments worldwide. Their treatment increasingly relies +on rehabilitation therapies, in which smart walkers are being introduced to empower the user’s recovery +state and autonomy, while reducing the clinicians effort. +For that, these should be able to decode hu- +man motion and needs, as early as possible. Current walkers decode motion intention using information +gathered from wearable or embedded sensors, namely inertial units, force sensors, hall sensors, and lasers, +whose main limitations imply an expensive solution or hinder the perception of human movement. Smart +walkers commonly lack an advanced and seamless human-robot interaction, which intuitively and promptly +understands human motions. A contactless approach is proposed in this work, addressing human motion +decoding as an early action recognition/detection problematic, using RGB-D cameras. We studied different +deep learning-based algorithms, organised in three different approaches, to process lower body RGB-D video +sequences, recorded from an embedded camera of a smart walker, and classify them into 4 classes (stop, walk, +turn right/left). A custom dataset involving 15 healthy participants walking with the walker device was +acquired and prepared, resulting in 28800 balanced RGB-D frames, to train and evaluate the deep learning +networks. The best results were attained by a convolutional neural network with a channel-wise attention +mechanism, reaching accuracy values of 99.61% and above 93%, for offline early detection/recognition and +trial simulations, respectively. Following the hypothesis that human lower body features encode prominent +information, fostering a more robust prediction towards real-time applications, the algorithm focus was also +quantitatively evaluated using Dice metric, leading to values slightly higher than 30%. Promising results +were attained for early action detection as a human motion decoding strategy, with enhancements in the +focus of the proposed architectures. +Keywords: +Deep learning, early action detection, early action recognition, human motion decoding, +human-robot interaction, RGB-D video, smart walker +1. Introduction +In 2018, over a billion people were estimated to live with some form of disability (WHO, 2021). This +is caused by ageing population and by an increase in chronic health conditions (WHO, 2021). Tumours, +aneurysms, strokes and cerebellar ataxia are prominent causes of gait and posture impairments (Mikola- +jczyk et al., 2018; Bonney et al., 2016; Jonsdottir and Ferrarin, 2018; Celik et al., 2021), which can have +severe repercussions in strength, sensation, and movement, leading to lack of stability and increased risk of +falls (Mikolajczyk et al., 2018). Rehabilitation therapies reveal promising results tackling these impairments +Email address: cristina@dei.uminho.pt (Cristina P. Santos) +January 16, 2023 +arXiv:2301.05575v1 [cs.CV] 13 Jan 2023 + +(Milne et al., 2017). Gait rehabilitation requires long periods of intense physical exercise, presenting chal- +lenges for clinicians, due to the high demand of physical effort and inter-patient variability. Additionally, +it may also be affected by the clinician experience and inter-clinician variability (Mikolajczyk et al., 2018), +making this therapy more time-consuming and prone to error. +To overcome these limitations, assistive +technologies have emerged as effective means to increase the patient’s independence and participation in +their rehabilitation therapies (WHO, 2011). Assistive technologies include Smart Walkers (SWs), which are +intended to be used by or with humans. SWs no longer serve as just conventional physical supporters, but +comprehend other intelligent functionalities to promote an efficient Human-Robot Interaction (HRI). These +devices should be able to decode human motion and needs, as early as possible, which would be essential +for a seamless HRI. This would be clinically relevant since it would enable a more natural and anticipated +assistance, encouraging patients to take an active role in rehabilitation exercises or therapy sessions (Zhao +et al., 2020). However, the interaction entailed by motion decoding should result from the device built-in +sensors in order to maximise intuitiveness and technology acceptance. +Human motion decoding in SWs can be achieved through the use of wearable sensors, such as Inertial +Measurement Units (IMU) (Weon and Lee, 2018). However, these have to be placed on the user’s body, +which hampers the clinician’s task and the patient’s movement, making rehabilitation more time consum- +ing. Additionally, these can suffer electromagnetic interference from the walker’s motors. Several sensors +embedded in SWs have also been used for this purpose, although entailing other limitations that hinder +rehabilitation. +For instance, force sensors (Cheng and Wu, 2017; Sierra et al., 2018) present a reduced +long-term effectiveness (Paulo et al., 2017), infrared sensors (Paulo et al., 2015) can be easily corrupted +by light conditions and a handlebar specially designed with hall sensors (Park et al., 2019) implies specific +movements besides the natural gait, increasing the patient’s cognitive load. Additionally, preprocessing may +also be required to clean the output signals, for instance, when resorting to IMU or force sensors. +RGB-D cameras can be used along with SWs (Palermo et al., 2021; André et al., 2020). +Although +fostering a cheap, intuitive, and contactless solution, without interfering with the user’s gait, these are not +usually explored for the purpose of motion decoding. This can be due to the challenges inherent to image +analysis, especially in realistic environments. Deep Learning (DL) methods have shown robustness tackling +Human Action Recognition (HAR) (Yeaser et al., 2020) and, more recently, Human Action Prediction +(HAP) (Chalen and Vintimilla, 2019) from RGB-D images or videos. While the former tackles the issue +of current action recognition or detection, the latter intends to anticipate the action’s ending (early action +recognition/detection), seeing only a small part of the action, or even its beginning (action anticipation), +taking a step towards forecasting. Despite the progresses made, dealing with the hindrances of RGB-D +image analysis, HAR and HAP methods have not yet been duly explored for human motion decoding in +SWs. +Following the need of a more intuitive way of decoding motion in SWs, robust to realistic environments +and capable of perceiving natural gait movements without hindering them, this work innovatively addresses +motion decoding as a HAP problem. We propose the use of DL-based algorithms for online early detection +and recognition of walking directions, from RGB-D videos. These videos, recorded by a SW embedded +camera (Lopes et al., 2021), focus on the user’s lower body, which we hypothesise to encode the most +relevant motion information. Inspired by literature on HAP, five DL algorithms were implemented within +three approaches. These approaches are responsible for implementing two different strategies: classification +or segmentation-classification, in an attempt to evaluate and enhance the algorithm’s focus on the human +body, along with its performance. Different inputs from the RGB-D videos were evaluated, encoding the +videos’ temporal information into one single image. Therefore, this solution avoids not only extra sensors +and a cognitive load to the user, but also computational-expensive complementary tasks, such as pose +estimation, and additional computational complexity by not resorting to spatio-temporal DL models. To +efficiently predict walking directions directly from camera streams, three key-performance requirements were +also established to evaluate the DL algorithms: +• Accurately recognise and detect 4 actions (Stop, Walk, Turn Right (TR) and Turn Left (TL)), while +only seeing small temporal sequences to increase the efficiency of the algorithm’s response in online +scenarios. +January 16, 2023 + +• Extract human-centred features, when detecting these actions, to avoid background bias and motions +as consequence of camera’s movements. This will help to achieve a reliable performance for real-time +applications, as the SW can only move after detecting the action. +• Face online detection with a maximum admissible delay of 0.64s, which corresponds to the determined +medium duration of one healthy step, while walking at the fastest velocity assumed by the walker +(1m/s). This step time was determined in laboratory experiments and it is in accordance with Müller +et al. (2017). +This paper is organised as follows: Section 2 presents relevant related work, critically discussing its +advantages and limitations. Section 3 details all the methods for data acquisition and data preparation, +along with the devised DL approaches, including model architectures and the involved algorithms. Section +4 summarises the implementation details for training and evaluation of the proposed models. Section 5 and +Section 6 present and critically discuss all the obtained results, respectively. Finally, Section 7 summarises +the findings of this work and proposes future research insights. +2. Related Work +2.1. Human Motion Decoding in Smart Walkers +Most of the current SWs decode human motion directly, demanding some level of physical intervention +from the user and/or an extra load of cognitive effort. This direct mode has been typically implemented with +specially designed handlebars, including force/pressure/load sensors (Huang et al., 2005; Rodriguez-losada, +2008; Spenko et al., 2006; Jiménez et al., 2019; Cheng and Wu, 2017), infrared cameras and Light Emitting +Diodes (Paulo et al., 2017) or Hall sensors (Park et al., 2019), which require specific hand motions to encode +each walking direction. In an indirect mode, the walker becomes responsible for analysing the end-user’s +movement, inferring, from this, the walking directions. LIDAR sensors combined with wearable ones (Weon +and Lee, 2018) have been used to analyse the kinematics of lower limbs and measure feet orientation. Page +et al. (2015) also resorted to a depth camera for feet position and orientation detection. Lv et al. (2020) used +multi-channel proximity sensors to determine each leg’s distance and velocity. These studies highlighted the +relevance of lower body features to naturally infer the user’s walking directions. Nonetheless, the presented +sensory technologies incur in additional costs for the motion detection task, including increased expenses (e.g. +LIDAR) and number of sensors, reducing technology acceptance (e.g. IMU and proximity sensors), signal +corruption by environmental conditions (e.g. depth camera by sunlight) or electromagnetic interference, +while causing long-term discomfort and hampering movement (e.g. IMU). +Vision-based approaches should benefit the task of indirectly decoding human motion, avoiding the +aforementioned limitations. Particularly, through the use of RGB-D cameras, usually built into SWs for other +functionalities (e.g., gait and posture assessment (Palermo et al., 2021)). Shen et al. (2020) used a RGB-D +camera to perceive the intended walking direction from the torso kinematics. +An hybrid Proportional- +Integral-Derivative (PID) controller with integrated digital practical differentiator was then implemented +to calculate the desired wheel rotation angles, being able to track the subject in forward and turning +movements. However, the use of upper body information entailed a disadvantage: the upper body swaying +(Shen et al., 2020). Chalvatzaki et al. (2019) complements RGB-D data with 2D laser data to perform +real-time gait analysis, as well as to track the user’s Centre-of-Mass position and velocity. This, assembled +with the desired human-related robot coupling parameters, became the input used to forecast the human +motion and the evolution of these parameters. The i-Walk system (Chalvatzaki et al., 2020) points out the +potential of using RGB-D cameras, in particular coupled with DL models, to attain an effective activity and +gesture recognition, but not specifically to drive the walker. Although promising, these vision-based motion +decoding approaches (Shen et al., 2020; Chalvatzaki et al., 2019, 2020) require additional computational tasks +to estimate human poses or parameters from the acquired RGB videos. This can lead to error propagation, +when feeding these inputs to a following motion decoding algorithm. +To the best knowledge of the authors, no current SWs present a solution that decodes human motion +directly from camera streams, decreasing the complexity inherent to human pose-based solutions, while +January 16, 2023 + +enabling a seamless and intuitive HRI. This work contributes for this purpose by proposing DL-based +approaches to early detect and recognise walking directions through lower-body RGB-D videos, while walking +with a robotic walker. +2.2. Vision-based DL Methods for HAR/HAP +Recent studies have shown great potential for HAR (Berardini et al., 2020; Li et al., 2016) and HAP +(Aliakbarian et al., 2017; Canuto et al., 2021; Gao et al., 2017), resorting to DL algorithms fed with RGB- +based inputs. Convolutional Neural Network (CNN) models, optionally combined with Long Short-Term +Memory (LSTM), are the commonly used architectures in these tasks, usually resorting to video chunks +or sequences of frames as input. The most common methods are based on colour (RGB) data (Berardini +et al., 2020), on a combination of colour and depth (RGB-D) data (Jalal et al., 2017), or on skeleton data +(Li et al., 2016). +Concerning HAR, Berardini et al. (2020) applied a DL solution to automatically recognise falls in stacks +of seven RGB frames. The model architecture consisted on a CNN model as feature extractor, namely +the VGG16, and a Bidirectional LSTM as feature classifier. Li et al. (2016) also implemented a LSTM +model, aiming at Online Action Detection (OAD) through skeleton information. Moreover, simultaneously +to frame-wise action (and background) classification, they proposed an estimation of the start and end +frames of the current action, based on the definition of Gaussian-like curves for each action. As for De Geest +and Tuytelaars (2018), a different OAD proposal was made, using LSTM to model long term dependencies +between actions, since human actions can imply a certain order (e.g. standing, after being sited). However, +these can also be random or environment-depending, as in free walking trajectories. +When handling HAP, there are two main lines of work in the literature: i) directly predicting the future +frames’ class or the current one, before the action ends (classification) (Aliakbarian et al., 2017; Canuto +et al., 2021; Guo et al., 2020; Ke et al., 2019) or ii) generating future visual representations that are further +classified (regression followed by classification) (Gao et al., 2017; Vondrick et al., 2016; Shi et al., 2018). +Moreover, a common focus of these studies is the design of novel loss functions that can reduce the predictive +generalisation error (Aliakbarian et al., 2017; Gao et al., 2017; Shi et al., 2018). +Vondrick et al. (2016) resorted to visual representations as promising prediction targets. The proposed +model consisted on AlexNet architecture, implementing five convolutional layers followed by five Fully Con- +nected (FC) layers, with ReLu activations throughout the architecture. To deal with the model uncertainty, +the last three FC layers presented K networks, so each frame resulted in K future visual representations, +one for each plausible future. Approaches like Vondrick et al. (2016) anticipate only a representation of a +fixed future time, based on a single past frame’s representation, dismissing the history and temporal trend. +Gao et al. (2017) proposed instead a Reinforced Encoder-Decoder network which took multiple history rep- +resentations as input and learned to anticipate a sequence of future ones. The video was segmented into +chunks of six frames, each processed by a CNN model to extract a chunk representation. This was then fed +to a LSTM-based Encoder-Decoder, outputting a prediction of the future video chunks’ representations, in +a supervised manner. The classification of these future representations was handled by two fully connected +layers. +Despite all the research and developments, as well as the ability to use unlabelled videos for training, +generating future representations is still defying, as well as time-consuming and prompt to error accumulation +(Ke et al., 2019). Additionally, the learned representation may not be related to the action itself, as it can be +influenced by background or other variables (Aliakbarian et al., 2017). Direct action prediction approaches +avoid these limitations, by exploiting different forms of features and/or tailored losses to directly predict +future classes. +Aliakbarian et al. (2017) resorted to the VGG16 model, connecting it to two different branches, one +to compute context-aware features, encoding global information about the scene, and another to compute +action-aware features. These features were then sequentially introduced in a multi-stage LSTM. The extrac- +tion of action-aware features relied on Class Activation Maps (CAMs) (Zhou et al., 2016), which indicate +the regions in the input frame that most contributed to the class prediction. Nonetheless, this implied the +training of CNN models before using them to compute CAMs, in an offline manner, which were then fed to +January 16, 2023 + +Figure 1: General workflow, including all the main methods implemented. GT stands for Ground-Truth. +the multi-frame classification LSTM model. Besides not being suitable for online applications, this use of +CAMs does not guarantee these maps always focus on the action’s relevant elements. For example, similar +human activities may drive the model to extract features from the surroundings (e.g. background movement, +due to camera motion, objects, among others), as it is difficult to distinguish the human fine-grained move- +ments. In Aliakbarian et al. (2017), CAMs improvement was not taken into consideration during training, +to assure the model’s weights would be learnt in a way that would enhance action-centred feature extraction +and improve the model’s focus. These were just used, after training, for feature extraction. +In this sense, a new line of approaches is also emerging, namely the use of transformers (Girdhar et al., +2019; Liu et al., 2019) and attention mechanisms (Ke et al., 2019; Qiao et al., 2020; Wu et al., 2020). +Commonly for fine-grained action recognition, the frame or sequence of frames incorporates irrelevant or +redundant information, with no discriminatory property. So, these algorithms guide the model to use atten- +tional regions, instead of the whole frame, enhance local features and attain selectively feature fusion. For +example, Wu et al. (2020) implemented channel-wise and spatial attention mechanisms, along with baseline +CNNs (VGG16 and ResNet-50) and LSTM. Additionally, when comparing to LSTM, transformers can be a +lighter and maybe more suitable alternative for online performances (Kozlov et al., 2020). Nonetheless, the +study of these algorithms for video analysis and HAR/HAP is still a newly tackled field. +Overall, the presented DL methods for HAR/HAP typically target a broad spectrum of actions (e.g. +blowing candles, falling, drinking), recorded with static cameras (Berardini et al., 2020; Aliakbarian et al., +2017), rather than specific walking directions (as stop, walking straight, turning right and left). The latter +actions are the most used and relevant to specify and control the walker’s trajectory (Paulo et al., 2017; Park +et al., 2019; Weon and Lee, 2018), but demand no objects or interactions. The differences between them +rely on human positions or fine-grained movements, incurring on a more challenging problem. Additionally, +when resorting to mobile embedded cameras in SWs, the videos will include background motions and will +be commonly recorded in a front-view perspective, hindering the perception of turning events. +3. Methodology +This section describes the methodology followed in this work, from data acquisition and preparation, +to the devised DL-based algorithms for model evaluation. Figure 1 illustrates the overall pipeline of the +developed work and the inherent methodology. +3.1. Data Acquisition +Facing the lack of suitable public datasets for HAR or HAP with SWs, this work integrated the collection +of a custom dataset, which included walking and turnings as different actions. This was recorded with mobile +cameras of a SW, always maintaining a front-view perspective of the user. Data acquisition was conducted +under the ethical procedures of the Ethics Committee in Life and Health Sciences (CEICVS 147/2021), +following the Helsinki Declaration and the Oviedo Convention. All participants gave their informed consent +to be part of the study. Data were collected at the School of Engineering of University of Minho, outside a +controlled laboratory space. +January 16, 2023 + +ection3.4 +DL Approaches +RGB frames +RGB input +Input Design +Section 3.6 and 3.7 +Section 3.1 +Section 3.2 +Evaluation +GG16 +Single-frame Classification +ResNet-50 +Approach +Dataset +Selected approach +Attention Mechanism +Post-processing and Trial simulations +Data Acquisition +and model +Preparation + Quantitative focus evaluation +Section 3.5 +Human Masks +GT masks +Depth frames +Segmentation-Classification +UNE +Generation +Approach +Adapted UNETFigure 2: Mobile acquisition setup, where a laptop (C) running the Xsens MVN software and its acquisition base (E) are +placed over the SW (A), moving along with the robotic device. The user is equipped with the Xsens sensors (B), hidden +bellow a layer of clothing. +A mobile acquisition setup was used for this purpose, as presented in Palermo et al. (2021). +This +setup was composed by: i) WALKit Smart Walker (Figure 2A) used in rehabilitation (Moreira et al., 2019; +Lopes et al., 2021), which integrates a RGB-D camera (Orbbec 3D Technology International Inc., USA) to +assess the patient’s gait, recording the user’s legs and feet at a frame rate of 30 Hz; and ii) Xsens MVN +Awinda (Xsens Technologies B.V., The Netherlands), which includes wearable IMUs (Figure 2B) and a base +station (Figure 2E) connected to a laptop running the MVN software from Xsens (Figure 2C). This system +recorded data at 60 Hz and has already been used before in other studies (Palermo et al., 2021). Its interest +in this work was to provide relevant information to accurately place the Ground-Truth (GT) labels of the +different actions, considering specific gait events (e.g., heel strike and toe-off (Kurai et al., 2019)). Data +synchronization was ensured by sending a digital pulse from WALKit SW to Xsens MVN Awinda. +This mobile setup allowed for real-time data acquisition through pre-defined circuits, which are explained +in Section 3.1.1. Additionally, it enabled data to be recorded in realistic scenarios, with non-ideal light +conditions and dynamic backgrounds to maximize data variability. +3.1.1. Dataset +The dataset consisted of RGB-D images acquired from fifteen healthy adults (nine males and six females), +with average age of 25.27±2.54 years old, body mass of 65.15±9.22 kg and height of 168.47±7.42 cm. +Considering the aim of decoding human motion in SWs, the walking trajectories should be performed +and recorded as naturally as possible, without causing abnormalities in the user’s gait. WALKit SW allows +to be used in a passive mode, but since it is heavy, this would produce non-natural slower and larger +steps. Therefore, an automatic trajectory mode was developed, allowing the device to actively follow given +trajectories, only as a recording device, creating velocity commands for each wheel according to the desired +linear velocity for the SW and the turning strategy (angle and radius). This algorithm enabled the execution +of four designed circuits, according to the turning direction (right or left) and curvature radius (wide and +tight). +A circuit comprised a sequence of i) 10s standing, ii) 3-meter walking, iii) 90º turn, iv) 3-meter walking, +and v) 10s standing. Each participant performed 2 valid trials per circuit and per gait speed: 0.5 m/s, 0.7 +m/s, and 1 m/s. These speeds were selected according to the walker’s most commonly used velocity range, +as well as considering the typical self-selected slow, normal, and fast walking speeds for healthy subjects +(O’Callaghan et al., 2020). Different light conditions were held for each trial, increasing the environment and +movement variability (e.g. backgrounds, lighting, velocities, turn features), while improving the statistical +significance of the data. +January 16, 2023 + +A +B +B +D +D +E +EThe circuit sequence remained unknown to the participants until the beginning of the trial, when a brief +explanation was given. The floor was marked with tape and signalised, during the recordings, with chairs +and staff people, so that participants could see and react to the circuit’s morphology. The marked spots were +positioned in a way that remained invisible to the SW’s cameras, during the performance of each respective +trial. Additionally, the overall circuit and trial’s order was randomised for each subject before the beginning +of the acquisition. +The participants were instructed not to just follow the smart walker, but to interact with it, along +the designed and marked circuit. This enabled realistic scenarios, while capturing the user’s intention as +naturally as possible. Some familiarisation trials were also performed before the real recording procedure, +encouraging the HRI. Overall, each subject performed 24 trials, taking no more than 1 minute each. +The labelling process was executed in real-time, along the data acquisition, in two different ways: i) +with joystick commands and ii) with velocity commands. The former relied on an external person using +joystick digital buttons, similar to Figueiredo et al. (2020), to mark transitional moments between actions, +according to the observation of clear feet movements. This produced GT labels responsible for denoting +the user’s interaction and intention (JOY labels). The latter represents the device’s actions, generating GT +labels mainly for action recognition, when the background is already changing accordingly to the performed +movements (VEL labels). +3.2. Dataset Preparation +Since the principal walking frequency is no higher than 2 Hz for gait speeds above 1 m/s (Pachi and Ji, +2005), all data was down-sampled to 15 Hz, covering more time of action with a lower number of similar +samples. Aiming the creation of a balanced dataset, 40 frames were extracted from each video sequence. +This number was selected considering the minimum number of frames obtained for the turn actions during +the trials performed by the participants. This extraction was performed avoiding the action’s boundaries +or transitions to prevent possible bias in the labelling procedure, while aiming an effective early action +recognition. For this reason, the start of each class was marked with the latest of the two labels (JOY and +VEL). In total, the final dataset contained 28800 RGB-D frames. +To simulate real scenarios (Section 3.7), a dataset containing the data of the test participants was created, +including the complete RGB trial videos, towards online early action detection. The foot contacts obtained +with Xsens MVN Awinda were used to better position the JOY labels, enabling more accurate labels to +mark action transitions. +3.3. Input Design +Literature revealed the use of windows of frames (Berardini et al., 2020) and/or other computed inputs, +as the optical flow (Simonyan and Zisserman, 2014). This paper proposes two types of input, computed +within a sliding window over temporal video clips: i) the difference between the last and first RGB frames +(DIF input) and ii) the sum of all RGB images (ADD input), in each temporal window. +The window length was empirically defined as 4 (0.27s), since it incorporates visible human motion +across all gait speeds, while never containing information from more than 1 different step. Examples to +better understand the correlation between the original frames and the resultant DIF and ADD images are +presented in Figure 3. +The designed inputs were optionally cropped, removing surrounding background, in an attempt for the +model to direct its focus to the user and his/hers fine-grained movements. The cropping procedure was +performed according to a Region of Interest (ROI), instead of using bounding box localization networks, +since the user constantly assumes a central position in the image (in the middle of the walker’s handles). +In total, four input forms were generated and studied in this work to recognise and detect human motions: +cropped and non-cropped DIF and ADD images. +3.4. Deep-Learning Approaches +In this work, we have implemented three different approaches (Approach 1, Approach 2, and Ap- +proach 3), as illustrated in Figure 4, which were trained and evaluated considering the four proposed input +forms (i.e., cropped and non-cropped DIF and cropped and non-cropped ADD images). +January 16, 2023 + +Figure 3: (a) A TR sequence extracted from our dataset. (b) DIF image, corresponding to the difference between the last +and the first window frames. (c) ADD image, computed from the sum of all window frames. +Regarding Approach 1, this consisted on single-frame classification. Here, the VGG16 model was used +without its top layers (Berardini et al., 2020), only with a global average pooling (GAP) followed by a +softmax classification layer, as it is illustrated in Figure 5. Batch normalization layers were also added +to improve the optimization and generalization performance. Moreover, the activation function of the last +convolution layer was changed from ReLU to hyperbolic tangent function (tanh), to decrease the clipping +of final feature map values and avoid possible vanishing gradients. The ResNet-50 model was also used, +considering its configuration presented in He et al. (2016). +Approach 2 consisted of the best backbone obtained in Approach 1 with a channel-wise attention +mechanism, as presented in Wu et al. (2020). This was tested as an attempt to increase the model focus +towards the user’s legs and feet. The attention mechanism follows the network’s last convolutional block and +creates a channel descriptor vector with the corresponding weights for each feature map. This vector was +computed through an average pooling and two convolutional layers, the first followed by a ReLU activation +function and the second by a sigmoid function. The obtained vector is then used to perform a multiplication +between each feature map and the corresponding computed weight, as it is illustrated in Figure 6. +As for Approach 3, UNET neural network (Ronneberger et al., 2015) was used to segment the user’s +legs and feet in order to obtain the optimized weights which could allow the neural network to focus on +that region of the user’s body. These optimized weights were used to pre-train a novel classification model. +This latter model integrates the UNET encoder, followed by two convolutional blocks. A dropout layer with +probability of 50% was added, followed by a GAP and a softmax layers, decoding human motion into the +four classes (stop, walk, TR, and TL). Figure 7 illustrates the neural network used as the basis of Approach +3. +As presented in Berardini et al. (2020), transfer learning technique was used for all approaches. The +backbone weights of VGG16 and ResNet-50 classification models were initialised to the weights of these same +state-of-the-art networks pre-trained on the general object classification benchmark, ImageNet (Deng et al., +2010), towards accuracy maximisation, while using large amounts of computational resources, which trains +the models to detect and produce general features. In the segmentation-classification approach, the UNET +encoder weights of the classification model were initialised with the ones learned in the previous training of +the UNET model for segmentation, while the remaining layers were initialised with the He normal function. +The first 16 layers of this encoder were also frozen during classification training. +January 16, 2023 + +Figure 4: Workflow of the devised approaches, showing how they were conducted. +Figure 5: Diagram of the implemented VGG16 architecture. +3.5. Human Masks Generation +To train the UNET neural network, human masks were computed through an adaptation of our algo- +rithm presented in André et al. (2020), using depth frames. This algorithm involved geometric and threshold +operations that removed the background, as well as the floor plane, to isolate the user. Depth frames can be +corrupted by the high exposure from sunlight, which, in extreme conditions, causes unrecognisable silhou- +ettes in the computed masks. These corrupted masks were duly removed through empirically established +thresholds. When facing minor corruptions (e.g., incomplete feet), the masks were corrected through classic +vision algorithms. These were processed accordingly to form the final GT masks, corresponding to the input +used for model training and evaluation (cropped or non-cropped DIF or ADD). Examples of the obtained +masks can be found in Appendix A.2. +3.6. Quantitative focus evaluation +Due to camera motion, background movements are also present in the input images (Figure 3), according +to the action performed by the SW. This can compromise early action detection in future real-time applica- +tions, as the model should be focused on lower body features to predict walking directions, specially during +their first occurrence, without overfitting to the background. +January 16, 2023 + +RGB input +DIF +CROPPEDDIF +ADD +CROPPEDADD +Approach 1 +Approach 2 +Approach 3 +CLASSIFICATION +CLASSIFICATION +SEGMENTATION +VGG16 & ResNet50 +Best backbone of +CLASSIFICATION +APPROACH +Approach 1 + Attention +mechanism +Evaluation +Model Selection +Evaluation +Evaluation +Final Approach +Selection +Trial SimulationsMaxPooling +MaxPooling +BatchNorm +BatchNorm + BatchNorm +MaxPooling + BatchNorm +BatchNorm + BatchNorm +MaxPooling +MaxPooling + BatchNorm + BatchNorm + BatchNorm +BatchNorm +BatchNorm +BatchNorm +Conv 5,2 +Conv 3,2 +Conv 2,1 +Conv 2,2 +Conv 3,3 +Conv 4,3 + Dropout +Conv 1,1 +Conv 3,1 +Conv 4,2 +Conv 5,1 +Conv 5,3 + Dropout +Softmax +Conv 1,2 + Dropout +Conv 4,1 + GAPFigure 6: Diagram of the implemented channel-wise attention mechanism as presented in Wu et al. (2020). +The channel- +attention weighted feature maps are then fed to the Global Average Pooling of the selected CNN model. +Figure 7: Schematic of the adapted UNET model for single-frame classification. MP stands for MaxPooling, GAP stands for +Global Average Pooling, and the tuples of numbers represent the respective kernel sizes and 0.5 the dropout rate. +An algorithm for consistent quantitative evaluation of this focus was implemented. Following Selvaraju +et al. (2019), grad-CAMs heatmaps were generated for each input image, over the model last convolutional +layer. These were then compared with the respective GT human masks, also used as segmentation labels. +Hence, the model focus was evaluated through the similarity between the grad-CAMs heatmaps and the GT +masks. +3.7. Trial simulations and Post-processing +The results from each approach were analysed and compared, towards the selection of the most promising +model architecture, and input form. These were then evaluated in trial simulations, assessing the perfor- +mance in online early action detection tasks. +To deal with the model uncertainty while predicting, a post-processing technique was designed and +implemented. This technique assumes a minimum action duration (2s, which equals to, at least, two complete +steps in every considered gait speed) and applies it to the previous predicted class, meaning that it only +allows for an action transition to happen if the previous predicted one lasted, at least, 2s. Additionally, +it also reduces the spectrum of possible transitions by not allowing a TR/TL to happen immediately after +a TL/TR event, respectively. +These conditions help avoiding prediction errors and are consistent with +rehabilitation therapy sessions. This post-processing technique also reduces possible on-off noise, without +introducing delays in the decision process. +January 16, 2023 + +descriptor vector +Feature Maps +AvgPooling +Weighted +Conv 1,2 +Conv 1,1 +Channelencoder block +classifier +Input Image +Base +Base +encoder block +Conv (3x3) +Conv (3x3) +MP(2x2) +ReLU +ReLU +encoder block +BatchNorm +BatchNorm +MP(2x2) +UNET encoder +Conv (3x3) +Conv (3x3) +encoder block +Tanh +ReLU +MP(2x2) +BatchNorm +MP (2x2) +encoder block +DropOut (0.5) +DropOut (0.5) +GAP +MP(2x2) +Softmax +classifier4. Experimental Protocol +This section details the experimental protocol to train and evaluate the proposed approaches, including +the dataset split and the training hardware used. +4.1. Dataset Split +The dataset split followed the 80-20 split configuration: 80% was used for training (12 participants) and +the remaining 20% was used for testing (3 participants). To control the training process, one participant +was left to the validation set, similar to Canuto et al. (2021). Table 1 describes each of these splits. For the +tasks involving the generated human masks, namely segmentation and the quantitative focus evaluation, +the dataset used was smaller, since some sequences were removed giving mask corruption reasons. +To perform the trial simulations, the test split was used to create a new dataset, as mentioned in Section +3.2. In total, this test dataset included 72 untrimmed videos from 3 participants (8, 11, 15, according to +Table 1). +Table 1: Constitution of each dataset split +Split +Participants IDs +Number of images +Classification +Segmentation +Train +[1,15) \{5, 8, 11} +19536 +13209 +Validation +5 +1776 +1295 +Test +8, 11, 15 +5328 +4736 +4.2. Training Details +CNN configuration: Reasonable hyperparameters were extracted from literature that tackles similar +tasks and models (Aliakbarian et al. (2017) for single-frame classification and Ronneberger et al. (2015) for +segmentation). For classification, the parameters were optimised using Mini Batch Gradient Descent, with +learning rate of 0.001, Nesterov momentum of 0.9 and batch size of 64. The initial learning rate was decayed +by 50% until a minimum of 1e−4, if the training loss did not improve within 4 epochs. As loss function, +the categorical cross-entropy was used. For segmentation, the Adam optimizer was used instead, with a +learning rate of 1e−4 and a batch size of 16. UNET convolution layers were initialised with He normal +weights initialization. As loss function, the binary cross-entropy was used. At the end of the training, the +best model was selected according to the validation F1-score or loss, for classification and segmentation, +respectively. +Data preparation: The generated input images were resized, from a resolution of 480x480 pixels to a +resolution of 224x224 pixels, preserving the images’ aspect ratio, to match the pre-trained networks’ input +size. For the cropped inputs, the original aspect ratio decreased with the cropping procedure and the image +was thus resized as much as possible, towards the target resolution. After this, padding was performed, +centring the image, to fulfil the 224x224 dimensions. Image augmentation was then used in order to avoid +overfitting, as well as to improve the model focus on the human body, despite its global position in the images. +Random alterations were applied to the image brightness and contrast, as well as spatial augmentations, such +as height and width shifts and zoom. Since directions are an important feature to distinguish between TR, +TL and straight walking, rotations were avoided. Additionally, random Gaussian blur was added. Finally, +the images where normalised between 0 and 1. +Training Hardware: All models were developed offline, with the collected data, using the Tensorflow +and Keras DL library, on a Python environment. +The depicted experiments and DL approaches were +trained in a computer with the following hardware specifications: GPU: 1x Nvidia GeForce RTX 3080 +Ti/PCle/SSE2; CPU: Intel(R) Core(TM) i9-10940X @3.3GHz (14 core, 28 threads); RAM: 65,5 GB. +January 16, 2023 + +4.3. Performance Metrics +Classification was evaluated using common metrics for this task, namely accuracy (ACC), precision, +recall, and F1-score (Berardini et al., 2020; Aliakbarian et al., 2017). +These metrics are defined in the +equations below, where TP, TN, FP, FN refer to the true/false positive/negative observations and n stands +for the total number of classes (in this case, 4). +ACC = +n +� +i=1 +TPi + TNi +TPi + TNi + FPi + FNi +(1) +Precision = 1 +n +n +� +i=1 +TPi +TPi + FPi +(2) +Recall = 1 +n +n +� +i=1 +TPi +TPi + FNi +(3) +F1 − score = 2 ∗ Precision × Recall +Precision + Recall +(4) +Intersection over Union (IoU) and Dice (Dice) were computed for segmentation and quantitative focus +evaluation, as shown by the following equations (TP, FP and FN refer to the true positive and false +positive/negative observations). +IoU = +TP +TP + FP + FN +(5) +Dice = +2 × TP +(TP + FP) + (TP + FN) +(6) +Inspired by Baptista-Rios et al. (2020), OAD metrics were implemented to better evaluate the perfor- +mance in online simulations, such as instantaneous accuracy (IA), instantaneous precision (IP), instantaneous +weighted accuracy (wIA) and instantaneous calibrated precision (cIP). These are represented in the follow- +ing equations, were t corresponds to the time instant, K to the the total population considered until time t, +n to the number of classes and TP, TN, FP, FN refer to the seen true/false positive/negative observations +overall classes. +IA = +1 +K × n +t +� +j=0 +TPj + TNj +(7) +IP = +�t +j=0 TPj +�t +j=0 TPj + FPj +(8) +wIA = +1 +K × n[w × +t +� +j=0 +TPj + 1 +w × +t +� +j=0 +TNj] +(9) +cIP = +w × �t +j=0 TPj +w × �t +j=0 TPj + �t +j=0 FPj +(10) +These metrics evaluate the model performance as the frames are acquired, without having to wait to an +unknown end. Additionally, wIA and cIP condition the value of a true observation to the total negatives vs. +total positives ratio (w), which is dynamic and always based only on the seen portion of the video. +5. Results +Starting from the CNN models for single-frame classification (approach 1), to an attention mechanism +(approach 2) and the segmentation-classification approach (approach 3), the influence of the various +forms of input is studied and the classification models are evaluated in two aspects: i) the accuracy of the +predicted labels; and ii) model focus on the human body region. +January 16, 2023 + +5.1. Single-frame Classification Approaches +Table 2 shows the validation results, obtained for approaches 1 and 2 considering single-frame classi- +fication. The respective training curves can be found in Appendix A.1. According to Table 2, ResNet-50 +outperformed the VGG16, except for the cropped DIF input. The DIF revealed here a better performance, +followed by the ADD input, which was only worst by an overall maximum margin of approximately 2%. +Adding a channel-wise attention mechanism to ResNet-50 further enhanced its classification perfor- +mances, specially for the cropped inputs. The F1-score was increased by 3.98% and 3.85%, for the cropped +DIF and ADD inputs, respectively. This mechanism also changed the influence exerted by the different +types of images. With this model, ADD input surpassed the accuracy and F1-score values obtained for the +DIF input, by a maximum margin of 1.3%. +Table 2: Validation results of the VGG16 and ResNet-50 (without and with a channel-wise attention mechanism), as well as +the training time +Input Type +Crop +ACC (%) +F1-score (%) +Precision (%) +Recall (%) +Training Time (h) +VGG16 (approach 1) +DIF +False +97.02 +96.80 +97.38 +96.23 +4.45 +DIF +True +94.76 +95.02 +95.66 +94.37 +4.47 +ADD +False +95.50 +96.28 +97.79 +94.82 +4.44 +ADD +True +94.48 +94.53 +94.99 +94.03 +4.54 +ResNet-50 (approach 1) +DIF +False +98.42 +98.27 +98.52 +98.03 +4.45 +DIF +True +94.37 +94.34 +95.24 +93.47 +4.47 +ADD +False +96.34 +96.26 +96.65 +95.83 +4.44 +ADD +True +94.87 +95.76 +97.05 +94.48 +4.46 +ResNet-50 with attention (approach 2) +DIF +False +99.04 +99.03 +99.04 +99.04 +2.93 +DIF +True +98.31 +98.32 +98.37 +98.25 +2.76 +ADD +False +99.38 +99.39 +99.38 +99.38 +3.86 +ADD +True +99.61 +99.61 +99.61 +99.61 +4.17 +Table 3 shows the results obtained when evaluating each model’s grad-CAMs. Contrarily to the quanti- +tative classification results for VGG16 and ResNet-50, the cropped inputs presented better focus, meaning +a higher similarity between the model’s grad-CAMs and the GT masks. In average, cropping part of the +background from the inputs increased the Dice and IoU metrics by 4.72% and 3.16%, for the DIF input, and +by 9.09% and 5.98%, for the ADD input, respectively. The cropped ADD led to a better focus, achieving +its best results with the ResNet-50. This model showed an overall average boost of 7.03% in Dice and 4.96% +in IoU metric, when compared to the VGG16. +The attention mechanism improved the focus of the ResNet-50 for every input, even if just for a small +margin (<5%). The greatest improvement was achieved by the non-cropped ADD input (4.20% in Dice). +However, this one still led to lower metrics than the non-cropped DIF input, as the latter achieved Dice +values 4.07% higher than the non-cropped ADD. Contrarily to previous results, cropped DIF attained the +highest similarity between their GT masks and the grad-CAMs obtained from the ResNet-50 model with +attention. Nonetheless, the difference relative to the cropped ADD corresponds only to a small percentage +(1.11%). Additionally, the cropped ADD presented the lowest values of standard deviation, entailing a more +consistent focus across the dataset, with smaller fluctuations and more representative IoU and Dice values. +Table 3: Quantative evaluation results, in percentage, for validation grad-CAMs, when predicting with the VGG16 and ResNet- +50 models (without and with an attention mechanism) +Input +VGG16 +ResNet-50 +ResNet-50 with attention +Type +Crop +Dice (±std) +IoU (±std) +Dice (±std) +IoU (±std) +Dice (±std) +IoU (±std) +DIF +False +16.16 (±9.64) +9.10 (±5.87) +29.06 (±13.26) +17.70 (±9.01) +30.37 (±12.86) +18.59 (±9.13) +DIF +True +22.57 (±14.26) +13.44 (±8.95) +32.09 (±11.57) +19.67 (±8.22) +32.90 (±9.13) +20.04 (±6.43) +ADD +False +20.19 (±6.84) +11.39 (±4.21) +22.10 (±16.00) +13.36 (±10.51) +26.30 (±14.71) +15.99 (±10.04) +ADD +True +28.34 (±9.17) +16.84 (±6.26) +32.13 (±13.34) +19.89 (±9.48) +31.79 (±8.54) +19.21 (±6.13) +January 16, 2023 + +5.2. Segmentation-Classification Approach +Table 4 shows the validation results obtained for segmentation, along with the training time. +The +segmentation and following classification training curves can be found in Appendix A.2. Considering these +results, all inputs seem to accurately segment the human body region, with Dice and IoU values higher +then 93%. Among these inputs, the cropped ADD stood out, although for a small margin (less than 1%), +presenting an IoU and Dice of 94.52% and 97.17%, respectively. The segmentation test results, along with +examples of segmented images can be found in Appendix A.2. +Table 4: Validation results, in percentage, of the UNET model, as well as the training time for 30 epochs +Input Type +Crop +Accuracy +Loss +IoU (±std) +Dice (±std) +Training Time (h) +DIF +False +98.80 +0.02 +93.81 (±1.96) +96.80 (±1.05) +1.37 +DIF +True +98.36 +0.03 +93.85 (±1.95) +96.81 (±1.05) +1.37 +ADD +False +98.94 +0.02 +94.34 (±1.71) +97.08 (±0.91) +1.38 +ADD +True +98.57 +0.03 +94.52 (±1.71) +97.17 (±0.91) +1.37 +Tables 5 and 6 present the validation results for classification, as well as the training time for 100 epochs, +and the quantitative evaluation of the model focus, respectively. The DIF input attained the highest values +of F1-score, namely 94.14% and 93.36% for non-cropped and cropped, respectively. However and once again, +the best focus was achieved with the cropped ADD input (Dice values with average of 28.17%). +Table 5: Validation results of the adapted UNET classification model, following the segmentation task, as well as the training +time for 100 epochs +Input Type +Crop +ACC (%) +Loss +F1-score +Precision (%) +Recall (%) +Training Time (h) +DIF +False +94.09 +0.16 +94.14 +94.29 +93.92 +4.48 +DIF +True +93.47 +0.17 +93.36 +93.70 +93.02 +4.53 +ADD +False +90.82 +0.24 +91.08 +92.07 +90.23 +4.49 +ADD +True +92.79 +0.27 +92.69 +93.08 +92.34 +4.43 +Table 6: Quantitative focus evaluation results, in percentage, of the validation grad-CAMs, when predicting with the adapted +UNET model for classification +Input Type +Crop +Dice (±std) +IoU (±std) +DIF +False +19.48 (±8.00) +11.01 (±5.06) +DIF +True +18.27 (±10.46) +10.44 (±6.80) +ADD +False +21.21 (±7.80) +12.08 (±4.88) +ADD +True +28.17 (±9.66) +16.75 (±6.35) +5.3. Analysis of the final approach +The best classification performance, as well as the most relevant and human-centred focus, were achieved +by the ResNet-50 model with a channel-wise attention mechanism (approach 2). Therefore, further evalu- +ation is conducted to analyse this model’s performance, in both offline and trial simulations, with an unseen +test dataset. +5.3.1. Offline testing +Table 7 presents the percentage of wrong classifications over the test set. Similarly to validation, the +cropped ADD revealed the most promising results, with the wrong predictions representing only 0.64% of +the dataset. +January 16, 2023 + +Figure 8: Confusion matrices for the ResNet-50 model with attention, over the 4 forms of input. +Table 7: Percentage of wrongly classified frames in the test set, by the ResNet-50 model with an attention mechanism +Input Type +Crop +Wrong predictions (%) +DIF +False +0.68 +DIF +True +2.38 +ADD +False +0.71 +ADD +True +0.64 +Figure 8 presents the obtained confusion matrices. These matrices reveal excellent results, specially for +the stop and TL classes, where the TP rate was never lower than 0.99. The cropped DIF was the least +favoured input by this architecture, with the lowest TP rate for the walk class (0.93). +Table 8 shows the results obtained from the focus evaluation of ResNet-50 model with attention. The +input influence over the model focus is in agreement with the one already verified in validation (Table 3). +Cropped DIF attained the higher similarity between their GT masks and the grad-CAMs, but with a very +small difference from the cropped ADD results (without surpassing 0.16%, considering both Dice and IoU). +The standard-deviation obtained for the cropped ADD was also smaller in comparison with that obtained +for the cropped DIF, similar to what was found for the validation set. +January 16, 2023 + +DIF input +ADD input +1200 +0.99 +0.0 +0.0 +0.01 +0.0 +0.0 +0.0 +1200 +1.0 +STOP +STOP +1000 +1000 +0.0 +0.99 +0.0 +0.01 +0.01 +0.98 +0.01 +0.01 +WALK +WALK +800 +800 +label +label +600 + 600 +0.0 +0.01 +0.99 +0.0 +0.0 +0.01 +0.99 +0.0 +TR +TR +400 +400 +200 +200 +0.0 +0.0 +0.0 +1.0 +0.0 +0.0 +0.0 +1.0 +TL +TL +TOP +STOP +WA +Predicted label +Predicted label +Cropped DIF input +Cropped ADD input +1200 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +1200 +1.0 +1.0 +STOP +STOP +1000 +1000 +0.0 +0.93 +0.02 +0.04 +0.0 +0.99 +0.0 +0.0 +WALK +WALK +800 +800 +label +label +600 + 600 +0.0 +0.01 +0.98 +0.0 +0.0 +0.01 +0.99 +0.0 +TR +TR +400 +400 +200 +200 +0.0 +0.0 +0.01 +0.99 +0.0 +0.0 +0.0 +1.0 +TL +TL +STOP +TOP +Predicted label +Predicted labelFigure 9: Plot of the GT (dashed line), predicted (dotted line) and post-processed predicted labels (solid line) for (a) Trial A +and (b) Trial B. Class IDs: 0=stop, 1=walk, 2=TR, 3=TL. +Figure 10: Plot of the values of the online metrics described in Section 4.3 for (a) Trial A and (b) Trial B, namely: instantaneous +accuracy (IA, solid line), instantaneous weighted accuracy (wIA, dashed line), instantaneous precision (IP, dashed line with +dots) and instantaneous calibrated precision (cIP, dotted line). +Table 8: Quantitative focus evaluation results, in percentage, of the test grad-CAMs, when predicting with the ResNet-50 +model with an attention mechanism +Input Type +Crop +Dice (±std) +IoU (±std) +DIF +False +29.97 (±14.56) +18.50 (±10.26) +DIF +True +32.38 (±10.32) +19.76 (±7.20) +ADD +False +22.14 (±13.02) +13.07 (±8.59) +ADD +True +32.30 (±8.83) +19.60 (±6.37) +5.3.2. Trial simulations +The cropped ADD, when fed to the ResNet-50 model with attention, achieved the best results across +validation and test, considering both classification power and focus evaluation. +Therefore, this was the +approach tested in trial simulations, along with the post-processing technique. Two representative trials, +corresponding to different conditions and extreme cases, are displayed: trial A) participant 11 performs a +TL at 0.5m/s (lowest gait speed); trial B) participant 15 performs a TR at 1m/s (fastest gait speed). +Figure 9 allows the comparison, at each instant, between the predictions (with and without post- +processing) and the GT classes. Figure 10 shows the temporal evolution of the online metrics described +in Section 4.3. +Table 9 shows, respecting the trials’ temporal order, the delays between the post-processed label and +the GT one, for each action transition. Negative values represent classes predicted earlier than their actual +start. +January 16, 2023 + +1.000 +1.00 +0.975 +0.99 +0.950 +e +2 0.98 +e +valu +val +0.925 +0.97 +tric +0.900 +Met +Met +0.96 +0.875 +0.95 +0.850 +0.94 +0.825 +0 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +Frame Index +Frame Index3.0 +3.0 +2.5 - +2.5 - +2.0 +2.0 +D +D +81.5. +1.5 +1.0 - +1.0 - +0.5 - +0.5 - +1 +0.0 +0.0 - +0 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +Frame ID +Frame IDFigure 11: Grad-CAMs visualisation, temporally ordered, for each one of the transitions in the slow trial (trial A). The green +and blue labels correspond to the first prediction and GT label, respectively, of the action that is beginning (P=predicted +class). +Table 9: Delays of the final predicted labels in relation to the respective GT labels, computed for each transition of the circuit +Time delay (s) +Trial +Walk +Turn +Walk +Stop +A +1.80 +0.80 +0.00 +-0.67 +B +0.20 +-0.20 +0.27 +0.27 +5.3.3. Grad-CAMs visualisation +Figure 11 presents the grad-CAMs visualisation for Trial A, where the model’s predicted labels corre- +spond exactly to the post-processed ones. For the beginning of each class, the visualisation starts at the +first frame of that action (for delayed predictions) or at the first correct prediction (for early predictions, +as it is the case of the stop class, in this trial) and ends at the first right prediction or first GT frame, +respectively. Note that these are not necessarily consecutive frames on the dataset, that depends on the +action delay registered in Table 9. Nonetheless, they serve as a good representation of the focus evolution +between the GT and its respective correct prediction (or vice-versa) and, for the delayed predicted labels, +it always includes two immediately preceding frames. +January 16, 2023 + +STOP +STOP +STOP +STOP +ALK +门 +0.9997414 +0.9936452 +0.9943064 +0.9462649 +61344504 +WALK +WALK +WALK +TALK +WALK +WALK +WALK +TL +.99775714 +0.9987877 +98631525 +0.7700368 +0.98371756 +2 +TL +TL +WALK +WALK +WALK +WALK +WALK +0.90710625 +0.99804914 +0.99995494 +0.9999778 +0.9999862 +3- WALK +WALK +WALK +STOP +STOF +STO +STOF +STOF +0.8881017 +0.98464704 +0.9998155 +0.99951065 +0.99981517 +4- STOP +STOP +STOPFigure 12: Grad-CAMs visualisation, temporally ordered, for each one of the transitions in the fast trial (trial B). The +green and blue labels correspond to the first prediction and GT label, respectively, of the action that is beginning (P=predicted +class). The orange ones correspond to the perturbations in the model’s predictions that do not correspond to the post-processed +predicted class. +The same applies to Figure 12, representing Trial B. This trial presents more on-off noise in the model’s +outcomes, specially in the transition from walk to turn (Figure 9b). Hence, its predictions do not always +correspond to the post-processed labels. As the latter constitutes the final predicted classes, the beginning or +end of these visualisations are depicted by the respective post-processed labels, for early and late predictions, +respectively. +Moreover, in the two first presented classes, a frame immediately after the correct post- +processed prediction/GT label was added for purposes of focus evolution assessment. +6. Discussion +The results obtained by the DL approaches are critically discussed in this section, pointing some limita- +tions and insights for possible improvements. +6.1. Tailored inputs and model’s focus evaluation +Distinct input forms influence the models performance differently. Despite leading, in most cases, to +lower classification metrics, cropping the images helps to direct the focus to the human ROI (see Table 3), +as it excludes a significant portion of background. Therefore, better classification results can be associated +with less reliable extracted features. +January 16, 2023 + +STOP +STOP +STOP +WALK +WALK +0.9939032 +0.9860512 +088574237 +0.8731011 +0.97841656 +WA +WALK +WALK +VALK +WALK +IR +IR +0.7609441 +0.91726273 +936562 +0.7841732 +0.9954194 +TR +2 +TR +TR +WALK +WALK +IR +IR +TR +TR +WALK +0.96653795 +0.98590267 +0.8900123 +.96647906 +0.8672215 +3- WALK +WALK +WALK +WALK +WAL +WALK +WALK +STOF +0.9733175 +0.9876581 +0.9695332 +0.6238496 +0.732093 +STOP +STOP +STOPA greater improvement in focus was registered when cropping the ADD input, which confirms that +non-cropped ADD includes more information about the background motion. +The fact that the highest +improvement rate in focus achieved by adding an attention mechanism to the ResNet-50 model was recorded +by the non-cropped ADD also supports this statement. According to the results, the cropped ADD was +the input that consistently presented high similarities between grad-CAMs and GT masks, raising the belief +that ADD images also encode more human body motion information. +From these comparisons, three aspects can be inferred: i) background motion may contain more evident +features, easier to extract, that can help the offline classification task. However, this should not be the main +focus of the model since these features are not reliable for detecting transitions, real-time applications or +even generalisations to other datasets, where the background is static; ii) a careful performance evaluation is +needed, since better classification results can be associated with non-ideal feature extractions and overfitting +to the background; and iii) despite not being a commonly used approach, cropping the inputs was a feasible +way to enhance the model focus and thus increased the reliability of the respective classification results. +Nevertheless, the higher registered improvement was not higher than 10% (see Table 3). This can be related +to the background characteristics, such as the presence of floor stripes enhancing background motion (Figures +11 and 12). +Literature on HAR/HAP normally uses sequences of RGB frames to provide the sense of temporal +motion. +Instead of this approach, the proposed inputs were able to provide (a significant part of) this +temporal information in one single frame, avoiding the use of Recurrent Neural Networks. With a larger +background variability in the dataset, while carefully avoiding pavement marks during the acquisitions, these +tailored inputs could become a more reliable method to induce action-aware feature extraction, avoiding +more efficiently the background features. +Nevertheless, the lower metrics obtained when evaluating the model focus can also be due to the quanti- +tative focus evaluation algorithm itself. The GT masks used here as a comparison term are binary, presenting +the highest score (1) for all the human area, but this region may not be equally important to decode human +motion. For example, feet and knees may present more orientation and position variations which indicate +the step’s direction, so it would be correct for the model to give higher focus to these particular regions. +For this reason, comparing heatmaps to these masks is correctly penalising FP, but it is also punishing +the model for not focusing on the complete lower body, including, for example, the more static upper leg +region. These effects can lower the values of the calculated Dice and IoU. The masks are already computed +in the tightest ROI possible to attenuate this effect, but it does not completely solve it. A possible solution +for this problem would be to change the GT masks pixel values, according to the input images. Thus, +as the higher pixel intensities in the input correspond to motions with larger amplitudes, while the lower +correspond to more static areas, this information could be used to scale the scores equal to 1 in the GT +masks, creating a sort of human motion masks. An even more accurate form of information to scale these +masks according to the body’s amplitude of motion, would be the human poses, from the Xsens data, for +instance. Nevertheless, this last option would unduly increase the computational expense and complexity +of this algorithm. Observing the grad-CAMs of the final selected algorithm (Figures 11 and 12), one could +also attempt to only generate masks of the user’s feet. +6.2. Single-frame Classification Approach +Based on the results presented in Table 2, it is possible to notice that ResNet-50 performed better than +VGG16, achieving higher F1-score values than VGG16 ([94.34%, 98.27%] over [94.53%, 96.80%], respec- +tively). Hence, this problem benefited from residual features, skip connections and deeper networks duly +initialised or pre-trained. Nevertheless, the difference between both performances was not that high (lower +than 1.47% in F1-score - Table 2), meaning that the task of early recognising human motions from the SW +dataset may also be approached by less deep models. Nonetheless, ResNet-50 achieved better classification +results and focused better on the human region (average boost of 7.03% in Dice and 4.96% in IoU), over all +input forms, and was, thus, the chosen model to be tested with an attention mechanism (approach 2). +The addition of the channel-wise attention mechanism enhanced not only the classification metrics, +improving the F1-score by an average of 2.93%, but also the similarity between grad-CAMs and GT masks, +with improvements until 4.20% in Dice, across all inputs. Only the model focus associated with the cropped +January 16, 2023 + +ADD input was slightly worse than the ones registered with the ResNet-50 baseline model (see Table 3). +However, since the difference is not that large (0.34% and 0.68% in Dice and IoU, respectively), this could +be due to small variations and, thus, was not considered as a relevant fact. +These results proved the importance of dealing and modelling the distinct learning abilities of the different +convolutional channels, not only to increase CNN performance, but also to improve the relevance of the +features extracted. However, the values presented in Table 3 for ResNet-50 with attention are not that +higher than the ones for the corresponding baseline model, specially for the cropped inputs, proving that +this channel-wise attention mechanism, although unequivocally beneficial to the classification task, still does +not completely correct its main focus, as Dice and IoU are still below 50% and most pixels in grad-CAMs +heatmaps do not correspond to the human body region. +Facing these facts, a spatial attention mechanism could also be designed for this problem, guiding the +model to use attentional regions, instead of the whole frame. As in Wu et al. (2020), also enhancing local +features by combining this with the channel-wise mechanism, could lead to better performances and, in +this case, more properly focused solutions. The spatial attention maps could even be compared with the +suggested human motion masks for focus evaluation or, in a more bold experiment, as part of the training +loss. +The cropped ADD appeared as the most promising input for this task and, when fed to the ResNet-50 +model with the attention mechanism, achieved the best overall results. +6.3. Segmentation-Classification Approach +Looking at the segmentation training curves (Figure A.16), one can see that, as the epochs advance, +there is a tendency for overfitting, given the small increase in the gap between the validation the training +losses. +Although apparently small, this can propagate to the following pre-trained classification model +and induce bad generalisation abilities or even worse cases of overfitting. That is why the segmentation +training was shorten to 30 epochs and the weights were chosen considering the minimal validation loss. +Despite the training reduction, the adapted UNET still revealed problems of weak generalisation (Figure +A.17). In agreement with these training curves, the classification metrics were worse than the ones achieved +by previous evaluated models, as the maximum F1-score was of 94.14% (Table 5), which is lower than +the minimum registered for the previous models (94.34%, for the baseline ResNet-50, shown in Table 2). +Nevertheless, these metrics were still above 90%. The severest cases of unrepresentative training dataset +and consequent generalisation issues were verified by the ADD input type, which is associated with lower +validation performances, namely 92.69% (cropped) and 91.08% (non-cropped form) of F1-score (Table 5). +When connecting the segmentation and classification results (Tables 4 and 5, along with Figure A.17), it +seems to exist an inverse relation between segmentation power and the classification generalisation ability. +This may mean that this cascade approach is leading the model to focus on input traits that are not +representative of the whole dataset, following the overfitting problems during segmentation. Cross-validation +should be performed to prove this statement. Even so, the focus on particular traits can be associated with +the fact that, despite the final aim of human motion decoding, the GT masks used are leading to the +segmentation of the whole body, including large clothes and more static human areas. Therefore, using the +aforementioned human motion masks as labels could decrease the chances of overfitting, while pursuing the +differential segmentation of the human body, according to its motion. This could enhance the weights used +to pre-train the classification model. Other options to help overcoming the overfitting problem consist on +experimenting other simpler segmentation models or even include spatial data augmentation. Moreover, the +number of frozen layers should also be studied and tuned. +As for the grad-CAMs evaluation (Table 6), the segmentation-classification approach was not the most +effective to attain its main goal: improving the extraction of human-centred features during classification. +The low results, along with their resemblance to the ones achieved by VGG16, point to an influence mainly +exerted by the input properties and not by the two-stage framework itself. +6.4. Trial simulations +The ResNet-50 model with a channel wise attention mechanism was the best model in both aspects: +classification rates and focus relevance, specially when fed with the cropped ADD input. +Testing this +January 16, 2023 + +approach in trial simulations led to general good performances, with the model being able to identify the +different consecutive walking events. +The model uncertainty revealed a greater prominence at higher gait speeds, in the form of on-off noise (see +Figure 9). However, these perturbations were easily smoothed by the proposed post-processing technique, +without adding time delays in the correct predictions. Despite presenting more noise, Trial B achieved +overall higher online metric values, over 94%, due to its lower time delays registered in transitions. +Table 9 shows, respecting the trials’ temporal order, the delays between the post-processed and the GT +labels, for each action transition. For the trial performed at 1 m/s, the delays are at least 0.37s lower +than the average step time for this gait speed (0.64s). For lower gait speeds, the time lags registered were +higher, confirming the greater challenge implied by early detecting slower and more subtle motion changes. +These delays could be further decreased for real-time applications, through a proper training procedure that +includes transitions in the dataset, but also through a data quality enhancement to further improve the +model focus (e.g. large variety of backgrounds without floor stripes/marks). +Overall, the results prove the chosen approach is suitable for early action recognition, achieving average +online metrics between 91.72% and 98.65%. However, it still needs improvements to early detect an action, +as it can be seen by the model performances at the transition inputs. Nevertheless, the obtained results +were still good, with not so critical delays, considering the fact that the neural network was not trained +with transitions. Hence, with the mentioned suggestions, specially the inclusion of transitional frames in +the training procedure, this performance could be further enhanced. For this to happen, one has to first +improve the labelling accuracy, decreasing the bias effect introduced by the person controlling the joystick. +6.4.1. Grad-CAMs visualisation +The displayed grad-CAMs (see Figures 11 and 12) showed that the model focus is not too deviated +from the human region, but it still considers some background information, specially the visible motion of +the floor stripes. The floor stripes were a non-ideal property of the acquisition environment, which visibly +affected the model training. As background motion is a consequence of the walker’s movement, and not the +user’s, this misleading focus can be among the causes of the registered time delays, when predicting each +action’s beginning. +For example, in Figure 12 (last row), the stop detection was delayed, as the walker kept moving after +the subject stopped. So the model must have considered the stripes and the large clothes motions, instead +of the user’s steadier positions. As the device decelerates, this background motion became less evident and +the model started to focus on the feet. In the slow trial, this deceleration phase is shorter and slower, +so the background motion stopped appearing in the RGB inputs before the human movement, allowing +the model to better perceive the progressive horizontal alignment of the feet (last row of Figure 11). This +situation is similar to the beginning of the first walking event at low gait speed (first row), where the walker +starts to slowly accelerate, so the background appears static, and the feet move slowly and closer to each +other, leading to a confusion between stop and walk classes. It seems the model cannot yet associate the +horizontally misalignment of the feet as a walking trait. +The turning event was anticipated in Trial B, which seemed like a good achievement for motion intention +decoding. Nonetheless, looking at the respective grad-CAMs (second row, in Figure 12), one can see that +this class was first predicted based on the vertical misalignment between the floor stripes. This helps to +visualise and understand the confusion and model uncertainty between these two classes (walk and TR/TL). +These visualisations showed that the model focus still needs to be improved in transitions, in order to +be integrated in a real-time control mechanism. Nevertheless, this focus appeared to be better than the +expected from its quantitative evaluation, with grad-CAMs concentrated around the feet, which can be due +to the GT masks used in that algorithm. +7. Conclusion +This work presents a novel way to decode motion intention in SWs. Three different approaches were +devised, two facing single-frame classification, and another facing a segmentation-classification approach. +January 16, 2023 + +For that, a custom dataset of 15 healthy participants was acquired with a smart robotic walker, considering +realistic scenarios and circuits. Each participant performed a total of 24 trials, each one containing three +of the target classes (stop, walk, turn right and turn left). Considering this, a balanced dataset of frames +containing 28800 RGB-D images was created, extracting 40 frames per video sequence, and used to train, +evaluate, and compare the proposed approaches. +Regarding the model architectures, state-of-the-art VGG16 and ResNet-50 were implemented in the first +approach and then, an attention mechanism was added to the best model (second approach). For the third +approach, the UNET neural network was used for segmentation and adapted for the following classification +task. Four different input forms were studied, namely cropped and non-cropped ADD and DIF images. +These were obtained considering a sliding window approach of 4 frames with a stride of 2, by summing the +four frames of the window (ADD) or subtracting the last frame from the first (DIF). This approach encoded +0.27s of motion information without using recurrent neural networks, which are a commonly used approach +in the literature. To evaluate the model performance, we considered standard metrics, namely accuracy, F1- +score, recall, and precision. For trial simulations assessments, OAD metrics were used, such as instantaneous +accuracy, instantaneous precision, instantaneous weighted accuracy, and instantaneous calibrated precision. +We also evaluated the model focus considering a novel method that quantitatively compares its grad-CAMs +with GT masks of the human body region. +Regarding the different inputs, we concluded that the non-cropped ADD input encodes more motion +information, but these, together with DIF images share a common disadvantage with the optical flow: in +realistic videos, these inputs also encode background motion which may deviate the model focus from the +human region. Cropping most part of the background surrounding the image’s ROI proved to have a major +impact on the model’s focus. We also verified that ResNet-50 with an attention mechanism, when fed with +cropped ADD inputs, attained the most promising results (offline accuracy and F1-score higher than 95%). +This enabled an enhancement in the model focus towards the human body region (Dice rounding 32%) when +compared to the other models, but still needs further improvements. +7.1. Limitations and future research insights +This work presents some limitations that should be considered in future research insights. Enhancing the +quality of the acquired data becomes necessary to train the algorithms with transitional inputs and improve +the relevance of the extracted features. Recording in a more controlled environment, without marked floors +or too bright conditions, would be relevant to not deviate the model focus to the background. Moreover, +the labelling procedure should also be improved (e.g. with the use of force sensors) to allow the inclusion +of transitions, without mislabelled samples. Data from pathological individuals should be acquired and the +use of transfer learning may be considered to endow the model with the ability to detect motion intentions +for this population. Regarding methodology, the method to evaluate the model focus can be refined. For +instance, the human masks can be improved by lowering the pixel values in more static human body areas, +creating human motion masks, along with the inclusion of false positives as a metric to accurately assess how +much the model is focusing on the background. Spatial attention mechanisms or self-supervised learning can +also be explored to improve the model focus. Additionally, tailored losses could also be tested to improve +the early detection ability, especially if the model keeps presenting significant time delays. Lastly, the use +of visual transformers could also be investigated to progressively obtain simpler models capable of learning +by observation. +Along with its potential for improvement, we hope this work can also serve as future benchmark and +encourage further investigations on decoding fine-grained human actions directly through visual information. +Acknowledgements +This work has been supported by the Fundação para a Ciência e Tecnologia (FCT) with the Reference +Scholarship under Grant 2020.05708.BD and under the national support to R&D units grant, through the +reference project UIDB/04436/2020 and UIDP/04436/2020. +January 16, 2023 + +Author Contributions +Carolina Gonçalves: Methodology, Investigation, Data curation, Formal analysis, Software, Writing - +Original Draft, Writing - Review & Editing; João M. Lopes: Methodology, Investigation, Data curation, +Software, Writing - Review & Editing, Funding acquisition; Sara Moccia: Methodology, Resources, Super- +vision, Validation, Writing - Review & Editing; Daniele Berardini: Software, Writing - Review & Editing; +Lucia Migliorelli: Software, Writing - Review & Editing; Cristina P. Santos: Conceptualization, Method- +ology, Resources, Supervision, Validation, Writing - Review & Editing, Project administration, Funding +acquisition. +References +Aliakbarian, M. S., Saleh, F. S., Salzmann, M., Fernando, B., Petersson, L., and Andersson, L. (2017). Encouraging LSTMs +to Anticipate Actions Very Early. Proceedings of the IEEE International Conference on Computer Vision, pages 280–289. +André, J., Lopes, J., Palermo, M., Gonçalves, D., Matias, A., Pereira, F., Afonso, J., Seabra, E., Cerqueira, J., and Santos, +C. (2020). Markerless gait analysis vision system for real-time gait monitoring. 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(2020). +Learning-Aided User Intent Estimation for Smart Rollators. +Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS, +pages 3178–3183. +Zhao, X., Zhu, Z., Liu, M., Zhao, C., Zhao, Y., Pan, J., Wang, Z., and Wu, C. (2020). A Smart Robotic Walker With Intelligent +Close-Proximity Interaction Capabilities for Elderly Mobility Safety. Frontiers in Neurorobotics, pages 1–17. +Zhou, B., Khosla, A., Lapedriza, A., Oliva, A., and Torralba, A. (2016). Learning Deep Features for Discriminative Localization. +Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 2921–2929. +Appendix A. +Appendix A.1. Single-frame Classification Approach +Training VGG16 and ResNet-50 architectures with each developed input, computed from the acquired +dataset, resulted in the training curves presented in Figure A.13. As one can see, the overall curves are +stable and with no signs of overfitting, reaching good results. It is noticeable that ResNet-50 learned faster +and provided some gains in loss and accuracy. +January 16, 2023 + +Figure A.13: Accuracy and loss training curves for VGG16 and ResNet-50 models. +January 16, 2023 + +DIF Input +DIF Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +9 0.4 - +0.3 - +0.3 - +0.2 +0.2 - +0.1 +0.1 +0.0 +0.0 - +20 +20 +40 +60 +100 +60 +80 +0 +80 +0 +40 +100 +Epoch # +Epoch # +Cropped DIF Input +Cropped DIF Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_oss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 - +20 +40 +60 +100 +0 +80 +20 +60 +80 +100 +0 +40 +Epoch # +Epoch # +ADD Input +ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.3 . +0.3 - +0.2 +0.2 - +0.1 +0.1 +0.0 +0.0 - +40 +20 +20 +60 +80 +100 +0 +60 +80 +0 +40 +100 +Epoch # +Epoch # +Cropped ADD Input +Cropped ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +20 +60 +80 +100 +20 +80 +0 +40 +0 +40 +60 +100 +Epoch # +Epoch #Figure A.14 shows the obtained training curves for the ResNet-50 model with a channel-wise attention +mechanism. +Figure A.14: Accuracy and loss training curves for ResNet-50 model with an attention mechanism. +Table A.10 compares the grad-CAMs evaluation over the unseen test set for the VGG16 and ResNet-50 +(without and with attention) models. +Table A.10: Quantative evaluation results, in percentage, for test grad-CAMs, when predicting with the VGG16 and ResNet-50 +models (without and with an attention mechanism) +Input +VGG16 +ResNet-50 +ResNet-50 with attention +Type +Crop +Mean +Dice (±std) +Mean +IoU (±std) +Mean +Dice (±std) +Mean +IoU (±std) +Mean +Dice (±std) +Mean +IoU (±std) +DIF +False +17.18 (±8.60) +9.64 (±5.31) +26.01 (±15.19) +15.84 (±10.24) +29.97 (±14.56) +18.50 (±10.26) +DIF +True +26.39 (±13.64) +15.91 (±9.12) +28.38 (±13.05) +17.21 (±8.92) +32.38 (±10.32) +19.76 (±7.20) +ADD +False +20.99 (±7.75) +11.94 (±4.91) +16.93 (±14.07) +9.95 (±9.19) +22.14 (±13.02) +13.07 (±8.59) +ADD +True +28.62 (±10.90) +17.18 (±7.64) +29.37 (±12.54) +17.86 (±8.87) +32.30 (±8.83) +19.60 (±6.37) +Appendix A.2. Segmentation-Classification Approach +Examples of the generated masks, used as labels for segmentation and focus evaluation algorithm, are +given in Figure A.15. +Figures A.16 and A.17 display the segmentation and classification training curves, respectively, using the +UNET and adapted UNET models. +January 16, 2023 + +Train Loss/ACC: DlF Input +Train Loss/ACC: ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.6 +0.5 +0.5 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 - +0.1 +0.0 +0.0 +0 +10 +20 +30 +40 +50 +60 +70 +0 +20 +40 +60 +80 +Epoch # +Epoch # +Train Loss/ACC: Cropped DIF Input +Train Loss/ACC: Cropped ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.6 +0.5 +0.5 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 - +0.1 +0.0 +0.0 +0 +10 +20 +30 +40 +50 +60 +0 +20 +40 +60 +80 +100 +Epoch # +Epoch #Figure A.15: Examples of individually non-corrupted masks, along with their corresponding RGB inputs, for a window length +of 4 frames. The presented masks are already corrected. +Figure A.16: Accuracy and loss training curves for segmentation. +January 16, 2023 + +DIF Input +ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +Epoch # +Epoch # +Cropped DiF Input +Cropped ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +Epoch # +Epoch #Figure A.17: Accuracy and loss training curves for the adapted UNET model for classification. +To help visualise this model’s segmentation ability, Figures A.18, A.19, A.20 and A.21 show the best and +worst cases of segmented images, for each type of input. Notice that, for the cropped ADD, the segmentation +of the human body was very satisfactory, even in the worst case, although including some noise. Contrarily +to this, the other inputs revealed occlusions as the apparent main factor behind a worse segmentation. The +quantitative test results shown in Table A.11 indicate that the cropped ADD images were the easiest to +segment, followed by the non-cropped ADD, cropped DIF and, finally, non-cropped DIF images. +Table A.11: Evaluation results, in percentage, of the UNET model segmentation over the test set +Input Type +Crop +IoU (±std) +Dice (±std) +DIF +False +90.79 (±5.55) +95.08 (±3.27) +DIF +True +91.89 (±4.66) +95.71 (±2.64) +ADD +False +92.12 (±5.71) +95.80 (±3.45) +ADD +True +93.63 (±3.41) +96.68 (±1.87) +January 16, 2023 + +DIF Input +ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +Loss/Accuracy +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Epoch # +Epoch # +Cropped DIF Input +Cropped ADD Input +train_loss +train_loss +1.0 +1.0 +val_loss +val_loss +0.9 +0.9 +train_acc +train_acc +0.8 +0.8 +val_acc +val_acc +Loss/Accuracy +Loss/Accuracy +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Epoch # +Epoch #Figure A.18: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective non- +cropped DIF inputs. +Figure A.19: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective cropped +DIF inputs. +January 16, 2023 + +Figure A.20: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective non- +cropped ADD inputs. +Figure A.21: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective cropped +ADD inputs. +January 16, 2023 + diff --git a/PdE5T4oBgHgl3EQfYg_i/content/tmp_files/load_file.txt b/PdE5T4oBgHgl3EQfYg_i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ca08383ae6d419f9c2e81024c88b37c09a023f2 --- /dev/null +++ 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(CMEMS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' University of Minho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Guimarães,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Portugal bLABBELS - Associate Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Braga/Guimarães,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Portugal cThe BioRobotics Institute and Department of Excellence in Robotics and AI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Scuola Superiore Sant’Anna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Italy dDepartment of Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Università Politecnica delle Marche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Ancona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Italy eClinical Academic Center of Braga (2CA-Braga),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Braga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Portugal Abstract Gait disabilities are among the most frequent impairments worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Their treatment increasingly relies on rehabilitation therapies, in which smart walkers are being introduced to empower the user’s recovery state and autonomy, while reducing the clinicians effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For that, these should be able to decode hu- man motion and needs, as early as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Current walkers decode motion intention using information gathered from wearable or embedded sensors, namely inertial units, force sensors, hall sensors, and lasers, whose main limitations imply an expensive solution or hinder the perception of human movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Smart walkers commonly lack an advanced and seamless human-robot interaction, which intuitively and promptly understands human motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A contactless approach is proposed in this work, addressing human motion decoding as an early action recognition/detection problematic, using RGB-D cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' We studied different deep learning-based algorithms, organised in three different approaches, to process lower body RGB-D video sequences, recorded from an embedded camera of a smart walker, and classify them into 4 classes (stop, walk, turn right/left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A custom dataset involving 15 healthy participants walking with the walker device was acquired and prepared, resulting in 28800 balanced RGB-D frames, to train and evaluate the deep learning networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The best results were attained by a convolutional neural network with a channel-wise attention mechanism, reaching accuracy values of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='61% and above 93%, for offline early detection/recognition and trial simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Following the hypothesis that human lower body features encode prominent information, fostering a more robust prediction towards real-time applications, the algorithm focus was also quantitatively evaluated using Dice metric, leading to values slightly higher than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Promising results were attained for early action detection as a human motion decoding strategy, with enhancements in the focus of the proposed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Keywords: Deep learning, early action detection, early action recognition, human motion decoding, human-robot interaction, RGB-D video, smart walker 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Introduction In 2018, over a billion people were estimated to live with some form of disability (WHO, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This is caused by ageing population and by an increase in chronic health conditions (WHO, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Tumours, aneurysms, strokes and cerebellar ataxia are prominent causes of gait and posture impairments (Mikola- jczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Bonney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Jonsdottir and Ferrarin, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Celik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021), which can have severe repercussions in strength, sensation, and movement, leading to lack of stability and increased risk of falls (Mikolajczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Rehabilitation therapies reveal promising results tackling these impairments Email address: cristina@dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='uminho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='pt (Cristina P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Santos) January 16, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05575v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CV] 13 Jan 2023 (Milne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Gait rehabilitation requires long periods of intense physical exercise, presenting chal- lenges for clinicians, due to the high demand of physical effort and inter-patient variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, it may also be affected by the clinician experience and inter-clinician variability (Mikolajczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018), making this therapy more time-consuming and prone to error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To overcome these limitations, assistive technologies have emerged as effective means to increase the patient’s independence and participation in their rehabilitation therapies (WHO, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Assistive technologies include Smart Walkers (SWs), which are intended to be used by or with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' SWs no longer serve as just conventional physical supporters, but comprehend other intelligent functionalities to promote an efficient Human-Robot Interaction (HRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These devices should be able to decode human motion and needs, as early as possible, which would be essential for a seamless HRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This would be clinically relevant since it would enable a more natural and anticipated assistance, encouraging patients to take an active role in rehabilitation exercises or therapy sessions (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, the interaction entailed by motion decoding should result from the device built-in sensors in order to maximise intuitiveness and technology acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Human motion decoding in SWs can be achieved through the use of wearable sensors, such as Inertial Measurement Units (IMU) (Weon and Lee, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, these have to be placed on the user’s body, which hampers the clinician’s task and the patient’s movement, making rehabilitation more time consum- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, these can suffer electromagnetic interference from the walker’s motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Several sensors embedded in SWs have also been used for this purpose, although entailing other limitations that hinder rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For instance, force sensors (Cheng and Wu, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Sierra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018) present a reduced long-term effectiveness (Paulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017), infrared sensors (Paulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2015) can be easily corrupted by light conditions and a handlebar specially designed with hall sensors (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019) implies specific movements besides the natural gait, increasing the patient’s cognitive load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, preprocessing may also be required to clean the output signals, for instance, when resorting to IMU or force sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' RGB-D cameras can be used along with SWs (Palermo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' André et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Although fostering a cheap, intuitive, and contactless solution, without interfering with the user’s gait, these are not usually explored for the purpose of motion decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This can be due to the challenges inherent to image analysis, especially in realistic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Deep Learning (DL) methods have shown robustness tackling Human Action Recognition (HAR) (Yeaser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020) and, more recently, Human Action Prediction (HAP) (Chalen and Vintimilla, 2019) from RGB-D images or videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' While the former tackles the issue of current action recognition or detection, the latter intends to anticipate the action’s ending (early action recognition/detection), seeing only a small part of the action, or even its beginning (action anticipation), taking a step towards forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Despite the progresses made, dealing with the hindrances of RGB-D image analysis, HAR and HAP methods have not yet been duly explored for human motion decoding in SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Following the need of a more intuitive way of decoding motion in SWs, robust to realistic environments and capable of perceiving natural gait movements without hindering them, this work innovatively addresses motion decoding as a HAP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' We propose the use of DL-based algorithms for online early detection and recognition of walking directions, from RGB-D videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These videos, recorded by a SW embedded camera (Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021), focus on the user’s lower body, which we hypothesise to encode the most relevant motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Inspired by literature on HAP, five DL algorithms were implemented within three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These approaches are responsible for implementing two different strategies: classification or segmentation-classification, in an attempt to evaluate and enhance the algorithm’s focus on the human body, along with its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Different inputs from the RGB-D videos were evaluated, encoding the videos’ temporal information into one single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, this solution avoids not only extra sensors and a cognitive load to the user, but also computational-expensive complementary tasks, such as pose estimation, and additional computational complexity by not resorting to spatio-temporal DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To efficiently predict walking directions directly from camera streams, three key-performance requirements were also established to evaluate the DL algorithms: Accurately recognise and detect 4 actions (Stop, Walk, Turn Right (TR) and Turn Left (TL)), while only seeing small temporal sequences to increase the efficiency of the algorithm’s response in online scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Extract human-centred features, when detecting these actions, to avoid background bias and motions as consequence of camera’s movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This will help to achieve a reliable performance for real-time applications, as the SW can only move after detecting the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Face online detection with a maximum admissible delay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64s, which corresponds to the determined medium duration of one healthy step, while walking at the fastest velocity assumed by the walker (1m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This step time was determined in laboratory experiments and it is in accordance with Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This paper is organised as follows: Section 2 presents relevant related work, critically discussing its advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Section 3 details all the methods for data acquisition and data preparation, along with the devised DL approaches, including model architectures and the involved algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Section 4 summarises the implementation details for training and evaluation of the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Section 5 and Section 6 present and critically discuss all the obtained results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Finally, Section 7 summarises the findings of this work and proposes future research insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Human Motion Decoding in Smart Walkers Most of the current SWs decode human motion directly, demanding some level of physical intervention from the user and/or an extra load of cognitive effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This direct mode has been typically implemented with specially designed handlebars, including force/pressure/load sensors (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Rodriguez-losada, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Spenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Jiménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Cheng and Wu, 2017), infrared cameras and Light Emitting Diodes (Paulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017) or Hall sensors (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019), which require specific hand motions to encode each walking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In an indirect mode, the walker becomes responsible for analysing the end-user’s movement, inferring, from this, the walking directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' LIDAR sensors combined with wearable ones (Weon and Lee, 2018) have been used to analyse the kinematics of lower limbs and measure feet orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2015) also resorted to a depth camera for feet position and orientation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020) used multi-channel proximity sensors to determine each leg’s distance and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These studies highlighted the relevance of lower body features to naturally infer the user’s walking directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, the presented sensory technologies incur in additional costs for the motion detection task, including increased expenses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' LIDAR) and number of sensors, reducing technology acceptance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' IMU and proximity sensors), signal corruption by environmental conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' depth camera by sunlight) or electromagnetic interference, while causing long-term discomfort and hampering movement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' IMU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Vision-based approaches should benefit the task of indirectly decoding human motion, avoiding the aforementioned limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Particularly, through the use of RGB-D cameras, usually built into SWs for other functionalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', gait and posture assessment (Palermo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020) used a RGB-D camera to perceive the intended walking direction from the torso kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' An hybrid Proportional- Integral-Derivative (PID) controller with integrated digital practical differentiator was then implemented to calculate the desired wheel rotation angles, being able to track the subject in forward and turning movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, the use of upper body information entailed a disadvantage: the upper body swaying (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Chalvatzaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2019) complements RGB-D data with 2D laser data to perform real-time gait analysis, as well as to track the user’s Centre-of-Mass position and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This, assembled with the desired human-related robot coupling parameters, became the input used to forecast the human motion and the evolution of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The i-Walk system (Chalvatzaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020) points out the potential of using RGB-D cameras, in particular coupled with DL models, to attain an effective activity and gesture recognition, but not specifically to drive the walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Although promising, these vision-based motion decoding approaches (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Chalvatzaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019, 2020) require additional computational tasks to estimate human poses or parameters from the acquired RGB videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This can lead to error propagation, when feeding these inputs to a following motion decoding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To the best knowledge of the authors, no current SWs present a solution that decodes human motion directly from camera streams, decreasing the complexity inherent to human pose-based solutions, while January 16, 2023 enabling a seamless and intuitive HRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This work contributes for this purpose by proposing DL-based approaches to early detect and recognise walking directions through lower-body RGB-D videos, while walking with a robotic walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Vision-based DL Methods for HAR/HAP Recent studies have shown great potential for HAR (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2016) and HAP (Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017), resorting to DL algorithms fed with RGB- based inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Convolutional Neural Network (CNN) models, optionally combined with Long Short-Term Memory (LSTM), are the commonly used architectures in these tasks, usually resorting to video chunks or sequences of frames as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The most common methods are based on colour (RGB) data (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020), on a combination of colour and depth (RGB-D) data (Jalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017), or on skeleton data (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Concerning HAR, Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020) applied a DL solution to automatically recognise falls in stacks of seven RGB frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The model architecture consisted on a CNN model as feature extractor, namely the VGG16, and a Bidirectional LSTM as feature classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2016) also implemented a LSTM model, aiming at Online Action Detection (OAD) through skeleton information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, simultaneously to frame-wise action (and background) classification, they proposed an estimation of the start and end frames of the current action, based on the definition of Gaussian-like curves for each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As for De Geest and Tuytelaars (2018), a different OAD proposal was made, using LSTM to model long term dependencies between actions, since human actions can imply a certain order (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' standing, after being sited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, these can also be random or environment-depending, as in free walking trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' When handling HAP, there are two main lines of work in the literature: i) directly predicting the future frames’ class or the current one, before the action ends (classification) (Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019) or ii) generating future visual representations that are further classified (regression followed by classification) (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Vondrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, a common focus of these studies is the design of novel loss functions that can reduce the predictive generalisation error (Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Vondrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2016) resorted to visual representations as promising prediction targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The proposed model consisted on AlexNet architecture, implementing five convolutional layers followed by five Fully Con- nected (FC) layers, with ReLu activations throughout the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To deal with the model uncertainty, the last three FC layers presented K networks, so each frame resulted in K future visual representations, one for each plausible future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Approaches like Vondrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2016) anticipate only a representation of a fixed future time, based on a single past frame’s representation, dismissing the history and temporal trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2017) proposed instead a Reinforced Encoder-Decoder network which took multiple history rep- resentations as input and learned to anticipate a sequence of future ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The video was segmented into chunks of six frames, each processed by a CNN model to extract a chunk representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This was then fed to a LSTM-based Encoder-Decoder, outputting a prediction of the future video chunks’ representations, in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The classification of these future representations was handled by two fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Despite all the research and developments, as well as the ability to use unlabelled videos for training, generating future representations is still defying, as well as time-consuming and prompt to error accumulation (Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, the learned representation may not be related to the action itself, as it can be influenced by background or other variables (Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Direct action prediction approaches avoid these limitations, by exploiting different forms of features and/or tailored losses to directly predict future classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2017) resorted to the VGG16 model, connecting it to two different branches, one to compute context-aware features, encoding global information about the scene, and another to compute action-aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These features were then sequentially introduced in a multi-stage LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The extrac- tion of action-aware features relied on Class Activation Maps (CAMs) (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2016), which indicate the regions in the input frame that most contributed to the class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, this implied the training of CNN models before using them to compute CAMs, in an offline manner, which were then fed to January 16, 2023 Figure 1: General workflow, including all the main methods implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' GT stands for Ground-Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' the multi-frame classification LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Besides not being suitable for online applications, this use of CAMs does not guarantee these maps always focus on the action’s relevant elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For example, similar human activities may drive the model to extract features from the surroundings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' background movement, due to camera motion, objects, among others), as it is difficult to distinguish the human fine-grained move- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2017), CAMs improvement was not taken into consideration during training, to assure the model’s weights would be learnt in a way that would enhance action-centred feature extraction and improve the model’s focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These were just used, after training, for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In this sense, a new line of approaches is also emerging, namely the use of transformers (Girdhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019) and attention mechanisms (Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Commonly for fine-grained action recognition, the frame or sequence of frames incorporates irrelevant or redundant information, with no discriminatory property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' So, these algorithms guide the model to use atten- tional regions, instead of the whole frame, enhance local features and attain selectively feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For example, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020) implemented channel-wise and spatial attention mechanisms, along with baseline CNNs (VGG16 and ResNet-50) and LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, when comparing to LSTM, transformers can be a lighter and maybe more suitable alternative for online performances (Kozlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, the study of these algorithms for video analysis and HAR/HAP is still a newly tackled field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Overall, the presented DL methods for HAR/HAP typically target a broad spectrum of actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' blowing candles, falling, drinking), recorded with static cameras (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017), rather than specific walking directions (as stop, walking straight, turning right and left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The latter actions are the most used and relevant to specify and control the walker’s trajectory (Paulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Weon and Lee, 2018), but demand no objects or interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The differences between them rely on human positions or fine-grained movements, incurring on a more challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, when resorting to mobile embedded cameras in SWs, the videos will include background motions and will be commonly recorded in a front-view perspective, hindering the perception of turning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Methodology This section describes the methodology followed in this work, from data acquisition and preparation, to the devised DL-based algorithms for model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 1 illustrates the overall pipeline of the developed work and the inherent methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data Acquisition Facing the lack of suitable public datasets for HAR or HAP with SWs, this work integrated the collection of a custom dataset, which included walking and turnings as different actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This was recorded with mobile cameras of a SW, always maintaining a front-view perspective of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data acquisition was conducted under the ethical procedures of the Ethics Committee in Life and Health Sciences (CEICVS 147/2021), following the Helsinki Declaration and the Oviedo Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' All participants gave their informed consent to be part of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data were collected at the School of Engineering of University of Minho, outside a controlled laboratory space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 ection3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='4 DL Approaches RGB frames RGB input Input Design Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7 Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Evaluation GG16 Single-frame Classification ResNet-50 Approach Dataset Selected approach Attention Mechanism Post-processing and Trial simulations Data Acquisition and model Preparation Quantitative focus evaluation Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 Human Masks GT masks Depth frames Segmentation-Classification UNE Generation Approach Adapted UNETFigure 2: Mobile acquisition setup, where a laptop (C) running the Xsens MVN software and its acquisition base (E) are placed over the SW (A), moving along with the robotic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The user is equipped with the Xsens sensors (B), hidden bellow a layer of clothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A mobile acquisition setup was used for this purpose, as presented in Palermo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This setup was composed by: i) WALKit Smart Walker (Figure 2A) used in rehabilitation (Moreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021), which integrates a RGB-D camera (Orbbec 3D Technology International Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', USA) to assess the patient’s gait, recording the user’s legs and feet at a frame rate of 30 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' and ii) Xsens MVN Awinda (Xsens Technologies B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', The Netherlands), which includes wearable IMUs (Figure 2B) and a base station (Figure 2E) connected to a laptop running the MVN software from Xsens (Figure 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This system recorded data at 60 Hz and has already been used before in other studies (Palermo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Its interest in this work was to provide relevant information to accurately place the Ground-Truth (GT) labels of the different actions, considering specific gait events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', heel strike and toe-off (Kurai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data synchronization was ensured by sending a digital pulse from WALKit SW to Xsens MVN Awinda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This mobile setup allowed for real-time data acquisition through pre-defined circuits, which are explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, it enabled data to be recorded in realistic scenarios, with non-ideal light conditions and dynamic backgrounds to maximize data variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Dataset The dataset consisted of RGB-D images acquired from fifteen healthy adults (nine males and six females), with average age of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='54 years old, body mass of 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='15±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='22 kg and height of 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='42 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Considering the aim of decoding human motion in SWs, the walking trajectories should be performed and recorded as naturally as possible, without causing abnormalities in the user’s gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' WALKit SW allows to be used in a passive mode, but since it is heavy, this would produce non-natural slower and larger steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, an automatic trajectory mode was developed, allowing the device to actively follow given trajectories, only as a recording device, creating velocity commands for each wheel according to the desired linear velocity for the SW and the turning strategy (angle and radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This algorithm enabled the execution of four designed circuits, according to the turning direction (right or left) and curvature radius (wide and tight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A circuit comprised a sequence of i) 10s standing, ii) 3-meter walking, iii) 90º turn, iv) 3-meter walking, and v) 10s standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Each participant performed 2 valid trials per circuit and per gait speed: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 m/s, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7 m/s, and 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These speeds were selected according to the walker’s most commonly used velocity range, as well as considering the typical self-selected slow, normal, and fast walking speeds for healthy subjects (O’Callaghan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Different light conditions were held for each trial, increasing the environment and movement variability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' backgrounds, lighting, velocities, turn features), while improving the statistical significance of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 A B B D D E EThe circuit sequence remained unknown to the participants until the beginning of the trial, when a brief explanation was given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The floor was marked with tape and signalised, during the recordings, with chairs and staff people, so that participants could see and react to the circuit’s morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The marked spots were positioned in a way that remained invisible to the SW’s cameras, during the performance of each respective trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, the overall circuit and trial’s order was randomised for each subject before the beginning of the acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The participants were instructed not to just follow the smart walker, but to interact with it, along the designed and marked circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This enabled realistic scenarios, while capturing the user’s intention as naturally as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Some familiarisation trials were also performed before the real recording procedure, encouraging the HRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Overall, each subject performed 24 trials, taking no more than 1 minute each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The labelling process was executed in real-time, along the data acquisition, in two different ways: i) with joystick commands and ii) with velocity commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The former relied on an external person using joystick digital buttons, similar to Figueiredo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020), to mark transitional moments between actions, according to the observation of clear feet movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This produced GT labels responsible for denoting the user’s interaction and intention (JOY labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The latter represents the device’s actions, generating GT labels mainly for action recognition, when the background is already changing accordingly to the performed movements (VEL labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Dataset Preparation Since the principal walking frequency is no higher than 2 Hz for gait speeds above 1 m/s (Pachi and Ji, 2005), all data was down-sampled to 15 Hz, covering more time of action with a lower number of similar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Aiming the creation of a balanced dataset, 40 frames were extracted from each video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This number was selected considering the minimum number of frames obtained for the turn actions during the trials performed by the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This extraction was performed avoiding the action’s boundaries or transitions to prevent possible bias in the labelling procedure, while aiming an effective early action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For this reason, the start of each class was marked with the latest of the two labels (JOY and VEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In total, the final dataset contained 28800 RGB-D frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To simulate real scenarios (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7), a dataset containing the data of the test participants was created, including the complete RGB trial videos, towards online early action detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The foot contacts obtained with Xsens MVN Awinda were used to better position the JOY labels, enabling more accurate labels to mark action transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Input Design Literature revealed the use of windows of frames (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020) and/or other computed inputs, as the optical flow (Simonyan and Zisserman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This paper proposes two types of input, computed within a sliding window over temporal video clips: i) the difference between the last and first RGB frames (DIF input) and ii) the sum of all RGB images (ADD input), in each temporal window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The window length was empirically defined as 4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27s), since it incorporates visible human motion across all gait speeds, while never containing information from more than 1 different step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Examples to better understand the correlation between the original frames and the resultant DIF and ADD images are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The designed inputs were optionally cropped, removing surrounding background, in an attempt for the model to direct its focus to the user and his/hers fine-grained movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The cropping procedure was performed according to a Region of Interest (ROI), instead of using bounding box localization networks, since the user constantly assumes a central position in the image (in the middle of the walker’s handles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In total, four input forms were generated and studied in this work to recognise and detect human motions: cropped and non-cropped DIF and ADD images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Deep-Learning Approaches In this work, we have implemented three different approaches (Approach 1, Approach 2, and Ap- proach 3), as illustrated in Figure 4, which were trained and evaluated considering the four proposed input forms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', cropped and non-cropped DIF and cropped and non-cropped ADD images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Figure 3: (a) A TR sequence extracted from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (b) DIF image, corresponding to the difference between the last and the first window frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (c) ADD image, computed from the sum of all window frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Regarding Approach 1, this consisted on single-frame classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Here, the VGG16 model was used without its top layers (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020), only with a global average pooling (GAP) followed by a softmax classification layer, as it is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Batch normalization layers were also added to improve the optimization and generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, the activation function of the last convolution layer was changed from ReLU to hyperbolic tangent function (tanh), to decrease the clipping of final feature map values and avoid possible vanishing gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The ResNet-50 model was also used, considering its configuration presented in He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Approach 2 consisted of the best backbone obtained in Approach 1 with a channel-wise attention mechanism, as presented in Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This was tested as an attempt to increase the model focus towards the user’s legs and feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The attention mechanism follows the network’s last convolutional block and creates a channel descriptor vector with the corresponding weights for each feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This vector was computed through an average pooling and two convolutional layers, the first followed by a ReLU activation function and the second by a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The obtained vector is then used to perform a multiplication between each feature map and the corresponding computed weight, as it is illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As for Approach 3, UNET neural network (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2015) was used to segment the user’s legs and feet in order to obtain the optimized weights which could allow the neural network to focus on that region of the user’s body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These optimized weights were used to pre-train a novel classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This latter model integrates the UNET encoder, followed by two convolutional blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A dropout layer with probability of 50% was added, followed by a GAP and a softmax layers, decoding human motion into the four classes (stop, walk, TR, and TL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 7 illustrates the neural network used as the basis of Approach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As presented in Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020), transfer learning technique was used for all approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The backbone weights of VGG16 and ResNet-50 classification models were initialised to the weights of these same state-of-the-art networks pre-trained on the general object classification benchmark, ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2010), towards accuracy maximisation, while using large amounts of computational resources, which trains the models to detect and produce general features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In the segmentation-classification approach, the UNET encoder weights of the classification model were initialised with the ones learned in the previous training of the UNET model for segmentation, while the remaining layers were initialised with the He normal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The first 16 layers of this encoder were also frozen during classification training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Figure 4: Workflow of the devised approaches, showing how they were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 5: Diagram of the implemented VGG16 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Human Masks Generation To train the UNET neural network, human masks were computed through an adaptation of our algo- rithm presented in André et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020), using depth frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This algorithm involved geometric and threshold operations that removed the background, as well as the floor plane, to isolate the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Depth frames can be corrupted by the high exposure from sunlight, which, in extreme conditions, causes unrecognisable silhou- ettes in the computed masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These corrupted masks were duly removed through empirically established thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' When facing minor corruptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', incomplete feet), the masks were corrected through classic vision algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These were processed accordingly to form the final GT masks, corresponding to the input used for model training and evaluation (cropped or non-cropped DIF or ADD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Examples of the obtained masks can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Quantitative focus evaluation Due to camera motion, background movements are also present in the input images (Figure 3), according to the action performed by the SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This can compromise early action detection in future real-time applica- tions, as the model should be focused on lower body features to predict walking directions, specially during their first occurrence, without overfitting to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='RGB input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='DIF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CROPPEDDIF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='ADD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CROPPEDADD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Approach 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Approach 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Approach 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='SEGMENTATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='VGG16 & ResNet50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Best backbone of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='APPROACH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Approach 1 + Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='mechanism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Evaluation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Model Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Evaluation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Evaluation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Final Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Trial SimulationsMaxPooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='MaxPooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='MaxPooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='MaxPooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='MaxPooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='Conv 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Conv 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Conv 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 Conv 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Conv 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 Conv 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 Dropout Conv 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 Conv 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 Conv 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Conv 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 Conv 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 Dropout Softmax Conv 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 Dropout Conv 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 GAPFigure 6: Diagram of the implemented channel-wise attention mechanism as presented in Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The channel- attention weighted feature maps are then fed to the Global Average Pooling of the selected CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 7: Schematic of the adapted UNET model for single-frame classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' MP stands for MaxPooling, GAP stands for Global Average Pooling, and the tuples of numbers represent the respective kernel sizes and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 the dropout rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' An algorithm for consistent quantitative evaluation of this focus was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Following Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2019), grad-CAMs heatmaps were generated for each input image, over the model last convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These were then compared with the respective GT human masks, also used as segmentation labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Hence, the model focus was evaluated through the similarity between the grad-CAMs heatmaps and the GT masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Trial simulations and Post-processing The results from each approach were analysed and compared, towards the selection of the most promising model architecture, and input form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These were then evaluated in trial simulations, assessing the perfor- mance in online early action detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To deal with the model uncertainty while predicting, a post-processing technique was designed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This technique assumes a minimum action duration (2s, which equals to, at least, two complete steps in every considered gait speed) and applies it to the previous predicted class, meaning that it only allows for an action transition to happen if the previous predicted one lasted, at least, 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, it also reduces the spectrum of possible transitions by not allowing a TR/TL to happen immediately after a TL/TR event, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These conditions help avoiding prediction errors and are consistent with rehabilitation therapy sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This post-processing technique also reduces possible on-off noise, without introducing delays in the decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 descriptor vector Feature Maps AvgPooling Weighted Conv 1,2 Conv 1,1 Channelencoder block classifier Input Image Base Base encoder block Conv (3x3) Conv (3x3) MP(2x2) ReLU ReLU encoder block BatchNorm BatchNorm MP(2x2) UNET encoder Conv (3x3) Conv (3x3) encoder block Tanh ReLU MP(2x2) BatchNorm MP (2x2) encoder block DropOut (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5) DropOut (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5) GAP MP(2x2) Softmax classifier4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Experimental Protocol This section details the experimental protocol to train and evaluate the proposed approaches, including the dataset split and the training hardware used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Dataset Split The dataset split followed the 80-20 split configuration: 80% was used for training (12 participants) and the remaining 20% was used for testing (3 participants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To control the training process, one participant was left to the validation set, similar to Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 1 describes each of these splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For the tasks involving the generated human masks, namely segmentation and the quantitative focus evaluation, the dataset used was smaller, since some sequences were removed giving mask corruption reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To perform the trial simulations, the test split was used to create a new dataset, as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In total, this test dataset included 72 untrimmed videos from 3 participants (8, 11, 15, according to Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 1: Constitution of each dataset split Split Participants IDs Number of images Classification Segmentation Train [1,15) \\{5, 8, 11} 19536 13209 Validation 5 1776 1295 Test 8, 11, 15 5328 4736 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Training Details CNN configuration: Reasonable hyperparameters were extracted from literature that tackles similar tasks and models (Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2017) for single-frame classification and Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2015) for segmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For classification, the parameters were optimised using Mini Batch Gradient Descent, with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='001, Nesterov momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 and batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The initial learning rate was decayed by 50% until a minimum of 1e−4, if the training loss did not improve within 4 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As loss function, the categorical cross-entropy was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For segmentation, the Adam optimizer was used instead, with a learning rate of 1e−4 and a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' UNET convolution layers were initialised with He normal weights initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As loss function, the binary cross-entropy was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' At the end of the training, the best model was selected according to the validation F1-score or loss, for classification and segmentation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data preparation: The generated input images were resized, from a resolution of 480x480 pixels to a resolution of 224x224 pixels, preserving the images’ aspect ratio, to match the pre-trained networks’ input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For the cropped inputs, the original aspect ratio decreased with the cropping procedure and the image was thus resized as much as possible, towards the target resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' After this, padding was performed, centring the image, to fulfil the 224x224 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Image augmentation was then used in order to avoid overfitting, as well as to improve the model focus on the human body, despite its global position in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Random alterations were applied to the image brightness and contrast, as well as spatial augmentations, such as height and width shifts and zoom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Since directions are an important feature to distinguish between TR, TL and straight walking, rotations were avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, random Gaussian blur was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Finally, the images where normalised between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Training Hardware: All models were developed offline, with the collected data, using the Tensorflow and Keras DL library, on a Python environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The depicted experiments and DL approaches were trained in a computer with the following hardware specifications: GPU: 1x Nvidia GeForce RTX 3080 Ti/PCle/SSE2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' CPU: Intel(R) Core(TM) i9-10940X @3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3GHz (14 core, 28 threads);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' RAM: 65,5 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Performance Metrics Classification was evaluated using common metrics for this task, namely accuracy (ACC), precision, recall, and F1-score (Berardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Aliakbarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These metrics are defined in the equations below, where TP, TN, FP, FN refer to the true/false positive/negative observations and n stands for the total number of classes (in this case, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' ACC = n � i=1 TPi + TNi TPi + TNi + FPi + FNi (1) Precision = 1 n n � i=1 TPi TPi + FPi (2) Recall = 1 n n � i=1 TPi TPi + FNi (3) F1 − score = 2 ∗ Precision × Recall Precision + Recall (4) Intersection over Union (IoU) and Dice (Dice) were computed for segmentation and quantitative focus evaluation, as shown by the following equations (TP, FP and FN refer to the true positive and false positive/negative observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' IoU = TP TP + FP + FN (5) Dice = 2 × TP (TP + FP) + (TP + FN) (6) Inspired by Baptista-Rios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020), OAD metrics were implemented to better evaluate the perfor- mance in online simulations, such as instantaneous accuracy (IA), instantaneous precision (IP), instantaneous weighted accuracy (wIA) and instantaneous calibrated precision (cIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These are represented in the follow- ing equations, were t corresponds to the time instant, K to the the total population considered until time t, n to the number of classes and TP, TN, FP, FN refer to the seen true/false positive/negative observations overall classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' IA = 1 K × n t � j=0 TPj + TNj (7) IP = �t j=0 TPj �t j=0 TPj + FPj (8) wIA = 1 K × n[w × t � j=0 TPj + 1 w × t � j=0 TNj] (9) cIP = w × �t j=0 TPj w × �t j=0 TPj + �t j=0 FPj (10) These metrics evaluate the model performance as the frames are acquired, without having to wait to an unknown end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, wIA and cIP condition the value of a true observation to the total negatives vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' total positives ratio (w), which is dynamic and always based only on the seen portion of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Results Starting from the CNN models for single-frame classification (approach 1), to an attention mechanism (approach 2) and the segmentation-classification approach (approach 3), the influence of the various forms of input is studied and the classification models are evaluated in two aspects: i) the accuracy of the predicted labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' and ii) model focus on the human body region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Single-frame Classification Approaches Table 2 shows the validation results, obtained for approaches 1 and 2 considering single-frame classi- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The respective training curves can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' According to Table 2, ResNet-50 outperformed the VGG16, except for the cropped DIF input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The DIF revealed here a better performance, followed by the ADD input, which was only worst by an overall maximum margin of approximately 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Adding a channel-wise attention mechanism to ResNet-50 further enhanced its classification perfor- mances, specially for the cropped inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The F1-score was increased by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='98% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='85%, for the cropped DIF and ADD inputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This mechanism also changed the influence exerted by the different types of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' With this model, ADD input surpassed the accuracy and F1-score values obtained for the DIF input, by a maximum margin of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 2: Validation results of the VGG16 and ResNet-50 (without and with a channel-wise attention mechanism), as well as the training time Input Type Crop ACC (%) F1-score (%) Precision (%) Recall (%) Training Time (h) VGG16 (approach 1) DIF False 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='45 DIF True 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='76 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='66 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47 ADD False 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='50 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='28 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='79 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='44 ADD True 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='48 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='53 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='54 ResNet-50 (approach 1) DIF False 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='42 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='52 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='45 DIF True 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='24 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47 ADD False 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='65 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='44 ADD True 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='87 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='76 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='46 ResNet-50 with attention (approach 2) DIF False 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='93 DIF True 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='31 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='32 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='76 ADD False 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='39 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='86 ADD True 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='61 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='61 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='61 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17 Table 3 shows the results obtained when evaluating each model’s grad-CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Contrarily to the quanti- tative classification results for VGG16 and ResNet-50, the cropped inputs presented better focus, meaning a higher similarity between the model’s grad-CAMs and the GT masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In average, cropping part of the background from the inputs increased the Dice and IoU metrics by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='72% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16%, for the DIF input, and by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='09% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='98%, for the ADD input, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The cropped ADD led to a better focus, achieving its best results with the ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This model showed an overall average boost of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03% in Dice and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='96% in IoU metric, when compared to the VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The attention mechanism improved the focus of the ResNet-50 for every input, even if just for a small margin (<5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The greatest improvement was achieved by the non-cropped ADD input (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20% in Dice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, this one still led to lower metrics than the non-cropped DIF input, as the latter achieved Dice values 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='07% higher than the non-cropped ADD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Contrarily to previous results, cropped DIF attained the highest similarity between their GT masks and the grad-CAMs obtained from the ResNet-50 model with attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, the difference relative to the cropped ADD corresponds only to a small percentage (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='11%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, the cropped ADD presented the lowest values of standard deviation, entailing a more consistent focus across the dataset, with smaller fluctuations and more representative IoU and Dice values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 3: Quantative evaluation results, in percentage, for validation grad-CAMs, when predicting with the VGG16 and ResNet- 50 models (without and with an attention mechanism) Input VGG16 ResNet-50 ResNet-50 with attention Type Crop Dice (±std) IoU (±std) Dice (±std) IoU (±std) Dice (±std) IoU (±std) DIF False 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='10 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='87) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='06 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='70 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='01) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='86) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='59 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13) DIF True 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='57 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='44 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='95) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='09 (±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='57) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='67 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='22) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='90 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='04 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='43) ADD False 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='19 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='84) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='39 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='10 (±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='00) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='36 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='51) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='30 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='04) ADD True 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='84 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='89 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='48) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='79 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='54) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13) January 16, 2023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Segmentation-Classification Approach Table 4 shows the validation results obtained for segmentation, along with the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The segmentation and following classification training curves can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Considering these results, all inputs seem to accurately segment the human body region, with Dice and IoU values higher then 93%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Among these inputs, the cropped ADD stood out, although for a small margin (less than 1%), presenting an IoU and Dice of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='52% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The segmentation test results, along with examples of segmented images can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 4: Validation results, in percentage, of the UNET model, as well as the training time for 30 epochs Input Type Crop Accuracy Loss IoU (±std) Dice (±std) Training Time (h) DIF False 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='81 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='96) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 DIF True 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='85 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='95) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='81 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 ADD False 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='91) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 ADD True 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='52 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='91) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 Tables 5 and 6 present the validation results for classification, as well as the training time for 100 epochs, and the quantitative evaluation of the model focus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The DIF input attained the highest values of F1-score, namely 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14% and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='36% for non-cropped and cropped, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However and once again, the best focus was achieved with the cropped ADD input (Dice values with average of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 5: Validation results of the adapted UNET classification model, following the segmentation task, as well as the training time for 100 epochs Input Type Crop ACC (%) Loss F1-score Precision (%) Recall (%) Training Time (h) DIF False 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='29 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='48 DIF True 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='36 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='70 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='53 ADD False 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='24 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='07 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='49 ADD True 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='69 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='43 Table 6: Quantitative focus evaluation results, in percentage, of the validation grad-CAMs, when predicting with the adapted UNET model for classification Input Type Crop Dice (±std) IoU (±std) DIF False 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='48 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='00) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='01 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='06) DIF True 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='46) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='44 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80) ADD False 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='88) ADD True 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='66) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='75 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='35) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Analysis of the final approach The best classification performance, as well as the most relevant and human-centred focus, were achieved by the ResNet-50 model with a channel-wise attention mechanism (approach 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, further evalu- ation is conducted to analyse this model’s performance, in both offline and trial simulations, with an unseen test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Offline testing Table 7 presents the percentage of wrong classifications over the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Similarly to validation, the cropped ADD revealed the most promising results, with the wrong predictions representing only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64% of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Figure 8: Confusion matrices for the ResNet-50 model with attention, over the 4 forms of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 7: Percentage of wrongly classified frames in the test set, by the ResNet-50 model with an attention mechanism Input Type Crop Wrong predictions (%) DIF False 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='68 DIF True 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 ADD False 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71 ADD True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64 Figure 8 presents the obtained confusion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These matrices reveal excellent results, specially for the stop and TL classes, where the TP rate was never lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The cropped DIF was the least favoured input by this architecture, with the lowest TP rate for the walk class (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 8 shows the results obtained from the focus evaluation of ResNet-50 model with attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The input influence over the model focus is in agreement with the one already verified in validation (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Cropped DIF attained the higher similarity between their GT masks and the grad-CAMs, but with a very small difference from the cropped ADD results (without surpassing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16%, considering both Dice and IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The standard-deviation obtained for the cropped ADD was also smaller in comparison with that obtained for the cropped DIF, similar to what was found for the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 DIF input ADD input 1200 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 TR TR 400 400 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 TL TL STOP TOP Predicted label Predicted labelFigure 9: Plot of the GT (dashed line), predicted (dotted line) and post-processed predicted labels (solid line) for (a) Trial A and (b) Trial B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Class IDs: 0=stop, 1=walk, 2=TR, 3=TL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 10: Plot of the values of the online metrics described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 for (a) Trial A and (b) Trial B, namely: instantaneous accuracy (IA, solid line), instantaneous weighted accuracy (wIA, dashed line), instantaneous precision (IP, dashed line with dots) and instantaneous calibrated precision (cIP, dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 8: Quantitative focus evaluation results, in percentage, of the test grad-CAMs, when predicting with the ResNet-50 model with an attention mechanism Input Type Crop Dice (±std) IoU (±std) DIF False 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='97 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='56) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='50 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26) DIF True 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='32) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='76 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20) ADD False 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='07 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='59) ADD True 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='30 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='83) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='60 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Trial simulations The cropped ADD, when fed to the ResNet-50 model with attention, achieved the best results across validation and test, considering both classification power and focus evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, this was the approach tested in trial simulations, along with the post-processing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Two representative trials, corresponding to different conditions and extreme cases, are displayed: trial A) participant 11 performs a TL at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5m/s (lowest gait speed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' trial B) participant 15 performs a TR at 1m/s (fastest gait speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 9 allows the comparison, at each instant, between the predictions (with and without post- processing) and the GT classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure 10 shows the temporal evolution of the online metrics described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 9 shows, respecting the trials’ temporal order, the delays between the post-processed label and the GT one, for each action transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Negative values represent classes predicted earlier than their actual start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='5 - 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 - 0 100 200 300 400 500 600 0 100 200 300 400 Frame ID Frame IDFigure 11: Grad-CAMs visualisation, temporally ordered, for each one of the transitions in the slow trial (trial A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The green and blue labels correspond to the first prediction and GT label, respectively, of the action that is beginning (P=predicted class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 9: Delays of the final predicted labels in relation to the respective GT labels, computed for each transition of the circuit Time delay (s) Trial Walk Turn Walk Stop A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='67 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Grad-CAMs visualisation Figure 11 presents the grad-CAMs visualisation for Trial A, where the model’s predicted labels corre- spond exactly to the post-processed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For the beginning of each class, the visualisation starts at the first frame of that action (for delayed predictions) or at the first correct prediction (for early predictions, as it is the case of the stop class, in this trial) and ends at the first right prediction or first GT frame, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Note that these are not necessarily consecutive frames on the dataset, that depends on the action delay registered in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, they serve as a good representation of the focus evolution between the GT and its respective correct prediction (or vice-versa) and, for the delayed predicted labels, it always includes two immediately preceding frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 STOP STOP STOP STOP ALK 门 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9997414 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='98371756 2 TL TL WALK WALK WALK WALK WALK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='90710625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99804914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99995494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9999778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9999862 3- WALK WALK WALK STOP STOF STO STOF STOF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8881017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='98464704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9998155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99951065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99981517 4- STOP STOP STOPFigure 12: Grad-CAMs visualisation, temporally ordered, for each one of the transitions in the fast trial (trial B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The green and blue labels correspond to the first prediction and GT label, respectively, of the action that is beginning (P=predicted class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The orange ones correspond to the perturbations in the model’s predictions that do not correspond to the post-processed predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The same applies to Figure 12, representing Trial B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This trial presents more on-off noise in the model’s outcomes, specially in the transition from walk to turn (Figure 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Hence, its predictions do not always correspond to the post-processed labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As the latter constitutes the final predicted classes, the beginning or end of these visualisations are depicted by the respective post-processed labels, for early and late predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, in the two first presented classes, a frame immediately after the correct post- processed prediction/GT label was added for purposes of focus evolution assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Discussion The results obtained by the DL approaches are critically discussed in this section, pointing some limita- tions and insights for possible improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Tailored inputs and model’s focus evaluation Distinct input forms influence the models performance differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Despite leading, in most cases, to lower classification metrics, cropping the images helps to direct the focus to the human ROI (see Table 3), as it excludes a significant portion of background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, better classification results can be associated with less reliable extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 STOP STOP STOP WALK WALK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9939032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9860512 088574237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8731011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='97841656 WA WALK WALK VALK WALK IR IR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7609441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='91726273 936562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='7841732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9954194 TR 2 TR TR WALK WALK IR IR TR TR WALK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='96653795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='98590267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8900123 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='96647906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8672215 3- WALK WALK WALK WALK WAL WALK WALK STOF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9733175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9876581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9695332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='6238496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='732093 STOP STOP STOPA greater improvement in focus was registered when cropping the ADD input, which confirms that non-cropped ADD includes more information about the background motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The fact that the highest improvement rate in focus achieved by adding an attention mechanism to the ResNet-50 model was recorded by the non-cropped ADD also supports this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' According to the results, the cropped ADD was the input that consistently presented high similarities between grad-CAMs and GT masks, raising the belief that ADD images also encode more human body motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' From these comparisons, three aspects can be inferred: i) background motion may contain more evident features, easier to extract, that can help the offline classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, this should not be the main focus of the model since these features are not reliable for detecting transitions, real-time applications or even generalisations to other datasets, where the background is static;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' ii) a careful performance evaluation is needed, since better classification results can be associated with non-ideal feature extractions and overfitting to the background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' and iii) despite not being a commonly used approach, cropping the inputs was a feasible way to enhance the model focus and thus increased the reliability of the respective classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, the higher registered improvement was not higher than 10% (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This can be related to the background characteristics, such as the presence of floor stripes enhancing background motion (Figures 11 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Literature on HAR/HAP normally uses sequences of RGB frames to provide the sense of temporal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Instead of this approach, the proposed inputs were able to provide (a significant part of) this temporal information in one single frame, avoiding the use of Recurrent Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' With a larger background variability in the dataset, while carefully avoiding pavement marks during the acquisitions, these tailored inputs could become a more reliable method to induce action-aware feature extraction, avoiding more efficiently the background features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, the lower metrics obtained when evaluating the model focus can also be due to the quanti- tative focus evaluation algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The GT masks used here as a comparison term are binary, presenting the highest score (1) for all the human area, but this region may not be equally important to decode human motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For example, feet and knees may present more orientation and position variations which indicate the step’s direction, so it would be correct for the model to give higher focus to these particular regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For this reason, comparing heatmaps to these masks is correctly penalising FP, but it is also punishing the model for not focusing on the complete lower body, including, for example, the more static upper leg region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These effects can lower the values of the calculated Dice and IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The masks are already computed in the tightest ROI possible to attenuate this effect, but it does not completely solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' A possible solution for this problem would be to change the GT masks pixel values, according to the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Thus, as the higher pixel intensities in the input correspond to motions with larger amplitudes, while the lower correspond to more static areas, this information could be used to scale the scores equal to 1 in the GT masks, creating a sort of human motion masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' An even more accurate form of information to scale these masks according to the body’s amplitude of motion, would be the human poses, from the Xsens data, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, this last option would unduly increase the computational expense and complexity of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Observing the grad-CAMs of the final selected algorithm (Figures 11 and 12), one could also attempt to only generate masks of the user’s feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Single-frame Classification Approach Based on the results presented in Table 2, it is possible to notice that ResNet-50 performed better than VGG16, achieving higher F1-score values than VGG16 ([94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34%, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27%] over [94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='53%, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80%], respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Hence, this problem benefited from residual features, skip connections and deeper networks duly initialised or pre-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, the difference between both performances was not that high (lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='47% in F1-score - Table 2), meaning that the task of early recognising human motions from the SW dataset may also be approached by less deep models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, ResNet-50 achieved better classification results and focused better on the human region (average boost of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='03% in Dice and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='96% in IoU), over all input forms, and was, thus, the chosen model to be tested with an attention mechanism (approach 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The addition of the channel-wise attention mechanism enhanced not only the classification metrics, improving the F1-score by an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='93%, but also the similarity between grad-CAMs and GT masks, with improvements until 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20% in Dice, across all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Only the model focus associated with the cropped January 16, 2023 ADD input was slightly worse than the ones registered with the ResNet-50 baseline model (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, since the difference is not that large (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='68% in Dice and IoU, respectively), this could be due to small variations and, thus, was not considered as a relevant fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These results proved the importance of dealing and modelling the distinct learning abilities of the different convolutional channels, not only to increase CNN performance, but also to improve the relevance of the features extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, the values presented in Table 3 for ResNet-50 with attention are not that higher than the ones for the corresponding baseline model, specially for the cropped inputs, proving that this channel-wise attention mechanism, although unequivocally beneficial to the classification task, still does not completely correct its main focus, as Dice and IoU are still below 50% and most pixels in grad-CAMs heatmaps do not correspond to the human body region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Facing these facts, a spatial attention mechanism could also be designed for this problem, guiding the model to use attentional regions, instead of the whole frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As in Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2020), also enhancing local features by combining this with the channel-wise mechanism, could lead to better performances and, in this case, more properly focused solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The spatial attention maps could even be compared with the suggested human motion masks for focus evaluation or, in a more bold experiment, as part of the training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The cropped ADD appeared as the most promising input for this task and, when fed to the ResNet-50 model with the attention mechanism, achieved the best overall results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Segmentation-Classification Approach Looking at the segmentation training curves (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16), one can see that, as the epochs advance, there is a tendency for overfitting, given the small increase in the gap between the validation the training losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Although apparently small, this can propagate to the following pre-trained classification model and induce bad generalisation abilities or even worse cases of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' That is why the segmentation training was shorten to 30 epochs and the weights were chosen considering the minimal validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Despite the training reduction, the adapted UNET still revealed problems of weak generalisation (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In agreement with these training curves, the classification metrics were worse than the ones achieved by previous evaluated models, as the maximum F1-score was of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14% (Table 5), which is lower than the minimum registered for the previous models (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='34%, for the baseline ResNet-50, shown in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, these metrics were still above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The severest cases of unrepresentative training dataset and consequent generalisation issues were verified by the ADD input type, which is associated with lower validation performances, namely 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='69% (cropped) and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08% (non-cropped form) of F1-score (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' When connecting the segmentation and classification results (Tables 4 and 5, along with Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17), it seems to exist an inverse relation between segmentation power and the classification generalisation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This may mean that this cascade approach is leading the model to focus on input traits that are not representative of the whole dataset, following the overfitting problems during segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Cross-validation should be performed to prove this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Even so, the focus on particular traits can be associated with the fact that, despite the final aim of human motion decoding, the GT masks used are leading to the segmentation of the whole body, including large clothes and more static human areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Therefore, using the aforementioned human motion masks as labels could decrease the chances of overfitting, while pursuing the differential segmentation of the human body, according to its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This could enhance the weights used to pre-train the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Other options to help overcoming the overfitting problem consist on experimenting other simpler segmentation models or even include spatial data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, the number of frozen layers should also be studied and tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As for the grad-CAMs evaluation (Table 6), the segmentation-classification approach was not the most effective to attain its main goal: improving the extraction of human-centred features during classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The low results, along with their resemblance to the ones achieved by VGG16, point to an influence mainly exerted by the input properties and not by the two-stage framework itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Trial simulations The ResNet-50 model with a channel wise attention mechanism was the best model in both aspects: classification rates and focus relevance, specially when fed with the cropped ADD input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Testing this January 16, 2023 approach in trial simulations led to general good performances, with the model being able to identify the different consecutive walking events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The model uncertainty revealed a greater prominence at higher gait speeds, in the form of on-off noise (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, these perturbations were easily smoothed by the proposed post-processing technique, without adding time delays in the correct predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Despite presenting more noise, Trial B achieved overall higher online metric values, over 94%, due to its lower time delays registered in transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table 9 shows, respecting the trials’ temporal order, the delays between the post-processed and the GT labels, for each action transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For the trial performed at 1 m/s, the delays are at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37s lower than the average step time for this gait speed (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For lower gait speeds, the time lags registered were higher, confirming the greater challenge implied by early detecting slower and more subtle motion changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These delays could be further decreased for real-time applications, through a proper training procedure that includes transitions in the dataset, but also through a data quality enhancement to further improve the model focus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' large variety of backgrounds without floor stripes/marks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Overall, the results prove the chosen approach is suitable for early action recognition, achieving average online metrics between 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='72% and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' However, it still needs improvements to early detect an action, as it can be seen by the model performances at the transition inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, the obtained results were still good, with not so critical delays, considering the fact that the neural network was not trained with transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Hence, with the mentioned suggestions, specially the inclusion of transitional frames in the training procedure, this performance could be further enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For this to happen, one has to first improve the labelling accuracy, decreasing the bias effect introduced by the person controlling the joystick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Grad-CAMs visualisation The displayed grad-CAMs (see Figures 11 and 12) showed that the model focus is not too deviated from the human region, but it still considers some background information, specially the visible motion of the floor stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The floor stripes were a non-ideal property of the acquisition environment, which visibly affected the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As background motion is a consequence of the walker’s movement, and not the user’s, this misleading focus can be among the causes of the registered time delays, when predicting each action’s beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For example, in Figure 12 (last row), the stop detection was delayed, as the walker kept moving after the subject stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' So the model must have considered the stripes and the large clothes motions, instead of the user’s steadier positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As the device decelerates, this background motion became less evident and the model started to focus on the feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' In the slow trial, this deceleration phase is shorter and slower, so the background motion stopped appearing in the RGB inputs before the human movement, allowing the model to better perceive the progressive horizontal alignment of the feet (last row of Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This situation is similar to the beginning of the first walking event at low gait speed (first row), where the walker starts to slowly accelerate, so the background appears static, and the feet move slowly and closer to each other, leading to a confusion between stop and walk classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' It seems the model cannot yet associate the horizontally misalignment of the feet as a walking trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The turning event was anticipated in Trial B, which seemed like a good achievement for motion intention decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nonetheless, looking at the respective grad-CAMs (second row, in Figure 12), one can see that this class was first predicted based on the vertical misalignment between the floor stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This helps to visualise and understand the confusion and model uncertainty between these two classes (walk and TR/TL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These visualisations showed that the model focus still needs to be improved in transitions, in order to be integrated in a real-time control mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Nevertheless, this focus appeared to be better than the expected from its quantitative evaluation, with grad-CAMs concentrated around the feet, which can be due to the GT masks used in that algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Conclusion This work presents a novel way to decode motion intention in SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Three different approaches were devised, two facing single-frame classification, and another facing a segmentation-classification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 For that, a custom dataset of 15 healthy participants was acquired with a smart robotic walker, considering realistic scenarios and circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Each participant performed a total of 24 trials, each one containing three of the target classes (stop, walk, turn right and turn left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Considering this, a balanced dataset of frames containing 28800 RGB-D images was created, extracting 40 frames per video sequence, and used to train, evaluate, and compare the proposed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Regarding the model architectures, state-of-the-art VGG16 and ResNet-50 were implemented in the first approach and then, an attention mechanism was added to the best model (second approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For the third approach, the UNET neural network was used for segmentation and adapted for the following classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Four different input forms were studied, namely cropped and non-cropped ADD and DIF images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' These were obtained considering a sliding window approach of 4 frames with a stride of 2, by summing the four frames of the window (ADD) or subtracting the last frame from the first (DIF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This approach encoded 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27s of motion information without using recurrent neural networks, which are a commonly used approach in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To evaluate the model performance, we considered standard metrics, namely accuracy, F1- score, recall, and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For trial simulations assessments, OAD metrics were used, such as instantaneous accuracy, instantaneous precision, instantaneous weighted accuracy, and instantaneous calibrated precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' We also evaluated the model focus considering a novel method that quantitatively compares its grad-CAMs with GT masks of the human body region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Regarding the different inputs, we concluded that the non-cropped ADD input encodes more motion information, but these, together with DIF images share a common disadvantage with the optical flow: in realistic videos, these inputs also encode background motion which may deviate the model focus from the human region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Cropping most part of the background surrounding the image’s ROI proved to have a major impact on the model’s focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' We also verified that ResNet-50 with an attention mechanism, when fed with cropped ADD inputs, attained the most promising results (offline accuracy and F1-score higher than 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' This enabled an enhancement in the model focus towards the human body region (Dice rounding 32%) when compared to the other models, but still needs further improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Limitations and future research insights This work presents some limitations that should be considered in future research insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Enhancing the quality of the acquired data becomes necessary to train the algorithms with transitional inputs and improve the relevance of the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Recording in a more controlled environment, without marked floors or too bright conditions, would be relevant to not deviate the model focus to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Moreover, the labelling procedure should also be improved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' with the use of force sensors) to allow the inclusion of transitions, without mislabelled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Data from pathological individuals should be acquired and the use of transfer learning may be considered to endow the model with the ability to detect motion intentions for this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Regarding methodology, the method to evaluate the model focus can be refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' For instance, the human masks can be improved by lowering the pixel values in more static human body areas, creating human motion masks, along with the inclusion of false positives as a metric to accurately assess how much the model is focusing on the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Spatial attention mechanisms or self-supervised learning can also be explored to improve the model focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Additionally, tailored losses could also be tested to improve the early detection ability, especially if the model keeps presenting significant time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Lastly, the use of visual transformers could also be investigated to progressively obtain simpler models capable of learning by observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Along with its potential for improvement, we hope this work can also serve as future benchmark and encourage further investigations on decoding fine-grained human actions directly through visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Acknowledgements This work has been supported by the Fundação para a Ciência e Tecnologia (FCT) with the Reference Scholarship under Grant 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='BD and under the national support to R&D units grant, through the reference project UIDB/04436/2020 and UIDP/04436/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Author Contributions Carolina Gonçalves: Methodology, Investigation, Data curation, Formal analysis, Software, Writing - Original Draft, Writing - Review & Editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' João M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Lopes: Methodology, Investigation, Data curation, Software, Writing - Review & Editing, Funding acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Sara Moccia: Methodology, Resources, Super- vision, Validation, Writing - Review & Editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Daniele Berardini: Software, Writing - Review & Editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Lucia Migliorelli: Software, Writing - Review & Editing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Cristina P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Santos: Conceptualization, Method- ology, Resources, Supervision, Validation, Writing - Review & Editing, Project administration, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', Oliva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=', and Torralba, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Learning Deep Features for Discriminative Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 2921–2929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Single-frame Classification Approach Training VGG16 and ResNet-50 architectures with each developed input, computed from the acquired dataset, resulted in the training curves presented in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' As one can see, the overall curves are stable and with no signs of overfitting, reaching good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' It is noticeable that ResNet-50 learned faster and provided some gains in loss and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='13: Accuracy and loss training curves for VGG16 and ResNet-50 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 DIF Input DIF Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': 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0 40 100 Epoch # Epoch # Cropped DIF Input Cropped DIF Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 val_loss val_oss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 train_acc train_acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 val_acc val_acc 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 - 20 40 60 100 0 80 20 60 80 100 0 40 Epoch # Epoch # ADD Input ADD Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 val_loss val_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 train_acc train_acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 val_acc val_acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 - 40 20 20 60 80 100 0 60 80 0 40 100 Epoch # Epoch # Cropped ADD Input Cropped ADD Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 val_loss val_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 train_acc train_acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='8 val_acc val_acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 20 60 80 100 20 80 0 40 0 40 60 100 Epoch # Epoch #Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14 shows the obtained training curves for the ResNet-50 model with a channel-wise attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14: Accuracy and loss training curves for ResNet-50 model with an attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='10 compares the grad-CAMs evaluation over the unseen test set for the VGG16 and ResNet-50 (without and with attention) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='10: Quantative evaluation results, in percentage, for test grad-CAMs, when predicting with the VGG16 and ResNet-50 models (without and with an attention mechanism) Input VGG16 ResNet-50 ResNet-50 with attention Type Crop Mean Dice (±std) Mean IoU (±std) Mean Dice (±std) Mean IoU (±std) Mean Dice (±std) Mean IoU (±std) DIF False 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='18 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='60) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='31) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='01 (±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='19) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='84 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='24) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='97 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='56) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='50 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='26) DIF True 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='39 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='91 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='12) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='05) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='92) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='38 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='32) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='76 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20) ADD False 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='99 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='75) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='94 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='91) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='93 (±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='07) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='95 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='19) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='14 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='02) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='07 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='59) ADD True 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='62 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='90) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='18 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='54) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='86 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='87) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='30 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='83) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='60 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='37) Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Segmentation-Classification Approach Examples of the generated masks, used as labels for segmentation and focus evaluation algorithm, are given in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='17 display the segmentation and classification training curves, respectively, using the UNET and adapted UNET models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Train Loss/ACC: DlF Input Train Loss/ACC: ADD Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 val_loss val_loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='9 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='15: Examples of individually non-corrupted masks, along with their corresponding RGB inputs, for a window length of 4 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The presented masks are already corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='16: Accuracy and loss training curves for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 DIF Input ADD Input train_loss train_loss 1.' metadata={'source': 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+page_content='17: Accuracy and loss training curves for the adapted UNET model for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' To help visualise this model’s segmentation ability, Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='18, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='19, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21 show the best and worst cases of segmented images, for each type of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Notice that, for the cropped ADD, the segmentation of the human body was very satisfactory, even in the worst case, although including some noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Contrarily to this, the other inputs revealed occlusions as the apparent main factor behind a worse segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' The quantitative test results shown in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='11 indicate that the cropped ADD images were the easiest to segment, followed by the non-cropped ADD, cropped DIF and, finally, non-cropped DIF images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='11: Evaluation results, in percentage, of the UNET model segmentation over the test set Input Type Crop IoU (±std) Dice (±std) DIF False 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='79 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='55) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='08 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='27) DIF True 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='89 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='66) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='64) ADD False 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='12 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='71) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='80 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='45) ADD True 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='63 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='41) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='68 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='87) January 16, 2023 DIF Input ADD Input train_loss train_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='0 0 20 40 60 80 100 0 20 40 60 80 100 Epoch # Epoch #Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='18: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective non- cropped DIF inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='19: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective cropped DIF inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='20: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective non- cropped ADD inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content='21: Examples of the best (upper) and worst (lower row) cases of segmented images, along with the respective cropped ADD inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf'} +page_content=' January 16, 2023' 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Bhubaneswar, Jatni 752050, India +2Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India +Maximally entangled single particle states (MESPS) are opening new possibilities in quantum +technologies as they have the potential to encode more information and are robust to decoherence +compared to their non-local two-particle counterparts. Herein, using discrete-time quantum walks +on k-cycles where k ∈ {3, 4, 5, 8} and by using either a single coin or effective-single coin or two coins +in various deterministic sequences, we generate MESPS for recurring time steps. These sequences +beget ordered quantum walks and yield MESPS with periods 4, 6, 9, 12, and 15. +For the first +time, we reveal single coins such as Hadamard, which can generate periodic MESPS (with periods +4 and 12) on 4 and 8-cycles. This scheme is resource-saving with possibly the most straightforward +experimental realization since the same coin is applied at each time step. +Introduction.— Hybrid or single particle entangle- +ment (SPE) refers to the entanglement between differ- +ent degrees of freedom such as spatial mode, polariza- +tion, and orbital angular momentum belonging to the +same particle[1]. +This local entanglement enables en- +coding more information at the single particle level, is +more robust against decoherence, and has simpler ex- +perimental implementation than its non-local bipartite +counterpart[1–3]. +SPE has significant applications in +photonic quantum information processing and analysis +of states of photons, elementary particles and quantum +liquids[1]. Quantum joining, a physical process that al- +lows the transfer of intra-particle entanglement between +photons into a single output photon’s hybrid entangle- +ment and its inverse, has been reported, and it has +applications in quantum networking[4]. +Photonic SPE +states are potentially advantageous in optical quantum +networks because they enable a more flexible network +with every photon transmitted via a suitable channel [5]. +SPE has also been used in experimental tests of non- +contextual hidden variable theories[1]. +After their introduction in [6], quantum walks(QWs), +apart from their use in quantum computation, have been +used in designing efficient quantum algorithms[7–9] as +well as in simulating dynamics of complex physical[10] +and biological systems[11]. +QWs can also be directly +implemented in laboratories without requiring a quan- +tum computer[12]. Experimentally, quantum walks have +already been realized with photons[3, 13, 14], trapped +ions[15, 16], superconducting qubits[17, 18], neutral +atoms[19, 20], NMR quantum information processors[21] +and ultra-cold Rubidium-87 atoms[22]. +The quantum +walker (or particle) is represented by a wave function +and obeys the quantum superposition principle, and this +makes QWs superior compared to their classical coun- +terparts [6]. +A discrete-time quantum walk (DTQW) +evolves by repeatedly applying two quantum opera- +∗ dineshkumar.quantum@gmail.com +† colin.nano@gmail.com +tors: coin and shift. +DTQW dynamics also works as +a sophisticated tool in engineering arbitrary quantum +states[23, 24]. DTQWs allow one to explore multi-path +quantum interference [20, 25], and numerous non-trivial +topological[26] and geometrical phenomena[27]. A quan- +tum walk can be described on a 1D or 2D lattice and +analogously on a cyclic graph with k sites (k-cycle). For +some detailed studies on quantum walks on k-cycles, see +Refs.[28–30]. Ref.[30] reports on the experimental imple- +mentation of QW on cyclic graphs with photons using +linear optical elements. A recent work [31] shows that it +is possible to design an ordered or periodic QW (i.e., the +walker returns to a particular position or site periodically +after a finite number of time steps) by combining two +chaotic or non-periodic quantum walks on 3− or 4−cycle +via Parrondo strategy[32]. Intriguingly, the emergence of +order from chaos and its inverse in QWs has applications +in quantum cryptography (secure encryption-decryption +protocols[31]), in designing new quantum algorithms and +in developing theory of quantum chaos control[33]. +Several manuscripts recently reported that DTQWs on +1D lines could be efficient tools to generate entangled +single-particle states(SPS) or SPE, see Refs.[2, 3, 34–39]. +Refs.[3, 35] report on experimental realizations of SPE +generation. One recent manuscript [34] shows that by +incorporating Parrondo sequences of coin operators in +DTQWs on 1D line, one can obtain phase-independent +SPE and, in one case, maximal SPE independent of the +initial state parameters for time steps of 3 and 5. +There has been no attempt to generate maximally +entangled SPS (MESPS) or, for that matter SPE in +cyclic graphs. +Also, seeing the versatility of DTQWs +and the preeminent applicability of SPE, exploring differ- +ent methods to generate highly or maximally entangled +SPS via DTQWs is an important task, as it would con- +tribute to extending the horizons of quantum technolo- +gies [1]. Our aim in this work is to study the propensity +of DTQWs on cyclic graphs in generating MESPS either +using a single coin or, an effective-single coin (i.e., coin +operator followed by Identity operator at successive time +steps) or two coins in a deterministic evolution operator +arXiv:2301.04501v1 [quant-ph] 11 Jan 2023 + +2 +sequence and, their relation to ordered QW dynamics. +QW model.— A DTQW on a k-cycle (Fig. 1), simi- +lar to that on a 1D lattice, is defined on a tensor prod- +uct of position (HP ) and coin (HC) Hilbert spaces, i.e., +H = HP ⊗HC. HC is associated with the coin (or qubit) +and has the computational basis {|0c⟩ , |1c⟩}, whereas +HP , is the k-dimensional position space for the walker +with computational basis {|xp⟩ : xp ∈ {0, 1, 2, ..., k − 1}}. +If the walker is initially localized at the site |0p⟩ in a +general superposition of the coin states, it is represented +by, +|ψi⟩ = |ψ(t = 0)⟩ = cos(θ/2) |0p0c⟩ + eiφ sin(θ/2) |0p1c⟩ , +(1) +with θ ∈ [0, π] and φ ∈ [0, 2π]. The unitary coin operator, +in general, is, +ˆC2(ρ, γ, η) = +� +√ρ +√1 − ρeiγ +√1 − ρeiη −√ρei(γ+η) +� +, +(2) +where 0 ≤ ρ ≤ 1 and 0 ≤ γ, η ≤ π . +The walker moves to the left by one site for coin state +|0c⟩ and to the right by one site for coin state |1c⟩. For +the walker on the k-cycle, we use the shift operator +ˆS = �1 +q=0 +�k−1 +j=0 |(j + 2q − 1(mod)k)p⟩ ⟨jp| ⊗ |qc⟩ ⟨qc| . +The full evolution now can be expressed as, +Uk(t) = ˆS.[Ik ⊗ ˆC2(ρ(t), γ(t), η(t))] , +(3) +where Ik is a k×k identity matrix. The time-evolution of +the system (quantum walker) after time steps t is then, +|ψ(t)⟩ = Uk(t) |ψ(t − 1)⟩ = Uk(t)Uk(t − 1)...Uk(1) |ψ(0)⟩ += +k−1 +� +j=0 +[α1(j, t) |jp, 0c⟩ + α2(j, t) |jp, 1c⟩] , +(4) +where, α1(j, t) and α2(j, t) are amplitudes for the states +|jp, 0c⟩ and |jp, 1c⟩ respectively. Further, the QW is said +to be ordered or periodic if the walker reverts to its initial +state after a time step, say t = N, irrespective of the +choice of the initial quantum state. For an ordered QW +with period N, we may write, +|ψ(N)⟩ = Uk(N)Uk(N − 1)...Uk(1) |ψi⟩ = |ψi⟩ . +(5) +For simplicity, if we apply a particular coin opera- +tor in the above QW evolution, i.e., Uk(t) = Uk(t − +1) = ...Uk(1) = Uk(say), then Eq. (5) is equivalent to, +U N +k |ψi⟩ = �2k +i=1 aiλN +i |λi⟩ , wherein the arbitrary |ψi⟩ is +expressed in terms of the eigenvalues {λi} and eigenvec- +tors {|λi⟩} of Uk, i.e., |ψi⟩ = �2k +i=1 ai |λi⟩. Clearly, from +Eq. (5), the condition of periodicity for the quantum walk +follows as: U N +k += I2k or λN +i += 1, +∀ i ∈ {1, 2, ..., 2k}. +Any unitary evolution operator which satisfies this con- +dition gives a periodic probability distribution for the +walker’s position, and such an operator is said to yield +ordered QW. Otherwise, the QW is said to be chaotic. +Furthermore, to simplify the problem of finding the eigen- +values of Uk and hence the periodicity of the QW on a +k-cycle, the 2×2 block circulant matrix Uk is block diag- +onalized by using commensurate Fourier matrix tool as +was done in Ref.[28]. Then the block diagonalized form +of Uk is given by FUkF †= diag[Uk,0, Uk,1, ..., Uk,k−1], +wherein F = F k⊗F 2 with F M being an M ×M commen- +surate Fourier matrix i.e., F M = (F M +m,n) = +1 +√ +M (e2πi mn +M ) +where m, n = 0, 1, ..., M − 1. +The periodicity condi- +tion is satisfied if the eigenvalues λ± +k,l of each block Uk,l +take the form of de Moivre numbers (e2πi mr +nr ) where +each (mr, nr) pair is coprime [28, 31] or, equivalently if, +λUk +k,l = +λ+ +k,l+λ− +k,l +2 += e2πi mr +nr , and N = LCM({nr}), with +r ∈ {1, 2, ..., 2k}. In Refs.[28, 29], examples of parameter +values for Uk that satisfy the periodicity condition have +been given(i.e., to obtain ordered QWs). We discuss a +unique analytical approach for obtaining values of such +parameters viz. {ρ, γ, η} yielding ordered QWs with de- +terministic (including Parrondo type) evolution operator +sequences in Results. +Measuring Entanglement.— The initial quantum state +in Eq. (1) is pure and evolves unitarily via DTQW. +We use Schmidt norm(S) to quantify the entangle- +ment between the coin and position degrees of freedom +of the time-evolved quantum state |ψ(t)⟩. +It is rel- +atively more convenient to calculate for QWs [2, 40– +42]. +Let ρψ be density operator for |ψ(t)⟩ i.e., ρψ = +|ψ(t)⟩ ⟨ψ(t)| and reduced density operator (ρc) for the +coin space can be defined by, ρc ≡ Trp(ρψ), where +the partial trace Trp is taken over the position de- +grees of freedom. +We write the eigenvalues of the re- +duced density matrix ρc as, E± = +1 +2 ± |⃗n| , where, +⃗n = +� +Re(Σjα1(j, t)α∗ +2(j, t)), Im(Σjα1(j, t)α∗ +2(j, t)), +1 +2Σj(|α1(j, t)|2 − |α2(j, t)|2) +� +. Finally, the Schmidt norm +is given by, S = +� +E− + +� +E+ +, and for the present +system with min(dim HP , dim HC) = 2, the Schmidt +norm for the case of a MESPS is +√ +2 [34]. +To check +whether our results are correct, we also calculate en- +tanglement entropy (E) which is the von-Neumann en- +tropy for the reduced density matrix ρc of coin state. +E is defined as E(ρc) += -Tr(ρclog2ρc) with 0 and +1 for separable and maximally entangled states (or, +MESPS) respectively, and can be calculated via, E = +−(E−)log2(E−)−(E+)log2(E+) . Indeed we see identical +results for MESPS via both the entanglement measures, +see Supplementary Material Sec. A. +Results.— We consider the following coin operators, +see Eq. (2), Hadamard coin ˆH = ˆC2(ρ = 1 +2, γ = 0, η = 0), +Grover or flip coin ˆX = ˆC2(ρ = 0, γ = 0, η = 0), and +Identity coin ˆI = ˆC2(ρ = 1, γ, η ∋ γ+η = π). We execute +several numerical experiments with these coin operators +and, also, by forming multiple deterministic coin evolu- +tion sequences[34] such as AkAkAk, AkBkAkAkBkAk..., +AkBkAkBk..., +AkBkBkAkBkBk..., +AkAkBkAkAkBk... + +3 +etc., where, Ak = Uk(ρ1, γ1, η1) = ˆS.[Ik ⊗ ˆC2(ρ1, γ1, η1)] +and Bk = Uk(ρ2, γ2, η2) = ˆS.[Ik ⊗ ˆC2(ρ2, γ2, η2)] . +If +ˆC2 = ˆH, then evolution operator is denoted as Hk = +Uk( 1 +2, 0, 0) = ˆS.[Ik ⊗ ˆH] . +Similarly, if ˆC2 = +ˆX, we +have evolution operator Xk = Uk(0, 0, 0) = ˆS.[Ik ⊗ ˆX] , +and if ˆC2 = ˆI, then evolution operator Ik = Uk(1, 0, π) += ˆS.[Ik ⊗ ˆI] . The primary idea behind such experiments +was to reveal evolution operator sequences involving ei- +ther two coins such as HkHkXk..., HkXkHkXk... etc. or, +effective-single coin (i.e., Ik with either Hk or Xk) such as +IkHkIk..., HkIkIk..., HkIkHkIk... etc., or, notably a sin- +gle coin such as HkHkHk... wherein the same coin ˆH +is applied at every time step of the QW, which yield +MESPS along with rendering ordered QWs. We first dis- +cuss single-coin evolution sequences, then effective single- +coin evolution sequences and finally, two-coin evolution +sequences to generate MESPS via DTQWs on the cyclic +graphs. +Note that from an experimental point of view, a QW +for a single coin evolution sequence like HkHkHk... is the +simplest in terms of experimental setup as it just uses the +same coin ˆH at each time step [30]. In other words, the +same setup will be sufficient for its realization in small +and significant time steps. Further, effective-single coin +evolution sequences like IkHkIk... or HkIkIk... consists of +just a single coin (here ˆH) and their implementation are +indeed resource-saving because no device is required for +Identity coin operation in the experimental realization of +the associated QW [3]. +FIG. 1: Even: 4-cycle (a) and 8-cycle (b) graphs; Odd: +3-cycle (c) and 5-cycle (d) graphs, with sites in green +dots. +MESPS on 4- and 8-cycle graphs.— We now consider +even i.e., 4-cycle (k = 4) and 8-cycle (k = 8) graphs, +as shown in Fig. 1. Fig. 2(red) shows the Schmidt norm +values as functions of time steps(t), generated by the sin- +gle coin evolution sequence H4H4H4... on the 4-cycle, +wherein each data point is an average of the Schmidt +norm (Sav) and the average is taken over θ with fixed +φ = π +2 and is evaluated as, Sav = ⟨ S +√ +2⟩ = 1 +π +� π +0 dθ +S +√ +2 . +Clearly, for maximal SPE or MESPS, Sav = 1. We ob- +serve that the single coin evolution sequence H4H4H4... +yields periodic MESPS at time steps t = 1, 5, 9, ... +with period 4. +Also, for 8-cycle case, single coin evo- +lution sequence H8H8H8... yields periodic MESPS at +t = 1, 13, 25, ... with period 12, shown in Fig. 2. +The +periodic behaviour of HkHkHk... in generating MESPS +is supported by its ordered QW dynamics on both k = 4 +and k = 8 cycle graphs, see Supplementary Material +H4H4H4 +H8H8H8 +C4C4C4 +10 +20 +30 +40 +50 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 2: Average Schmidt norm Sav versus time steps(t) +with single coin evolution sequences H4H4H4..., +C4C4C4... for 4-cycle and H8H8H8... for 8-cycle. +H4I4H4I4, H4X4H4X4 +I4H4I4 +H4I4I4 +H4H4X4 +0 +5 +10 +15 +20 +25 +30 +35 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 3: Sav versus time steps(t) with sequences: +I4H4I4..., H4I4I4..., H4I4H4I4..., H4H4X4..., +H4X4H4X4..., for 4-cycle. +Sec. A. +We now discuss effective-single coin evolution se- +quences I4H4I4..., H4I4I4... and H4I4H4I4... designed +from the coin set { ˆH, ˆI} with the 4-cycle. From Fig. 3 +we see that the Sav values generated via the sequence +I4H4I4... follow a periodic trend. +This observation is +well supported by the periodic probability distribution +P(x = 0) for the walker position at |0p⟩, in other words, +the sequence I4H4I4... not only generates MESPS at +t = 5, 7, 9, 17... with period 12 but also an ordered quan- +tum walk (see Supplementary Material Sec. A). Ana- +lytically one can also prove this by exploiting the peri- +odicity condition, beginning with the eigenvalues of the +U4,1-block of the evolution operator (U4)3, see Eq. 3, +λU4U4U4 +4,1 += 1 +2i√ρe +3 +2 i(γ+η)(e− 1 +2 i(γ+η) +e +1 +2 i(γ+η))(−3+2ρ+ +(e−i(η+γ) + ei(γ+η))ρ) . Similarly, the U4,1-block’s eigen- +values for the sequence I4H4I4 give, λI4H4I4 +4,1 += +i +√ +2 +. +Equating λU4U4U4 +4,1 +with λI4H4I4 +4,1 +for δ = (γ + η) = 0, +we get ρ = +2+ +√ +3 +4 +, which is an exact match with ρ +obtained in Ref.[28] for a periodic QW with period- +icity N += 24. +With this analytical description for + +(a) +(b) +(c) +(p)4 +I4H4I4... sequence giving an ordered QW, we observe +that single coin evolution sequence C4C4C4... (i.e., coin +ˆC2(ρ = 2+ +√ +3 +4 +, γ = 0, η = 0) applied at each time step) +generates MESPS with period 12 at t = 5, 17, 29, ..., see +Fig. 2. This periodicity is supported by the ordered QW +dynamics of the ˆC2( 2+ +√ +3 +4 +, 0, 0) coin, see Supplemental +Material Sec. A, again with N = 24. Moreover, effective +single coin evolution sequences H4I4I4... and H4I4H4I4... +yield periodic MESPS with periods 12 and 4 at time steps +t = 1, 3, 5, 13, ... and t = 1, 5, 9, ... respectively, see Fig. 3. +We also observe that two-coin evolution sequence +H4H4X4... gives recurring MESPS with period 6 at time +steps t = 1, 3, 7, 9, 13, ... (proof of this periodicity is in +Supplemental Material Sec. A), whereas the sequence +H4X4H4X4... gives recurring periodic MESPS with pe- +riod 4 at t = 1, 5, 9, ... , for 4-cycle, see Fig. 3. +H3I3I3 +H3I3H3I3 +H3H3X3 +H3X3H3X3 +0 +5 +10 +15 +20 +25 +30 +35 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 4: Sav versus time steps(t) with sequences: +H3I3I3..., H3I3H3I3..., H3H3X3..., H3X3H3X3..., for +3-cycle. +MESPS on 3- and 5-cycle graphs.— Moving now to +odd cycles, a 3-cycle (k = 3) and a 5-cycle (k = 5), see +Fig. 1. Unfortunately, for both 3 and 5−cycles, we do not +see periodic MESPS with single coin evolution sequences +HkHkHk..., XkXkXk..., IkIkIk..., however, HkHkHk... +generates MESPS at time step t = 1 but renders chaotic +QWs. In case of 3-cycle, the effective single coin evolution +sequence H3I3I3... yields periodic MESPS with period 6 +at t = 1, 2, 7, 8... , but the sequence I3H3I3... renders +ordered QWs without MESPS, whereas H3I3H3I3... ren- +ders chaotic QW with MESPS at t = 1, 2 (see Fig. 4). +Further, in the case of 5-cycle, we see that the evolu- +tion sequences I5H5I5...., H5I5I5..., and H5I5H5I5... do +not yield periodic MESPS. But these sequences generate +MESPS at t = 3, 4, t = 1, 2, 3, 4 and, t = 1, 2, 3 respec- +tively, as shown in Fig. 5 (also see Supplemental Material +Sec. A). +From Fig. 4, we observe that the two-coin evolution +sequences H3H3X3... and H3X3H3X3... generate recur- +ring as well as periodic MESPS respectively at t = +1, 3, 4, 6, 10, ...(with period 9) and t = 1, 5, 9, ...(with pe- +riod 4) via the DTQW on the 3-cycle (for proof of this +periodicity, see Supplementary Material Sec. A). More- +over, we see MESPS in the 5-cycle case via sequence +H5H5X5 +H5X5H5X5 +H5I5I5 +H5I5H5I5 +5 +10 +15 +20 +25 +30 +35 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 5: Sav versus time steps(t) with sequences: +H5I5I5..., H5I5H5I5..., H5H5X5..., H5X5H5X5..., for +5-cycle. +H5H5X5... at t = 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, ... with pe- +riod 15, while via H5X5H5X5... sequence at t = 1, 5, 9, ... +with period 4, see Fig. 5. +The above results on MESPS generation are juxta- +posed in Supplementary Material Sec. B, where one can +compare our proposed evolution sequences on these cyclic +graphs. +We see that employing H3H3X3..., H3I3I3..., +and H3X3H3X3... on a 3-cycle one can obtain MESPS +at all time steps up to 10, whereas on a 4-cycle their +analogues give MESPS at all odd time steps t ≤ 10, +see Figs. 3 and 4. +As these sequences also beget pe- +riodic QWs, thus, one obtains MESPS at larger time +steps (t > 10) as well. Moreover, it is interesting to ob- +serve that with just sequences H5H5X5... and H5I5I5..., +one can generate MESPS for all time steps t ≤ 10 +and also at larger t, on a 5-cycle, see Fig. 5. +In fact, +only H5H5X5... by itself yields MESPS at time steps +t = 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, ... with period 15. +It is for the first time we report the single coin evo- +lution sequences H4H4H4... and C4C4C4... which indi- +vidually yields periodic MESPS on a 4-cycle graph with +periods 4 and 12 respectively, see Fig. 2. Apart from that, +we showed that H8H8H8... generates periodic MESPS at +t = 1, 13, 25, ... (with period 12) on a 8-cycle. The gen- +eration of periodic MESPS with just a single coin such +as ˆH or ˆC2( 2+ +√ +3 +4 +, 0, 0) via DTQWs, is a milestone in +resource-saving SPE generation schemes and is also the +simplest scheme in the field of controlled entanglement +generation till date, as it would require the same experi- +mental setting for its realization for both small and large +time steps [3, 30]. +Conclusions.— This letter provides a novel scheme to +generate maximally entangled single particle states via +DTQWs on both odd (3,5)- and even (4,8)-cycle graphs, +with just a single coin and with both resource-saving +effective-single coin and two-coin deterministic evolu- +tion sequences. +It is shown that with a 4-cycle, the +single-coin evolution sequences: H4H4H4..., C4C4C4...; +the effective single coin evolution sequences: +I4H4I4, +H4I4I4..., H4I4H4I4...; and finally, the two-coin evolution + +5 +sequences: H4H4X4...,, H4X4H4X4... individually yield +periodic recurring MESPS with periods 4, 12, 12, 12, 4, +6 and 4 respectively. Moreover, these sequences render +the associated quantum walks periodic, and an analytical +proof of this result has been established. Finally, we show +MESPS generation via DTQWs on 3− and 5−cycles, too, +wherein we see that apart from begetting ordered QWs, +the sequences H3I3I3..., H3H3X3..., and H3X3H3X3..., +generate recurring MESPS with periods 6, 9, and 4 re- +spectively, for the 3-cycle case. In the 5-cycle case, the se- +quences H5H5X5... and H5X5H5X5... yield MESPS with +periods 15 and 4 respectively. In Supplementary Mate- +rial Sec. B, we summarize the proposed coin evolution +sequences to generate MESPS at time steps up to 10 and +beyond with the cyclic graphs. +One can experimentally implement our proposed +scheme using linear optical elements such as half-wave +plates (HWPs), quarter wave plates (QWPs) and polar- +izing beam splitters (PBSs), along with a fast switching +electro-optical modulator (EOM), wherein the photon’s +polarization degree of freedom encodes the coin state +with the position state is encoded into different time bins +of the photon [30, 37]. Evaluating the Schmidt norm re- +quires post-processing measurements like average polar- +izations of the photon by proper arrangement of an HWP +and QWP[37, 43]. +A comparison of our work with other relevant works +(DTQWs on 1D line) is shown in Supplementary Ma- +terial Sec. C. Apart from opening a unique avenue for +MESPS generation, our letter significantly outperforms +other schemes in terms of model simplicity and resource- +saving architecture and periodically yields MESPS at +both small and large time steps. +Our findings naturally raise new intriguing research +questions: What is the interplay between disorder and +entanglement (both hybrid and non-local) generation for +1D or higher dimensional walkers? +Can this scheme +be adapted to improve existing cryptography protocols +[31, 44]? +Investigations into these directions may lead +to significant results which foster new local or non-local +entanglement generation schemes. +Our presented work will significantly contribute to- +wards state-of-art controlled (maximal) entanglement +generation protocols, which is a fundamental resource in +quantum computing, teleportation and cryptography and +hence, a prerequisite to constructing reliable devices for +quantum information processing tasks. +Acknowledgement.— Colin Benjamin would like to +thank Science and Engineering Research Board (SERB) +for funding under the Core Research grant ”Josephson +junctions with strained Dirac materials and their appli- +cation in quantum information processing,” Grant No. +CRG/2019/006258. +[1] S. 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Janzing, Entropy of entanglement, Compendium of +Quantum Physics, 205–209 (2009). +[43] A. Peres, Quantum Theory: +Concepts and Methods +(Kluwer Academic, New York, 2002). +[44] C. Vlachou et al., Quantum walk public-key crypto- +graphic system, Int. J. Quantum Inf. 13, 1550050 (2015). +[45] Rong Zhang et al., Maximal coin-walker entanglement +in a ballistic quantum walk, Phys. Rev. A 105, 042216 +(2022) + +7 +SUPPLEMENTARY MATERIAL +Here in Sec. A, we provide some more details of our results, which includes the generation of maximally entangled +single-particle states (MESPS) and non-maximal single particle entanglement (SPE) via single-coin, effective-single +coin and two-coin evolution sequences. The proof for periodicity in QW dynamics and the occurrence of periodic +MESPS via two coin evolution sequences is also provided. We tabulated our proposed evolution sequences yielding +MESPS for comparison in Sec. B. Finally, we compare our work with other relevant works in Sec. C. +A. +MESPS and periodicity of QWs +4-cycle +The analytical and numerical studies on MESPS generation via DTQWs on cyclic graphs, dealt in the main text, for +the 4-cycle case with the single coin evolution sequence H4H4H4... and the effective-single (i.e., a single coin from the +experimentalist’s point of view) coin evolution sequences: I4H4I4..., H4I4I4..., H4I4H4I4..., yield recurring MESPS +with periods 4, 12, 12, and 4 respectively. +We also generate periodic MESPS with two-coin evolution sequences +H4H4X4..., and H4X4H4X4... with periods 6 and 4, respectively. +0 +10 +20 +30 +40 +50 t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sav +FIG. 6: Average Schmidt norm Sav (red) and average entanglement entropy Eav (blue) versus time steps (t) with +single-coin evolution sequence H4H4H4..., for 4-cycle. +Fig. 6 shows the Schmidt norm and entanglement entropy values as functions of time steps(t), generated by the +single coin evolution sequence H4H4H4... on the 4-cycle, wherein each data point is an average of the Schmidt norm +(Sav) or the entanglement entropy (Eav) and the average is taken over θ with fixed φ = +π +2 , and is evaluated as, +Sav = ⟨ S +√ +2⟩ = +1 +π +� π +0 dθ +S +√ +2 , or, Eav = ⟨E⟩ = +1 +π +� π +0 E dθ . Since both give identical results for MESPS, i.e., +Sav = Eav = 1, we only calculate Sav in the letter. One can also observe that the sequence H4H4H4... generates +MESPS at time steps t = 1, 5, 9, ... with period 4 (via any of the entanglement measures). Its ordered QW dynamics +support this periodicity, as shown in Fig. 7, which shows periodic probability distribution P(x = 0) for the walker +position at |0p⟩ with sequence H4H4H4... for 4-cycle, along with those via sequences H8H8H8... for 8-cycle and +C4C4C4... for 4-cycle. +In the main text, we observe that the two-coin evolution sequence H4H4X4... gives recurring MESPS periodic in +time for the 4-cycle. This periodicity is supported by ordered QW dynamics of H4H4X4 which is shown in Fig. 8. +Its analytical proof, following the steps involved in the periodicity condition, begins with the eigenvalues of the U4,1 +block of the evolution operator (U4)3, +λU4U4U4 +4,1 += 1 +2i√ρe +3 +2 i(γ+η)(e− 1 +2 i(γ+η) + e +1 +2 i(γ+η))(−3 + 2ρ + (e−i(η+γ) + ei(γ+η))ρ), +(6) +then for sequence H4H4X4, we have, +λH4H4X4 +4,1 += −i, +(7) + +8 +0 +10 +20 +30 +40 +50 t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P(x=0) +H4H4H4 +H8H8H8 +C4C4C4 +FIG. 7: Probability P(x = 0) of finding the walker at position |0p⟩ as function of time steps(t) with the single coin +evolution sequences H4H4H4..., C4C4C4... for 4-cycle, and H8H8H8... for 8-cycle. +from which we get for δ = (γ + η) = 0, ρ = 1 +4, which is an exact match to the value noted in Ref.[28] to generate an +ordered QW on a 4-cycle (with periodicity N = 12). Thus, H4H4X4... renders periodic QW as shown in Fig. 8. +0 +10 +20 +30 +40 +50 t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P(x=0) +I4H4I4 +H4H4X4 +H3H3X3 +FIG. 8: Probability P(x = 0) of finding the walker at position |0p⟩ as function of time steps(t) with the evolution +sequences I4H4I4..., H4H4X4... for 4-cycle, and H3H3X3... for 3-cycle. +The evolution sequences above yield better results as compared to the single coin evolution sequences X4X4X4... +and I4I4I4 since these do not generate MESPS. This is clearly seen in Fig. 9, where the average Schmidt norm (Sav) +values from the sequence H4H4H4... are compared with those from X4X4X4... and I4I4I4... sequences, for 4-cycle. +Though, X4X4X4... and I4I4I4 beget ordered QWs, they do not yield MESPS. +3-cycle and 5-cycle +Results for DTQW on 3- and 5-cycle graphs yield effective single-coin evolution sequences and two-coin evolution +sequences that result in MESPS, but no single-coin evolution sequence was found for the odd (3,5) cycle graphs. +However, sequence HkHkHk... gives MESPS only at time step t = 1 for k = 3 and k = 5-cycle graphs, see Figs. 10-11. +This is unlike the sequence H4H4H4... for the 4-cycle case, which gives MESPS at t = 1, 5, 9... with period 4, as +discussed in the main text. +Further, the effective single coin evolution sequence I3H3I3... renders ordered QWs without MESPS for 3-cycle, +whereas the evolution sequences I5H5I5... renders chaotic QWs but with MESPS at t = 3, 4 for 5-cycle, as shown in +Fig. 12. On the other hand, I4H4I4... begets ordered QWs as well as periodic MESPS at t = 5, 7, 9, 17, ... with period + +9 +H4H4H4 +X4X4X4 +I4I4I4 +10 +20 +30 +40 +50 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 9: Sav as function of time steps(t) with single coin evolution sequences: H4H4H4..., X4X4X4..., I4I4I4..., for +4-cycle graph. +H3H3H3 +X3X3X3 +I3I3I3 +10 +20 +30 +40 +50 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 10: Sav as function of time steps(t) with single coin evolution sequences: H3H3H3..., X3X3X3..., I3I3I3..., for +3-cycle graph. +12 (also see, Fig. 8). +In the main text, we also observe that the two-coin evolution sequences HkHkXk... and HkXkHkXk... yield recurring +and periodic MESPS for both 3- and 5-cycle graphs. A similar analytical proof for the periodic behaviour of H3H3X3... +on the 3-cycle can also be shown. Firstly, we get for the eigenvalues of the U3,1 block of the evolution operator (U3)3, +λU3U3U3 +3,1 += 1 +4 +√ρ(3(1 + i +√ +3)ei(η+γ)(ρ − 1) + 3i(i + +√ +3)e2i(η+γ)(ρ − 1) + 2ρ − 2e3i(η+γ)ρ), +(8) +and then for the sequence H3H3X3, we have, +λH3H3X3 +3,1 += −i +√ +3 +2 , +(9) +from which ρ = 0.550901, 0.15597 , which match the values noted in Ref.[28] to generate an ordered QW on a 3-cycle +(with periodicity N = 18). Thus, H3H3X3... renders periodic QW dynamics as shown in Fig. 8. + +10 +H5H5H5 +X5X5X5 +I5I5I5 +10 +20 +30 +40 +50 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 11: Sav as function of time steps(t) with single coin evolution sequences: H5H5H5..., X5X5X5..., I5I5I5..., for +5-cycle graph. +4-cycle +3-cycle +5-cycle +10 +20 +30 +40 +50 t +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sav +FIG. 12: Sav for effective single coin evolution sequence IkHkIk... up to 50 time steps(t), for k ∈ {3, 4, 5} i.e., 3-, 4- +and 5-cycle graphs. +B. +Our proposed evolution sequences +We summarize our proposed coin evolution sequences to generate recurring and periodic MESPS at time steps up +to 10 and beyond with the cyclic graphs in Table I + +11 +TABLE I: Evolution sequences to generate MESPS via DTQW up to 10-time steps and beyond periodically +DTQW on 4-cycle and 8-cycle +DTQW on 3-cycle and 5-cycle +Single coin evolution sequences: +H4H4H4... at t = 1, 5, 9, ... (P = 4) +C4C4C4... at t = 5, 17, 29, ... (P = 12) +H8H8H8... at t = 1, 13, 25, ... (P = 12) +Single coin evolution sequences: +H3H3H3... at t = 1 (Chaotic) +H5H5H5... at t = 1 (Chaotic) +Effective single coin evolution sequences: +I4H4I4... at t = 5, 7, 9, 17, ... (P = 12) +H4I4I4... at t = 1, 3, 5, 13, ... (P = 12) +H4I4H4I4... at t = 1, 5, 9, ... (P = 4) +Effective single coin evolution sequences: +I3H3I3... (No maximal SPE, periodic) +H3I3I3... at t = 1, 2, 7, 8, ... (P = 6) +H3I3H3I3... at t = 1, 2 (Chaotic) +I5H5I5... at t = 3, 4 (Chaotic) +H5I5I5... at t = 1, 2, 3, 4 (Chaotic) +H5I5H5I5... at t = 1, 2, 3 (Chaotic) +Two coin evolution sequences: +H4H4X4... at t = 1, 3, 7, 9, 13, ... (P = 6) +H4X4H4X4... at t = 1, 5, 9, ... (P = 4) +Two coin evolution sequences: +H3H3X3... at t = 1, 3, 4, 6, 10, ... (P = 9) +H3X3H3X3... at t = 1, 5, 9, ... (P = 4) +H5H5X5... at t = 1, 3 − 10, 12, 16, ... (P = 15) +H5X5H5X5... at t = 1, 5, 9, ... (P = 4) +t → time steps, P → Period at which the sequence yields maximal SPE +C. +A comparison of our scheme and results with other relevant works +We compare our scheme and results with other relevant works[2, 3, 34, 39, 45] in Table II. We first compare the +type of coin (evolution) sequences and the number of coin operators used. Here, we use single coin and effective +single coin evolution sequences(i.e., ’single’ from an experimental point of view) and deterministic evolution sequences +of two coin operators. Ref. [34] uses deterministic Parrondo type two-coin evolution sequences and has considered +four different coin operators. In Refs. [2, 39] coin sequences based on optimization and a deterministic sequence, +namely- universal entangler[39] are proposed. Ref. [3] uses a rigorous optimization scheme to obtain efficient coin +sequences with quantum process fidelity as the cost function. Ref.[45] deals with inhomogeneous QW, where they use +position-dependent coin operations. Experimental realization is straightforward with less number of coin operators +being involved and with a small number of sites. Given this, our proposed sequences (single, effective-single and two +coin evolution sequences) are as good as the entangling sequences of [2, 3, 34, 39, 45] and are much simpler to work +with. Moreover, our work involves only 3, 4or5−sites for the QW evolution, which is also resource-saving. +TABLE II: Comparison of the present work with other relevant works +Properties↓/Model→ +This Paper +(4-cycle with single coin: +ˆH or ˆC2(2+ +√ +3 +4 +, 0, 0)) +This Paper +(3,4,5-cycles with effective-single coin +or two-coin evolution sequences) +With Parrondo +sequences +Refs. [34] +With +Optimization +Ref. [3] +Analysis with +inhomogeneous-QW +Ref. [45] +With +Optimization +Ref. [2] +With +Optimization +Ref. [39] +No. of coin +operators used +One coin ˆH or ˆC2(ρ = 2+ +√ +3 +4 +, γ = 0, η = 0) Effective-single coin or, 2 coins +Two-coin sequences Hadamard and +Identity coins +2 coins +inhomogeneously +Full set of possible +coin operators +2 coins +Procedure +Simple, +single coin QW on cyclic graph +Simple, QW with deterministic +evolution sequences on cyclic graph +QW on 1D line with +Parrondo sequences +Optimization with +QW on 1D line +Inhomogenous QW +on 1D line +Basin hopping algorithm, +QW on 1D line +RL technique, +QW on 1D line +Maximally +entangled states +At time steps(t), +t = 1, 5, 9, ...(with H4H4H4..., P = 4); +moreover at, +t = 5, 17, 29, ...(with C4C4C4..., P = 12). +Both single-coin evolution +sequences H4H4H4... and C4C4C4... +individually yield periodic MESPS. +(P → Period at which the sequence yields +MESPS.) +At time steps(t), +t =1,3,4,6,10,... (H3H3X3..., P = 9); +t =1,2,7,8... (H3I3I3..., P = 6); +t = 1, 5, 9, ... (HkXkHkXk..., k = 3, 4, 5, P = 4); +moreover, at +t =1,3-10,12,16,... (H5H5X5..., P = 15); +t =1,2,3,4 (H5I5I5..., Chaotic); +t =5,7,9,17,... (I4H4I4..., P = 12), etc. +So MESPS ∀ t ≤ 10 and larger t. +And periodic emergence of MESPS. +At t = 3, 5 +and at asymptotic t +At t = 3 +and beyond +At any odd t and +asymptotically in even t +Almost at t = 10 +and beyond +Not achieved +Additionally, the focus on a small number of time steps in Ref. [2] shows maximal entanglement can be achieved in +10-time steps and beyond. Ref.[3] generates maximal entanglement via the optimization problem for any time step +beyond the second, whereas Ref.[45] shows maximal entanglement can be generated for any odd time steps, and in the +asymptotic limit for even steps. The method proposed in Ref. [34] gives MESPS in 3 and 5 time steps independent +of the initial states. Our scheme shows the generation of MESPS at all time steps(t) up to 10 as well as at a larger t +with periodic occurrence (see, Table I). + diff --git a/S9E3T4oBgHgl3EQfZwr5/content/tmp_files/load_file.txt b/S9E3T4oBgHgl3EQfZwr5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dac6a292d6cdec849e22635472532bf883af05f4 --- /dev/null +++ b/S9E3T4oBgHgl3EQfZwr5/content/tmp_files/load_file.txt @@ -0,0 +1,1031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf,len=1030 +page_content='Recurrent generation of maximally entangled single particle states via quantum walks on cyclic graphs Dinesh Kumar Panda1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' ∗ and Colin Benjamin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' † 1School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' National Institute of Science Education and Research Bhubaneswar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Jatni 752050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' India 2Homi Bhabha National Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Training School Complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Anushaktinagar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Mumbai 400094,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' India Maximally entangled single particle states (MESPS) are opening new possibilities in quantum technologies as they have the potential to encode more information and are robust to decoherence compared to their non-local two-particle counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Herein, using discrete-time quantum walks on k-cycles where k ∈ {3, 4, 5, 8} and by using either a single coin or effective-single coin or two coins in various deterministic sequences, we generate MESPS for recurring time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' These sequences beget ordered quantum walks and yield MESPS with periods 4, 6, 9, 12, and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' For the first time, we reveal single coins such as Hadamard, which can generate periodic MESPS (with periods 4 and 12) on 4 and 8-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This scheme is resource-saving with possibly the most straightforward experimental realization since the same coin is applied at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— Hybrid or single particle entangle- ment (SPE) refers to the entanglement between differ- ent degrees of freedom such as spatial mode, polariza- tion, and orbital angular momentum belonging to the same particle[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This local entanglement enables en- coding more information at the single particle level, is more robust against decoherence, and has simpler ex- perimental implementation than its non-local bipartite counterpart[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' SPE has significant applications in photonic quantum information processing and analysis of states of photons, elementary particles and quantum liquids[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Quantum joining, a physical process that al- lows the transfer of intra-particle entanglement between photons into a single output photon’s hybrid entangle- ment and its inverse, has been reported, and it has applications in quantum networking[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Photonic SPE states are potentially advantageous in optical quantum networks because they enable a more flexible network with every photon transmitted via a suitable channel [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' SPE has also been used in experimental tests of non- contextual hidden variable theories[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' After their introduction in [6], quantum walks(QWs), apart from their use in quantum computation, have been used in designing efficient quantum algorithms[7–9] as well as in simulating dynamics of complex physical[10] and biological systems[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' QWs can also be directly implemented in laboratories without requiring a quan- tum computer[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Experimentally, quantum walks have already been realized with photons[3, 13, 14], trapped ions[15, 16], superconducting qubits[17, 18], neutral atoms[19, 20], NMR quantum information processors[21] and ultra-cold Rubidium-87 atoms[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The quantum walker (or particle) is represented by a wave function and obeys the quantum superposition principle, and this makes QWs superior compared to their classical coun- terparts [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A discrete-time quantum walk (DTQW) evolves by repeatedly applying two quantum opera- ∗ dineshkumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='quantum@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='com † colin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='nano@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='com tors: coin and shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' DTQW dynamics also works as a sophisticated tool in engineering arbitrary quantum states[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' DTQWs allow one to explore multi-path quantum interference [20, 25], and numerous non-trivial topological[26] and geometrical phenomena[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A quan- tum walk can be described on a 1D or 2D lattice and analogously on a cyclic graph with k sites (k-cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' For some detailed studies on quantum walks on k-cycles, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='[28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [30] reports on the experimental imple- mentation of QW on cyclic graphs with photons using linear optical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A recent work [31] shows that it is possible to design an ordered or periodic QW (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', the walker returns to a particular position or site periodically after a finite number of time steps) by combining two chaotic or non-periodic quantum walks on 3− or 4−cycle via Parrondo strategy[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Intriguingly, the emergence of order from chaos and its inverse in QWs has applications in quantum cryptography (secure encryption-decryption protocols[31]), in designing new quantum algorithms and in developing theory of quantum chaos control[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Several manuscripts recently reported that DTQWs on 1D lines could be efficient tools to generate entangled single-particle states(SPS) or SPE, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [2, 3, 34–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [3, 35] report on experimental realizations of SPE generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' One recent manuscript [34] shows that by incorporating Parrondo sequences of coin operators in DTQWs on 1D line, one can obtain phase-independent SPE and, in one case, maximal SPE independent of the initial state parameters for time steps of 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' There has been no attempt to generate maximally entangled SPS (MESPS) or, for that matter SPE in cyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Also, seeing the versatility of DTQWs and the preeminent applicability of SPE, exploring differ- ent methods to generate highly or maximally entangled SPS via DTQWs is an important task, as it would con- tribute to extending the horizons of quantum technolo- gies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Our aim in this work is to study the propensity of DTQWs on cyclic graphs in generating MESPS either using a single coin or, an effective-single coin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', coin operator followed by Identity operator at successive time steps) or two coins in a deterministic evolution operator arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='04501v1 [quant-ph] 11 Jan 2023 2 sequence and, their relation to ordered QW dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' QW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— A DTQW on a k-cycle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 1), simi- lar to that on a 1D lattice, is defined on a tensor prod- uct of position (HP ) and coin (HC) Hilbert spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H = HP ⊗HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' HC is associated with the coin (or qubit) and has the computational basis {|0c⟩ , |1c⟩}, whereas HP , is the k-dimensional position space for the walker with computational basis {|xp⟩ : xp ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', k − 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' If the walker is initially localized at the site |0p⟩ in a general superposition of the coin states, it is represented by, |ψi⟩ = |ψ(t = 0)⟩ = cos(θ/2) |0p0c⟩ + eiφ sin(θ/2) |0p1c⟩ , (1) with θ ∈ [0, π] and φ ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The unitary coin operator, in general, is, ˆC2(ρ, γ, η) = � √ρ √1 − ρeiγ √1 − ρeiη −√ρei(γ+η) � , (2) where 0 ≤ ρ ≤ 1 and 0 ≤ γ, η ≤ π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The walker moves to the left by one site for coin state |0c⟩ and to the right by one site for coin state |1c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' For the walker on the k-cycle, we use the shift operator ˆS = �1 q=0 �k−1 j=0 |(j + 2q − 1(mod)k)p⟩ ⟨jp| ⊗ |qc⟩ ⟨qc| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The full evolution now can be expressed as, Uk(t) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆC2(ρ(t), γ(t), η(t))] , (3) where Ik is a k×k identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The time-evolution of the system (quantum walker) after time steps t is then, |ψ(t)⟩ = Uk(t) |ψ(t − 1)⟩ = Uk(t)Uk(t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='Uk(1) |ψ(0)⟩ = k−1 � j=0 [α1(j, t) |jp, 0c⟩ + α2(j, t) |jp, 1c⟩] , (4) where, α1(j, t) and α2(j, t) are amplitudes for the states |jp, 0c⟩ and |jp, 1c⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Further, the QW is said to be ordered or periodic if the walker reverts to its initial state after a time step, say t = N, irrespective of the choice of the initial quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' For an ordered QW with period N, we may write, |ψ(N)⟩ = Uk(N)Uk(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='Uk(1) |ψi⟩ = |ψi⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (5) For simplicity, if we apply a particular coin opera- tor in the above QW evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Uk(t) = Uk(t − 1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='Uk(1) = Uk(say), then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (5) is equivalent to, U N k |ψi⟩ = �2k i=1 aiλN i |λi⟩ , wherein the arbitrary |ψi⟩ is expressed in terms of the eigenvalues {λi} and eigenvec- tors {|λi⟩} of Uk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', |ψi⟩ = �2k i=1 ai |λi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Clearly, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (5), the condition of periodicity for the quantum walk follows as: U N k = I2k or λN i = 1, ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', 2k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Any unitary evolution operator which satisfies this con- dition gives a periodic probability distribution for the walker’s position, and such an operator is said to yield ordered QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Otherwise, the QW is said to be chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Furthermore, to simplify the problem of finding the eigen- values of Uk and hence the periodicity of the QW on a k-cycle, the 2×2 block circulant matrix Uk is block diag- onalized by using commensurate Fourier matrix tool as was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Then the block diagonalized form of Uk is given by FUkF †= diag[Uk,0, Uk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Uk,k−1], wherein F = F k⊗F 2 with F M being an M ×M commen- surate Fourier matrix i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', F M = (F M m,n) = 1 √ M (e2πi mn M ) where m, n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The periodicity condi- tion is satisfied if the eigenvalues λ± k,l of each block Uk,l take the form of de Moivre numbers (e2πi mr nr ) where each (mr, nr) pair is coprime [28, 31] or, equivalently if, λUk k,l = λ+ k,l+λ− k,l 2 = e2πi mr nr , and N = LCM({nr}), with r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', 2k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [28, 29], examples of parameter values for Uk that satisfy the periodicity condition have been given(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', to obtain ordered QWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We discuss a unique analytical approach for obtaining values of such parameters viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' {ρ, γ, η} yielding ordered QWs with de- terministic (including Parrondo type) evolution operator sequences in Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Measuring Entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— The initial quantum state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (1) is pure and evolves unitarily via DTQW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We use Schmidt norm(S) to quantify the entangle- ment between the coin and position degrees of freedom of the time-evolved quantum state |ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' It is rel- atively more convenient to calculate for QWs [2, 40– 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Let ρψ be density operator for |ψ(t)⟩ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', ρψ = |ψ(t)⟩ ⟨ψ(t)| and reduced density operator (ρc) for the coin space can be defined by, ρc ≡ Trp(ρψ), where the partial trace Trp is taken over the position de- grees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We write the eigenvalues of the re- duced density matrix ρc as, E± = 1 2 ± |⃗n| , where, ⃗n = � Re(Σjα1(j, t)α∗ 2(j, t)), Im(Σjα1(j, t)α∗ 2(j, t)), 1 2Σj(|α1(j, t)|2 − |α2(j, t)|2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Finally, the Schmidt norm is given by, S = � E− + � E+ , and for the present system with min(dim HP , dim HC) = 2, the Schmidt norm for the case of a MESPS is √ 2 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' To check whether our results are correct, we also calculate en- tanglement entropy (E) which is the von-Neumann en- tropy for the reduced density matrix ρc of coin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' E is defined as E(ρc) = -Tr(ρclog2ρc) with 0 and 1 for separable and maximally entangled states (or, MESPS) respectively, and can be calculated via, E = −(E−)log2(E−)−(E+)log2(E+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Indeed we see identical results for MESPS via both the entanglement measures, see Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— We consider the following coin operators, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (2), Hadamard coin ˆH = ˆC2(ρ = 1 2, γ = 0, η = 0), Grover or flip coin ˆX = ˆC2(ρ = 0, γ = 0, η = 0), and Identity coin ˆI = ˆC2(ρ = 1, γ, η ∋ γ+η = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We execute several numerical experiments with these coin operators and, also, by forming multiple deterministic coin evolu- tion sequences[34] such as AkAkAk, AkBkAkAkBkAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', AkBkAkBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', AkBkBkAkBkBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', AkAkBkAkAkBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', where, Ak = Uk(ρ1, γ1, η1) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆC2(ρ1, γ1, η1)] and Bk = Uk(ρ2, γ2, η2) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆC2(ρ2, γ2, η2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' If ˆC2 = ˆH, then evolution operator is denoted as Hk = Uk( 1 2, 0, 0) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆH] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Similarly, if ˆC2 = ˆX, we have evolution operator Xk = Uk(0, 0, 0) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆX] , and if ˆC2 = ˆI, then evolution operator Ik = Uk(1, 0, π) = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [Ik ⊗ ˆI] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The primary idea behind such experiments was to reveal evolution operator sequences involving ei- ther two coins such as HkHkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', HkXkHkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' or, effective-single coin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Ik with either Hk or Xk) such as IkHkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', HkIkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', HkIkHkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', or, notably a sin- gle coin such as HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' wherein the same coin ˆH is applied at every time step of the QW, which yield MESPS along with rendering ordered QWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We first dis- cuss single-coin evolution sequences, then effective single- coin evolution sequences and finally, two-coin evolution sequences to generate MESPS via DTQWs on the cyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Note that from an experimental point of view, a QW for a single coin evolution sequence like HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' is the simplest in terms of experimental setup as it just uses the same coin ˆH at each time step [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In other words, the same setup will be sufficient for its realization in small and significant time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Further, effective-single coin evolution sequences like IkHkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' or HkIkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' consists of just a single coin (here ˆH) and their implementation are indeed resource-saving because no device is required for Identity coin operation in the experimental realization of the associated QW [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 1: Even: 4-cycle (a) and 8-cycle (b) graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Odd: 3-cycle (c) and 5-cycle (d) graphs, with sites in green dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' MESPS on 4- and 8-cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— We now consider even i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', 4-cycle (k = 4) and 8-cycle (k = 8) graphs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2(red) shows the Schmidt norm values as functions of time steps(t), generated by the sin- gle coin evolution sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' on the 4-cycle, wherein each data point is an average of the Schmidt norm (Sav) and the average is taken over θ with fixed φ = π 2 and is evaluated as, Sav = ⟨ S √ 2⟩ = 1 π � π 0 dθ S √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Clearly, for maximal SPE or MESPS, Sav = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We ob- serve that the single coin evolution sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yields periodic MESPS at time steps t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Also, for 8-cycle case, single coin evo- lution sequence H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yields periodic MESPS at t = 1, 13, 25, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 12, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The periodic behaviour of HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' in generating MESPS is supported by its ordered QW dynamics on both k = 4 and k = 8 cycle graphs, see Supplementary Material H4H4H4 H8H8H8 C4C4C4 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2: Average Schmidt norm Sav versus time steps(t) with single coin evolution sequences H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 4-cycle and H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 8-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' H4I4H4I4, H4X4H4X4 I4H4I4 H4I4I4 H4H4X4 0 5 10 15 20 25 30 35 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3: Sav versus time steps(t) with sequences: I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4X4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We now discuss effective-single coin evolution se- quences I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' designed from the coin set { ˆH, ˆI} with the 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3 we see that the Sav values generated via the sequence I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' follow a periodic trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This observation is well supported by the periodic probability distribution P(x = 0) for the walker position at |0p⟩, in other words, the sequence I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' not only generates MESPS at t = 5, 7, 9, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 12 but also an ordered quan- tum walk (see Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ana- lytically one can also prove this by exploiting the peri- odicity condition, beginning with the eigenvalues of the U4,1-block of the evolution operator (U4)3, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3, λU4U4U4 4,1 = 1 2i√ρe 3 2 i(γ+η)(e− 1 2 i(γ+η) +e 1 2 i(γ+η))(−3+2ρ+ (e−i(η+γ) + ei(γ+η))ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Similarly, the U4,1-block’s eigen- values for the sequence I4H4I4 give, λI4H4I4 4,1 = i √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Equating λU4U4U4 4,1 with λI4H4I4 4,1 for δ = (γ + η) = 0, we get ρ = 2+ √ 3 4 , which is an exact match with ρ obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [28] for a periodic QW with period- icity N = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' With this analytical description for (a) (b) (c) (p)4 I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' sequence giving an ordered QW, we observe that single coin evolution sequence C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', coin ˆC2(ρ = 2+ √ 3 4 , γ = 0, η = 0) applied at each time step) generates MESPS with period 12 at t = 5, 17, 29, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This periodicity is supported by the ordered QW dynamics of the ˆC2( 2+ √ 3 4 , 0, 0) coin, see Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A, again with N = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Moreover, effective single coin evolution sequences H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yield periodic MESPS with periods 12 and 4 at time steps t = 1, 3, 5, 13, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We also observe that two-coin evolution sequence H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' gives recurring MESPS with period 6 at time steps t = 1, 3, 7, 9, 13, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (proof of this periodicity is in Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A), whereas the sequence H4X4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' gives recurring periodic MESPS with pe- riod 4 at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' , for 4-cycle, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' H3I3I3 H3I3H3I3 H3H3X3 H3X3H3X3 0 5 10 15 20 25 30 35 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 4: Sav versus time steps(t) with sequences: H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H3I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H3X3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 3-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' MESPS on 3- and 5-cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— Moving now to odd cycles, a 3-cycle (k = 3) and a 5-cycle (k = 5), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Unfortunately, for both 3 and 5−cycles, we do not see periodic MESPS with single coin evolution sequences HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', XkXkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', IkIkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', however, HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' generates MESPS at time step t = 1 but renders chaotic QWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In case of 3-cycle, the effective single coin evolution sequence H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yields periodic MESPS with period 6 at t = 1, 2, 7, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' , but the sequence I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' renders ordered QWs without MESPS, whereas H3I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' ren- ders chaotic QW with MESPS at t = 1, 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Further, in the case of 5-cycle, we see that the evolu- tion sequences I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='., H5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', and H5I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' do not yield periodic MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' But these sequences generate MESPS at t = 3, 4, t = 1, 2, 3, 4 and, t = 1, 2, 3 respec- tively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 5 (also see Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 4, we observe that the two-coin evolution sequences H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and H3X3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' generate recur- ring as well as periodic MESPS respectively at t = 1, 3, 4, 6, 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='(with period 9) and t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='(with pe- riod 4) via the DTQW on the 3-cycle (for proof of this periodicity, see Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' More- over, we see MESPS in the 5-cycle case via sequence H5H5X5 H5X5H5X5 H5I5I5 H5I5H5I5 5 10 15 20 25 30 35 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 5: Sav versus time steps(t) with sequences: H5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H5I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H5X5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 5-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with pe- riod 15, while via H5X5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' sequence at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 4, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The above results on MESPS generation are juxta- posed in Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' B, where one can compare our proposed evolution sequences on these cyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We see that employing H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', and H3X3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' on a 3-cycle one can obtain MESPS at all time steps up to 10, whereas on a 4-cycle their analogues give MESPS at all odd time steps t ≤ 10, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' As these sequences also beget pe- riodic QWs, thus, one obtains MESPS at larger time steps (t > 10) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Moreover, it is interesting to ob- serve that with just sequences H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and H5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', one can generate MESPS for all time steps t ≤ 10 and also at larger t, on a 5-cycle, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In fact, only H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' by itself yields MESPS at time steps t = 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' It is for the first time we report the single coin evo- lution sequences H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' which indi- vidually yields periodic MESPS on a 4-cycle graph with periods 4 and 12 respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Apart from that, we showed that H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' generates periodic MESPS at t = 1, 13, 25, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (with period 12) on a 8-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The gen- eration of periodic MESPS with just a single coin such as ˆH or ˆC2( 2+ √ 3 4 , 0, 0) via DTQWs, is a milestone in resource-saving SPE generation schemes and is also the simplest scheme in the field of controlled entanglement generation till date, as it would require the same experi- mental setting for its realization for both small and large time steps [3, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='— This letter provides a novel scheme to generate maximally entangled single particle states via DTQWs on both odd (3,5)- and even (4,8)-cycle graphs, with just a single coin and with both resource-saving effective-single coin and two-coin deterministic evolu- tion sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' It is shown that with a 4-cycle, the single-coin evolution sequences: H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' the effective single coin evolution sequences: I4H4I4, H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and finally, the two-coin evolution 5 sequences: H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=',, H4X4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' individually yield periodic recurring MESPS with periods 4, 12, 12, 12, 4, 6 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Moreover, these sequences render the associated quantum walks periodic, and an analytical proof of this result has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Finally, we show MESPS generation via DTQWs on 3− and 5−cycles, too, wherein we see that apart from begetting ordered QWs, the sequences H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', and H3X3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', generate recurring MESPS with periods 6, 9, and 4 re- spectively, for the 3-cycle case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In the 5-cycle case, the se- quences H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and H5X5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yield MESPS with periods 15 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In Supplementary Mate- rial Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' B, we summarize the proposed coin evolution sequences to generate MESPS at time steps up to 10 and beyond with the cyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' One can experimentally implement our proposed scheme using linear optical elements such as half-wave plates (HWPs), quarter wave plates (QWPs) and polar- izing beam splitters (PBSs), along with a fast switching electro-optical modulator (EOM), wherein the photon’s polarization degree of freedom encodes the coin state with the position state is encoded into different time bins of the photon [30, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Evaluating the Schmidt norm re- quires post-processing measurements like average polar- izations of the photon by proper arrangement of an HWP and QWP[37, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A comparison of our work with other relevant works (DTQWs on 1D line) is shown in Supplementary Ma- terial Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Apart from opening a unique avenue for MESPS generation, our letter significantly outperforms other schemes in terms of model simplicity and resource- saving architecture and periodically yields MESPS at both small and large time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Our findings naturally raise new intriguing research questions: What is the interplay between disorder and entanglement (both hybrid and non-local) generation for 1D or higher dimensional walkers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Can this scheme be adapted to improve existing cryptography protocols [31, 44]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Investigations into these directions may lead to significant results which foster new local or non-local entanglement generation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Our presented work will significantly contribute to- wards state-of-art controlled (maximal) entanglement generation protocols, which is a fundamental resource in quantum computing, teleportation and cryptography and hence, a prerequisite to constructing reliable devices for quantum information processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Vlachou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Quantum walk public-key crypto- graphic system, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 13, 1550050 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [45] Rong Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Maximal coin-walker entanglement in a ballistic quantum walk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A 105, 042216 (2022) 7 SUPPLEMENTARY MATERIAL Here in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A, we provide some more details of our results, which includes the generation of maximally entangled single-particle states (MESPS) and non-maximal single particle entanglement (SPE) via single-coin, effective-single coin and two-coin evolution sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The proof for periodicity in QW dynamics and the occurrence of periodic MESPS via two coin evolution sequences is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We tabulated our proposed evolution sequences yielding MESPS for comparison in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Finally, we compare our work with other relevant works in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' MESPS and periodicity of QWs 4-cycle The analytical and numerical studies on MESPS generation via DTQWs on cyclic graphs, dealt in the main text, for the 4-cycle case with the single coin evolution sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and the effective-single (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', a single coin from the experimentalist’s point of view) coin evolution sequences: I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', yield recurring MESPS with periods 4, 12, 12, and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We also generate periodic MESPS with two-coin evolution sequences H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', and H4X4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with periods 6 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 0 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 6: Average Schmidt norm Sav (red) and average entanglement entropy Eav (blue) versus time steps (t) with single-coin evolution sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 6 shows the Schmidt norm and entanglement entropy values as functions of time steps(t), generated by the single coin evolution sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' on the 4-cycle, wherein each data point is an average of the Schmidt norm (Sav) or the entanglement entropy (Eav) and the average is taken over θ with fixed φ = π 2 , and is evaluated as, Sav = ⟨ S √ 2⟩ = 1 π � π 0 dθ S √ 2 , or, Eav = ⟨E⟩ = 1 π � π 0 E dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Since both give identical results for MESPS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Sav = Eav = 1, we only calculate Sav in the letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' One can also observe that the sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' generates MESPS at time steps t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 4 (via any of the entanglement measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Its ordered QW dynamics support this periodicity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 7, which shows periodic probability distribution P(x = 0) for the walker position at |0p⟩ with sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 4-cycle, along with those via sequences H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 8-cycle and C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In the main text, we observe that the two-coin evolution sequence H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' gives recurring MESPS periodic in time for the 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This periodicity is supported by ordered QW dynamics of H4H4X4 which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Its analytical proof, following the steps involved in the periodicity condition, begins with the eigenvalues of the U4,1 block of the evolution operator (U4)3, λU4U4U4 4,1 = 1 2i√ρe 3 2 i(γ+η)(e− 1 2 i(γ+η) + e 1 2 i(γ+η))(−3 + 2ρ + (e−i(η+γ) + ei(γ+η))ρ), (6) then for sequence H4H4X4, we have, λH4H4X4 4,1 = −i, (7) 8 0 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 P(x=0) H4H4H4 H8H8H8 C4C4C4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 7: Probability P(x = 0) of finding the walker at position |0p⟩ as function of time steps(t) with the single coin evolution sequences H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 4-cycle, and H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 8-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' from which we get for δ = (γ + η) = 0, ρ = 1 4, which is an exact match to the value noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [28] to generate an ordered QW on a 4-cycle (with periodicity N = 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Thus, H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' renders periodic QW as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 0 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='0 P(x=0) I4H4I4 H4H4X4 H3H3X3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 8: Probability P(x = 0) of finding the walker at position |0p⟩ as function of time steps(t) with the evolution sequences I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 4-cycle, and H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for 3-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The evolution sequences above yield better results as compared to the single coin evolution sequences X4X4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and I4I4I4 since these do not generate MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 9, where the average Schmidt norm (Sav) values from the sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' are compared with those from X4X4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and I4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' sequences, for 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Though, X4X4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and I4I4I4 beget ordered QWs, they do not yield MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 3-cycle and 5-cycle Results for DTQW on 3- and 5-cycle graphs yield effective single-coin evolution sequences and two-coin evolution sequences that result in MESPS, but no single-coin evolution sequence was found for the odd (3,5) cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' However, sequence HkHkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' gives MESPS only at time step t = 1 for k = 3 and k = 5-cycle graphs, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 10-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' This is unlike the sequence H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' for the 4-cycle case, which gives MESPS at t = 1, 5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 4, as discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Further, the effective single coin evolution sequence I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' renders ordered QWs without MESPS for 3-cycle, whereas the evolution sequences I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' renders chaotic QWs but with MESPS at t = 3, 4 for 5-cycle, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' On the other hand, I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' begets ordered QWs as well as periodic MESPS at t = 5, 7, 9, 17, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' with period 9 H4H4H4 X4X4X4 I4I4I4 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 9: Sav as function of time steps(t) with single coin evolution sequences: H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', X4X4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', I4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 4-cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' H3H3H3 X3X3X3 I3I3I3 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 10: Sav as function of time steps(t) with single coin evolution sequences: H3H3H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', X3X3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', I3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 3-cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 12 (also see, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In the main text, we also observe that the two-coin evolution sequences HkHkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and HkXkHkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' yield recurring and periodic MESPS for both 3- and 5-cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A similar analytical proof for the periodic behaviour of H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' on the 3-cycle can also be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Firstly, we get for the eigenvalues of the U3,1 block of the evolution operator (U3)3, λU3U3U3 3,1 = 1 4 √ρ(3(1 + i √ 3)ei(η+γ)(ρ − 1) + 3i(i + √ 3)e2i(η+γ)(ρ − 1) + 2ρ − 2e3i(η+γ)ρ), (8) and then for the sequence H3H3X3, we have, λH3H3X3 3,1 = −i √ 3 2 , (9) from which ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='550901, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='15597 , which match the values noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [28] to generate an ordered QW on a 3-cycle (with periodicity N = 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Thus, H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' renders periodic QW dynamics as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 10 H5H5H5 X5X5X5 I5I5I5 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 11: Sav as function of time steps(t) with single coin evolution sequences: H5H5H5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', X5X5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', I5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', for 5-cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 4-cycle 3-cycle 5-cycle 10 20 30 40 50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='00 Sav FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 12: Sav for effective single coin evolution sequence IkHkIk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' up to 50 time steps(t), for k ∈ {3, 4, 5} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', 3-, 4- and 5-cycle graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Our proposed evolution sequences We summarize our proposed coin evolution sequences to generate recurring and periodic MESPS at time steps up to 10 and beyond with the cyclic graphs in Table I 11 TABLE I: Evolution sequences to generate MESPS via DTQW up to 10-time steps and beyond periodically DTQW on 4-cycle and 8-cycle DTQW on 3-cycle and 5-cycle Single coin evolution sequences: H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 4) C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 5, 17, 29, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 12) H8H8H8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 13, 25, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 12) Single coin evolution sequences: H3H3H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1 (Chaotic) H5H5H5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1 (Chaotic) Effective single coin evolution sequences: I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 5, 7, 9, 17, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 12) H4I4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 3, 5, 13, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 12) H4I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 4) Effective single coin evolution sequences: I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (No maximal SPE, periodic) H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 2, 7, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 6) H3I3H3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 2 (Chaotic) I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 3, 4 (Chaotic) H5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 2, 3, 4 (Chaotic) H5I5H5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 2, 3 (Chaotic) Two coin evolution sequences: H4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 3, 7, 9, 13, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 6) H4X4H4X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 4) Two coin evolution sequences: H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 3, 4, 6, 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 9) H3X3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 4) H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 3 − 10, 12, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 15) H5X5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' at t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P = 4) t → time steps, P → Period at which the sequence yields maximal SPE C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' A comparison of our scheme and results with other relevant works We compare our scheme and results with other relevant works[2, 3, 34, 39, 45] in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' We first compare the type of coin (evolution) sequences and the number of coin operators used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Here, we use single coin and effective single coin evolution sequences(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', ’single’ from an experimental point of view) and deterministic evolution sequences of two coin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [34] uses deterministic Parrondo type two-coin evolution sequences and has considered four different coin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [2, 39] coin sequences based on optimization and a deterministic sequence, namely- universal entangler[39] are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [3] uses a rigorous optimization scheme to obtain efficient coin sequences with quantum process fidelity as the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [45] deals with inhomogeneous QW, where they use position-dependent coin operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Experimental realization is straightforward with less number of coin operators being involved and with a small number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Given this, our proposed sequences (single, effective-single and two coin evolution sequences) are as good as the entangling sequences of [2, 3, 34, 39, 45] and are much simpler to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Moreover, our work involves only 3, 4or5−sites for the QW evolution, which is also resource-saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' TABLE II: Comparison of the present work with other relevant works Properties↓/Model→ This Paper (4-cycle with single coin: ˆH or ˆC2(2+ √ 3 4 , 0, 0)) This Paper (3,4,5-cycles with effective-single coin or two-coin evolution sequences) With Parrondo sequences Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [34] With Optimization Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [3] Analysis with inhomogeneous-QW Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [45] With Optimization Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [2] With Optimization Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [39] No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' of coin operators used One coin ˆH or ˆC2(ρ = 2+ √ 3 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' γ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' η = 0) Effective-single coin or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 2 coins Two-coin sequences Hadamard and Identity coins 2 coins inhomogeneously Full set of possible coin operators 2 coins Procedure Simple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' single coin QW on cyclic graph Simple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' QW with deterministic evolution sequences on cyclic graph QW on 1D line with Parrondo sequences Optimization with QW on 1D line Inhomogenous QW on 1D line Basin hopping algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' QW on 1D line RL technique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' QW on 1D line Maximally entangled states At time steps(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='(with H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' moreover at, t = 5, 17, 29, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='(with C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Both single-coin evolution sequences H4H4H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' and C4C4C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' individually yield periodic MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (P → Period at which the sequence yields MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=') At time steps(t), t =1,3,4,6,10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (H3H3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' t =1,2,7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (H3I3I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' t = 1, 5, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (HkXkHkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', k = 3, 4, 5, P = 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' moreover, at t =1,3-10,12,16,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (H5H5X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' t =1,2,3,4 (H5I5I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', Chaotic);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' t =5,7,9,17,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' (I4H4I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=', P = 12), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' So MESPS ∀ t ≤ 10 and larger t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' And periodic emergence of MESPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' At t = 3, 5 and at asymptotic t At t = 3 and beyond At any odd t and asymptotically in even t Almost at t = 10 and beyond Not achieved Additionally, the focus on a small number of time steps in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [2] shows maximal entanglement can be achieved in 10-time steps and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [3] generates maximal entanglement via the optimization problem for any time step beyond the second, whereas Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [45] shows maximal entanglement can be generated for any odd time steps, and in the asymptotic limit for even steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' The method proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' [34] gives MESPS in 3 and 5 time steps independent of the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} +page_content=' Our scheme shows the generation of MESPS at all time steps(t) up to 10 as well as at a larger t with periodic occurrence (see, Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E3T4oBgHgl3EQfZwr5/content/2301.04501v1.pdf'} diff --git a/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf b/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..357013b9f22414538540cd58a0ceeb66698e30dd --- /dev/null +++ b/TNAzT4oBgHgl3EQfXfzv/content/2301.01321v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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B. Baker,1 M. Burrows,2 Ch. Elster,1 K.D. Launey,2 P. Maris,3 G. Popa,1 and S. P. Weppner4 +1Institute of Nuclear and Particle Physics, and Department of +Physics and Astronomy, Ohio University, Athens, OH 45701, USA +2Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA +3Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA +4Natural Sciences, Eckerd College, St. +Petersburg, FL 33711, USA +The effective interaction between a nucleon and a nucleus is one of the most important ingredients +for reaction theories. Theoretical formulations were introduced early by Feshbach and Watson, and +efforts of deriving and computing those ‘optical potentials’ in a microscopic fashion have a long +tradition. However, only recently the leading order term in the Watson multiple scattering approach +could be calculated fully ab initio, meaning that the same nucleon-nucleon (NN) interaction enters +both the structure as well as the reaction pieces on equal footing. This allows the uncertainties +from the underlying chiral effective NN interaction to be systematically explored in nucleon-nucleus +elastic scattering observables. +In this contribution the main ingredients for arriving at the ab initio leading order of the effective +nucleon-nucleus interaction in the Watson approach will be reviewed. Concentrating on one specific +chiral NN interaction from the LENPIC collaboration and light nuclei with a 0+ ground state, the +leading order nucleon-nucleus interaction is calculated using up to the third chiral order (N2LO) in +the nucleon-nucleon potential, and elastic scattering observables are extracted. Then pointwise as +well as correlated uncertainty quantification is used for the estimation of the chiral truncation error. +Elastic scattering observables for 4He, 12C, and 16O for between 65 and 200 MeV projectile energy +will be analyzed. +I. +INTRODUCTION +Simplifying the many-body problem posed by scattering of a proton or neutron from a nucleus to a two-body +problem with an effective (optical) potential was introduced already by Bethe [1] in the 1930s, and its justification +summarized by Feshbach [2]. Since then differential cross sections as well as spin observables for elastic scattering +played an important role in either determining the parameters in phenomenological optical models for proton or +neutron scattering from nuclei or in testing validity and accuracy of microscopic models thereof. The theoretical +approach to elastic scattering from a nuclear target presented in this article is based on the ansatz of a multiple +scattering expansion that was pioneered by Watson [3, 4], made familiar by Kerman, McManus, and Thaler (KMT) [5]. +and refined further as spectator expansion [6–8]. Specifically, elastic scattering from stable nuclei has led in the 1990s +to a large body of work on microscopic optical potentials in which the nucleon-nucleon interaction and the density of +the nucleus were taken as input to rigorous calculations of first-order potentials, in either a Kerman-McManus-Thaler +(KMT) or a Watson expansion of the multiple scattering series (see e.g. [9–14]). Here the primary goal was a deeper +understanding of the reaction mechanism. However, a main disadvantage of that work was the lack of sophisticated +nuclear structure input compared to what is available today. +Recent developments of the nucleon-nucleon (NN) and three-nucleon (3N) interactions, derived from chiral effective +field theory, have yielded major progress [15–22]. These, together with the utilization of massively parallel computing +resources (e.g., see [23–27]), have placed ab initio large-scale simulations at the frontier of nuclear structure and reaction +explorations. Among other successful many-body theories, the ab initio no-core shell-model (NCSM) approach (see, +e.g., [28–31]), has over the last decade taken center stage in the development of microscopic tools for studying the +structure of atomic nuclei. +The NCSM concept combined with a symmetry-adapted (SA) basis in the ab initio +SA-NCSM [32] has further expanded the reach to the structure of intermediate-mass nuclei [33]. +Following the developments in nuclear structure theory, it is very natural to again consider rigorous calculations +of effective folding nucleon-nucleus (NA) potentials, since now the nuclear densities required as input for the folding +with the NN scattering amplitudes can be based on the same chiral NN interaction. This development also allows to +investigate effects of truncation uncertainties in the chiral expansion on NA scattering observables in a similar fashion +as already successfully performed in NN scattering (see e.g. [34–36]), nucleon-deuteron scattering [37], or structure +observables for light nuclei [31, 38]. +The theoretical and computational developments leading to ab initio NA effective interactions (in leading order in +the spectator expansion) are described in a serious of publications by the authors [39–43] and others (see e.g. [44–47]). +Thus the aim of this review is to shed light on truncation uncertainties in the chiral expansion, and within that +context give a perspective on intricacies of the spectator expansion as well as the explicit content of its leading order +term, which can now be calculated ab initio. +arXiv:2301.04293v1 [nucl-th] 11 Jan 2023 + +2 +Deriving ab initio optical potentials within a multiple scattering approach focuses on projectile energies at energies +about 80 MeV or higher, since the expectation is that at those energies the leading order term may already capture +the most important physics. Another recent ab initio approach starts from a formulation introduced by Feshbach [48] +and constructs optical potentials and elastic scattering observables within a Green’s function approach [49, 50]. For +elastic scattering from medium-mass nuclei the coupled-cluster method [51] and the SA-NCSM [52] approach have +been successfully implemented. These approaches are by design better suited for calculating scattering observables +at energies below about 20-30 MeV due to restrictions on the size of the model spaces which increase with increasing +projectile energy. In Ref. [53] an extensive overview of the status of the field of optical potentials and their need in +the rare-isotope era is given and the current status of ab initio approaches is discussed. We want to encourage the +reader to refer to this work, for more details. +II. +WATSON OPTICAL POTENTIAL WITHIN THE SPECTATOR EXPANSION +The standard starting point for describing elastic scattering of a single projectile from a target of A particles within +a multiple scattering approach is the separation of the Lippmann-Schwinger (LS) equation for the transition operator +T, +T = V + V G0(E)T +(1) +into two parts, namely an integral equation for T, +T = U + UG0(E)PT, +(2) +where U is the effective potential operator defined by a second integral equation, +U = V + V G0(E)QU. +(3) +Here P is a projection onto the ground state of the target, P = |Φ0⟩⟨Φ0| +⟨Φ0|Φ0⟩ , with P +Q = 1 and [G0(E), P] = 1. The free +propagator for the projectile and target system is given by G0(E) = (E − h0 − HA + iϵ)−1 where h0 is the kinetic +energy of the projectile and HA is the Hamiltonian of the target nucleus. The general solutions of the nuclear bound +state problem HA|Φ⟩ include the ground state, excited states and continuum states. For the scattering problem given +by the transition amplitude T the reference energy separating bound and continuum states is chosen such that the +ground state energy is set to zero. Thus energies referring to the target Hamiltonian in G0 are excitation energies of +the target. With these definitions the transition operator for elastic scattering may be redefined as Tel = PTP, in +which case Eq. (2) can be written as +Tel = PUP + PUPG0(E)Tel. +(4) +A. +Spectator expansion of the operator U +The transition operator for elastic scattering is given by a straightforward one-body integral equation, which of +course requires the knowledge of PUP, which is a many-body operator. For a brief review we follow the spectator +expansion of PUP as introduced in Ref. [54] in contrast to Ref. [6] where the expansion of T is considered. Following +those references, we assume the presence of two-body forces only for the present discussion. The extension to many- +body forces is not precluded by the formulation. With this assumption the operator U can be expanded as +U = +A +� +i=1 +Ui, +(5) +where Ui is given by +Ui = v0i + v0iG0(E)Q +A +� +j=1 +Uj, +(6) +provided that V = �A +i=1 v0i, where the two-body potential v0i acts between the projectile and the ith target nucleon. +Through the introduction of an operator τi which satisfies +τi = v0i + v0iG0(E)Qτi, +(7) + +3 +Eq. (6) can be rearranged as +Ui = τi + τiG0(E)Q +� +j̸=i +Uj. +(8) +This rearrangement process can be continued for all A target particles, so that the operator for the optical potential +can be expanded in a series of A terms of the form +U = +A +� +i=1 +τi + +A +� +i,j̸=i +τij + +A +� +i,j̸=i,k̸=i,j +τijk + · · · . +(9) +This is the Spectator Expansion for U , where each term is treated in turn. +The separation of the interactions +according to the number of interacting nucleons has a certain latitude, due to the many-body nature of G0(E), which +needs to be considered separately. In the following we will concentrate on the leading-order term, which is still a +many-body operator due the the presence of G0(E). The next-to-leading order term in this spectator expansion for +U has been formally derived and connected to standard three-body equations in Ref. [54]. +B. +Propagator expansion in the leading-order term of U +When using the leading-order term of the spectator expansion as given in Eq. (7), for elastic scattering only PτiP, +or equivalently ⟨Φ0|τi|Φ0⟩ needs be considered. With this in mind, Eq. (7) can be re-expressed as +τi = v0i + v0iG0(E)τi − v0iG0(E)Pτi = ˆτi − ˆτiG0(E)Pτi, +(10) +or +⟨Φ0|τi|Φ0⟩ = ⟨Φ0|ˆτi|Φ0⟩ − ⟨Φ0|ˆτi|Φ0⟩ +1 +(E − EA) − h0 + iε⟨Φ0|τi|Φ0⟩, +(11) +where ˆτi is defined as the solution of +ˆτi = v0i + v0iG0(E)ˆτi. +(12) +The combination of Eqs. (10) and (2) corresponds to the leading-order Watson optical potential [3, 4]. In ab initio +structure calculations the one-body densities or ground state wave functions for protons and neutrons are calculated +separately, so that Eq. (11) allows to combine e.g. for proton scattering of a nucleus the proton-neutron interaction +(ˆτi=pn) with the neutron one-body density and the proton-proton interaction with the proton one-body density. The +sum over i then adds both to obtain the driving term ⟨Φ0|ˆτi|Φ0⟩ the integral equation, Eq. (11). +If the projectile-target-nucleon interaction is assumed to be the same for all target nucleons and if iso-spin ef- +fects are neglected then the KMT approximation ( A−1 +A ⟨Φ0|ˆτi|Φ0⟩) can be derived from the leading-order Watson +potential [5]. When working with momentum space integral equations, the numerical implementation of Eq. (11) is +straightforward [40, 41, 45, 55]. Working in coordinate space with differential equations does not allow an equally +straightforward implementation, and thus the KMT prescription is the most favorable alternative. A comparison +between leading-order Watson potential and the KMT prescription is shown in Fig. 1 for elastic proton scattering +from 8He at 71 MeV laboratory kinetic energy. Despite the relatively large difference between the proton and neutron +densities for this nucleus the KMT prescription agrees with the exact Watson description very well up to momentum +transfers of about 2 fm−1. +Since Eq. (11) is a one-body integral equation, the principal problem is to find a solution of Eq. (12), which due +to many-body character of G0(E) is still a many-body integral equation, and in fact no more easily solved than the +starting point of Eq. (1). +For most practical calculations the so-called closure approximation to G0(E) is implemented [56] turning Eq. (12) +into a one-body integral equation. This approximation replaces HA by a constant that is interpreted as an average +excitation energy, and is justified when the projectile energy is large compared to typical excitation energies of the +nucleus. The closure approximation is very successfully applied for elastic scattering around 80 MeV and higher. +Going beyond the closure approximation in the spirit of the spectator expansion we want to single out one target +nucleon i and write G0(E) as +G0(E) = (E − h0 − HA + iε)−1 += (E − h0 − hi − +� +j̸=i +vij − Hi + iε)−1, +(13) + +4 +where the target Hamiltonian is expanded as HA = hi + � +j̸=i vij + Hi with vij being the interaction between +target nucleons i and j, and Hi being an (A-1)-body operator containing all higher order effects. Realizing that +� +j̸=i vij ≡ Wi and thus Hi = HA − hi − Wi does not have an explicit dependence on the ith particle, then Hi may +be replaced by an average energy Ei which is akin to the effective binding energy between the ith nucleon and the +A − 1 spectator. This is not an approximation since G0(E) may be regarded as +G0(E) = [(E − Ei) − h0 − hi − Wi − (Hi − Ei) + iε]−1 +(14) +and (Hi − Ei) should be set aside to be treated in the next order of the expansion of the propagator G0(E). In this +order of the expansion G0(E) becomes +Gi(E) = [(E − Ei) − h0 − hi − Wi + iε]−1, +(15) +and Eq. (12) reads +ˆτi = v0i + v0iGi(E)ˆτi. +(16) +In order to connect the above expression with the free NN amplitude +t0i = v0i + v0igit0i +(17) +with +gi = [(E − Ei) − h0 − hi + iε]−1. +(18) +algebraic relations between the resolvents lead to +ˆτi = t0i + t0iGiWigi(E)ˆτi. +(19) +Defining GiWi = giTi with Ti = Wi + WigiTi leads to +ˆτi = t0i + t0igiTigi ˆτi. +(20) +The three-body character of the above expression becomes more evident if one defines it as a set of coupled equations +as +ˆτi = t0i + t0igiXi +Xi = Tigi ˆτi. +(21) +Though the spectator expansion of the operator U in terms of active particles is defined in Eq. (9), we see that this +expansion is performed in terms of quantities which contain many-body propagators. Each of the ingredients τi, τij, +etc. may themselves be expanded in a spectator expansion, i.e. expanding the many-body propagator also according +to the number of active participants. The corrections to the propagator in the leading-order term of U contributions +that arise from the Q space, whereas the terms arising from the propagator remain in the P space at first order level. +Thus their contribution may be more relevant for elastic scattering. +In an explicit treatment of Gi(E) it is necessary to consider the explicit form of � +j̸=i vij = Wi, which is a priori a +two-body operator. In the framework of ab initio nuclear structure calculations this will involve two-body densities. +In earlier work [54, 57, 58] the quantity Wi was treated as one-body operator, specifically a mean-field potential. This +was a physically reasonable choice, though being outside the strict demands of the spectator expansion. However, +those studies revealed that the next order in the propagator expansion has little effect on elastic scattering observables +at energies larger than 100 MeV, while the description of differential cross section and spin-observables for elastic +scattering from 40Ca at 48 MeV showed considerable improvement with respect to experiment [57]. Obviously this +type of calculation will need to be explored within an ab initio approach. In Ref. [57] the energy Ei of Eq. (18) was +set to zero. +As illustrated in this section, deriving a multiple scattering expansion for elastic NA scattering means projecting +on the ground state of the target in order to obtain a Lippman-Schwinger type equation for the transition amplitude +and obtaining an operator U for the effective interaction, which is defined in the space Q = 1 − P. In this spirit, +the spectator expansion contains therefore two pieces, namely the expansion of the operator U in terms of active +particles in the scattering process as well as the expansion of target Hamiltonian HA in the propagator G0(E) in a +similar fashion. Thus it is very difficult to define a single expansion parameter which governs the convergence of the +expansion. + +5 +III. +LEADING ORDER AB INITIO OPTICAL POTENTIAL BASED ON A CHIRAL NN +INTERACTION +The leading order of the spectator expansion involves two active nucleons, the projectile and a target nucleon. +Therefore, the leading order is driven by the NN amplitude M , which in its most general form can be parameterized +in terms of Wolfenstein amplitudes [59–61], +M(q, KNN, ϵ) = A(q, KNN, ϵ)1 ⊗ 1 ++ iC(q, KNN, ϵ) +� +σ(0) · ˆn +� +⊗ 1 ++ iC(q, KNN, ϵ) 1 ⊗ +� +σ(i) · ˆn +� ++ M(q, KNN, ϵ)(σ(0) · ˆn) ⊗ (σ(i) · ˆn) ++ [G(q, KNN, ϵ) − H(q, KNN, ϵ)] (σ(0) · ˆq) ⊗ (σ(i) · ˆq) ++ [G(q, KNN, ϵ) + H(q, KNN, ϵ)] (σ(0) · ˆK) ⊗ (σ(i) · ˆK) ++ D(q, KNN, ϵ) +� +(σ(0) · ˆq) ⊗ (σ(i) · ˆK) + (σ(0) · ˆK) ⊗ (σ(i) · ˆq) +� +, +(22) +where σ(0) describes the spin of the projectile, and σ(i) the spin of the struck nucleon. The average momentum in +the NN frame is defined as KNN = 1 +2 (k′ +NN + kNN). The scalar functions A, C, M, G, H, and D are referred to +as Wolfenstein amplitudes and only depend on the scattering momenta and energy. Each term in Eq. (22) has two +components, namely a scalar function of two vector momenta and an energy and the coupling between the operators +of the projectile and the struck nucleon. The linear independent unit vectors ˆq, ˆK, and ˆn are defined in terms of the +momentum transfer and the average momentum as +ˆq = q +|q| , +ˆK = K +|K| , +ˆn = K × q +|K × q|, +(23) +and span the momentum vector space. With the exception of the momentum transfer q, which is invariant under frame +transformation, the vectors in Eq. (23) need to be considered in their respective frame in explicit calculations [41, 62]. +For the struck target nucleon the expectation values of the operator 1 and the scalar products of σ(i) with the linear +independent unit vectors of Eq. (23) need to be evaluated with the ground state wave functions of the respective +nucleus when calculating the leading-order NA effective interaction. Evaluating the expectation value of the operator +1 in the ground state of the nucleus results in the scalar nonlocal, translationally invariant one-body density that has +traditionally been used as input to microscopic or ab initio calculations of leading order effective interactions [11, 12, +40, 44]. The other operators from Eq. (23), namely (σ(i) · ˆn), (σ(i) · ˆq), and (σ(i) · ˆK) need to also be evaluated +for a leading-order ab initio NA effective interaction, in which the NN interaction is treated on equal footing in the +reaction and structure calculation. +Thus, the general expression for a nonlocal density needs to include the spin operator σ(i) explicitly, +ρKs +qs (p, p′) = +� +Φ′ +0 +����� +A +� +i=1 +δ3(pi − p)δ3(p′ +i − p′)σ(i)Ks +qs +����� Φ0 +� +, +(24) +where σ(i)Ks +qs +is the spherical representation of the spin operator and the wavefunction Φ0 (p1, ..., pA) = ⟨p1, ..., pA|Φ0⟩ +is defined in momentum space. Evaluating this expression for Ks = 0 gives the nonlocal one-body scalar density and +Ks = 1 becomes a nonlocal one-body spin density. +The Wolfenstein parameterization of Eq. (22) requires the evaluation of scalar products of the one-body spin +density with unit momentum vectors. Since those only depend on the momenta p and p′, those can be calculated +as ρKs (p, p′) · ˆn, ρKs (p, p′) · ˆq, and ρKs (p, p′) · ˆK. For the explicit calculation of ρKs (p, p′) · ˆn, we refer the reader +to [41, 62]. The scalar products (σ(i) · ˆq) and (σ(i) · ˆK) represent scalar products of a pseudo-vector and a vector, a +construct that is not invariant under parity transformations, and thus vanish when sandwiched between ground state +wave functions, which is explicitly shown in [62]. Thus the tensor contributions of the NN force only enter the leading +order effective NA interaction through the Wolfenstein amplitude M as long as elastic scattering is considered. When +e.g. transition amplitudes between states of different parity would be considered, the other tensor amplitudes will +contribute. +Currently contributions to elastic scattering observables due to the spin-projected one-body densities have only +been calculated for light nuclei with 0+ ground states, and it was found that this contribution is very small for nuclei +with equal proton and neutron numbers [41, 42]. This is likely different for nuclei with ground states of nonzero spin, + +6 +which was explored for 10B polarization transfer observables in Ref. [63, 64], where the authors assume a nuclear +structure which consists of a core and valence nucleons. The work of Ref. [45] extends the standard leading order +calculation to nonzero spin nuclei, however does not consider the inherent tensor contributions from the NN force in +their formulation. This leaves the importance of a consistent treatment of the NN force on elastic scattering from +nonzero spin nuclei still an open question. +The complete calculation of the leading-order effective interaction describing the scattering of a proton from a +nucleus in a 0+ ground state and which enters the integral Eq. (11) as driving term is given by +�Up(q, KNA, ϵ) = +(25) +� +α=n,p +� +d3Kη (q, K, KNA) Apα +� +q, 1 +2 +�A + 1 +A +KNA − K +� +; ϵ +� +ρKs=0 +α +� +P′, P +� ++ i(σ(0) · ˆn) +� +α=n,p +� +d3Kη (q, K, KNA) Cpα +� +q, 1 +2 +�A + 1 +A +KNA − K +� +; ϵ +� +ρKs=0 +α +� +P′, P +� ++ i +� +α=n,p +� +d3Kη (q, K, KNA) Cpα +� +q, 1 +2 +�A + 1 +A +KNA − K +� +; ϵ +� +Sn,α +� +P′, P +� +cos β ++ i(σ(0) · ˆn) +� +α=n,p +� +d3Kη (q, K, KNA) (−i)Mpα +� +q, 1 +2 +�A + 1 +A +KNA − K +� +; ϵ +� +Sn,α +� +P′, P +� +cos β. +The term η (q, K, KNA) is the Møller factor [65] describing the transformation from the NN frame to the NA frame. +The functions Apα, Cpα, and Mpα represent the NN interaction through Wolfenstein amplitudes [59]. +Since the +incoming proton can interact with either a proton or a neutron in the nucleus, the index α indicates the neutron (n) +and proton (p) contributions, which are calculated separately and then summed up. With respect to the nucleus, the +operator i(σ(0) · ˆn) represents the spin-orbit operator in momentum space with respect to the projectile. As such, +Eq. (25) exhibits the expected form of an interaction between a spin- 1 +2 projectile and a target nucleus in a J = 0 state +[66]. The momentum variables in the problem are given as +q = p′ − p = k′ − k, +(26) +K = 1 +2 (p′ + p) , +KNA = +A +A + 1 +� +(k′ + k) + 1 +2 (p′ + p) +� +, +P = K + A − 1 +A +q +2 , +P′ = K − A − 1 +A +q +2 . +The two quantities representing the structure of the nucleus are the scalar one-body density ρKs=0 +α +� +P′, P +� +and +the spin-projected momentum distribution Sn,α +� +P′, P +� += ρKs=1 � +P′, P +� +· ˆn. Both distributions are nonlocal and +translationally invariant. The reduced matrix elements entering the one-body densities are obtained within the NCSM +(SA-NCSM) in the center-of-mass frame of the nucleus. In order to employ them in calculating the leading-order +effective NA interaction, this center-of-mass variable must be removed. Within the framework of NCSM (SA-NCSM) +the technique for obtaining nonlocal and translationally invariant one-body densities is well developed [40, 44, 67–70]. +Lastly, the term cos β in Eq. (25) results from projecting ˆn from the NN frame to the NA frame. For further details, +see Ref. [41]. +IV. +CHIRAL TRUNCATION UNCERTAINTIES IN THE LEADING ORDER OPTICAL POTENTIAL +With the emergence of nuclear forces based on chiral effective field theory (EFT), we are presented with an opportu- +nity to study the nucleon-nucleus effective interaction as it develops order-by-order in a chiral EFT framework. Given +the hierarchical nature of chiral EFT, we can combine these order-by-order results to reliably estimate truncation +uncertainties associated with the higher chiral orders not included in the calculations. To this end, Refs. [35–37] first +implemented uncertainty quantification for the cases of NN and Nd scattering by assuming a quantity y(x) at a chiral +order k can be written as +yk(x) = yref(x) +k +� +n=0 +cn(x)Qn(x) +(27) + +7 +where yref(x) is a reference value that sets the scale of the problem and also includes the dimensions of the quantity +y(x) of interest. By construction, the coefficients cn(x) are dimensionless and are expected to be of order unity. The +remaining quantity Q(x) is the expansion parameter associated with the chiral EFT. The expansion parameter is +usually defined as +Q = 1 +Λb +max(Mπ, p) +(28) +where Λb is the breakdown scale of the EFT, Mπ is the pion mass, and p is the relevant momentum for the problem. +Various works [35–37] have identified the relevant momentum in different ways, but keeping with Ref. [43] we choose +the relevant momentum as the center-of-mass (c.m.) momentum in the nucleon-nucleus system +p2 +NA = ElabA2m2(Elab + 2m) +m2(A + 1)2 + 2AmElab +(29) +where Elab is the kinetic energy of the projectile in the laboratory frame, A is the target nucleus’s mass number, and +m is the mass of the nucleon. +Previous scattering works [36, 43] have noted that various results indicate, when identifying the relevant momentum, +the momentum transfer q should also be considered. That is, the expansion parameter would be more appropriately +defined as +Q = 1 +Λb +max(Mπ, pNA, q) +(30) +The momentum transfer in elastic scattering is defined as +q = 2pNA sin +�θc.m. +2 +� +(31) +where θc.m. is the scattering angle in the c.m. frame. Notably, including the momentum transfer in Eq. (30) makes +the expansion parameter a function of θc.m., even though the other momentum scales in Eq. (30) are independent +of the scattering angle. +When considering observables such as the differential cross section or analyzing power +that are functions of θc.m., this implies the expansion parameter will be larger at backward angles than at forward +angles. Furthermore, since the leading order of the spectator expansion is not applicable at low energies, we only +consider scattering at lab energies of 65 MeV or higher. As a result, the chiral expansion parameter becomes Q = +max(pNA, q)/Λb. This expansion parameter is shown in Fig. 2 for the case of A = 4 and Λb = 600 MeV. Because +of the factorization of the c.m. momentum, there is a universal scattering angle at which the momentum transfer q +begins to dominate the expansion parameter, regardless of the chosen Elab or nucleus. We will exploit this behavior +in later sections. +A. +Nuclear structure calculations +Prior to our detailed study of truncation uncertainties of a chiral NN interaction in elastic NA scattering observables +we need to choose a specific chiral NN interaction. Here we want to focus on the EKM chiral NN interaction [18, 19] +with a semi-local coordinate space regulator of R = 1 fm, which has a breakdown scale of Λb = 600 MeV. This +interaction gives a slightly better description of the ground state energies in the upper p-shell than a similar, more +recent interaction with a semi-local momentum space regulator. For consistency with the leading-order optical we +only use the NN potentials, omitting three-nucleon forces, which appear at N2LO in the chiral expansion, both in +the structure and the scattering part of the calculations. Including three-nucleon forces consistently in both, the +structure and scattering calculations requires going beyond the leading-order optical potential, and is beyond the +scope of this work. Though initial attempts of incorporating three-nucleon forces as an effective density-dependent +NN force in the scattering part have been presented [46], they can not yet be considered as systematic consideration of +three-nucleon forces in NA scattering. For similar reasons, we restrict most of our results to N2LO since three-nucleon +force contributions at N3LO and N4LO are significant [71]. +Next, the translationally-invariant one-body density needed for the scattering calculation can be obtained using +the NCSM approach, in which the nuclear wavefunction is expanded in Slater determinants of harmonic oscillator +basis functions [30]. Ideally, one uses a sufficiently large basis to ensure convergence of this expansion, but in practice +observables depend on both the many-body basis truncation, Nmax (defined as the total number of harmonic oscillator +quanta in the many-body system above the minimal configuration), and on the harmonic oscillator scale ¯hΩ. In Table I +we give the ground state binding energies and point-proton radii of 4He, 12C, and 16O obtained with the EKM chiral + +8 +NN potential [18, 19] with a semi-local coordinate space regulator of R = 1 fm (note that at N2LO we did not include +any three-nucleon forces). +For 4He we can obtain nearly converged results for both the binding energy and the proton radius, and these results +agree, to within their estimated numerical uncertainties (the first set of uncertainties in Table I), with Yakubovsky +calculations using the same NN potential [71]. However, for larger nuclei such as 12C and 16O we are more limited in +the Nmax values that can be reached on current computational resources. 1 +B. +Pointwise truncation uncertainties +To assess the relative size of chiral truncation uncertainties compared to other known uncertainties, e.g. the har- +monic oscillator parameters Nmax and ¯hΩ, we employ a pointwise truncation procedure and study reaction observables +that are not functional quantities, e.g. reaction cross sections at a specified laboratory energy. This pointwise approach +was previously implemented in Refs. [36, 43] and it starts by assuming the expansion parameter Q and reference scale +yref are known. From there, we can apply Eq. (27) to calculate the coefficients cn, which are treated as independent +draws from the same underlying distribution. The properties of this distribution can be learned from Bayesian tech- +niques and the posterior distribution for the prediction can be readily calculated with its associated credible intervals. +For more details, see Ref. [36]. +In order to estimate the chiral truncation uncertainties of the obtained ground state binding energies and radii, +we apply the pointwise approach with Q ≈ 0.3 as the effective expansion parameter, following Ref. [31]. +These +uncertainties are listed as the second set of uncertainties in Table I, starting from NLO. Here we see that for the +energies, the chiral uncertainties are at least of the same order as the estimated numerical uncertainties; however, the +uncertainties of the radii of 12C and 16O are clearly dominated by their systematic dependence on the basis parameter +¯hΩ. +To illustrate the pointwise approach for scattering observables, Fig. 3 shows the reaction cross sections for proton +scattering from 4He at 65 MeV and 16O at 100 MeV. For each case, the result is shown as a function of Nmax, and +variations with respect to ¯hΩ are indicated. While more obvious for the smaller nucleus where the NCSM can better +converge, in both cases the uncertainty resulting from the chiral truncation remains larger than the uncertainty arising +from the many-body method. To better illustrate this point, we present the reaction cross section for 4He with a +scale starting from 115 mb and with a range of only 45 mb, while using the full range of 600 mb for 16O. While larger +model spaces will better converge the NCSM results, smaller truncation uncertainties will only be achieved by higher +chiral orders, despite the noticeable dependence of the radii on the harmonic oscillator parameter ¯hΩ, in particular for +the heavier nuclei, in the current calculations. Note however that even at N3LO we anticipate the chiral truncation +uncertainties will be larger than the indicated variations with respect to the harmonic oscillator parameter ¯hΩ due to +the rather large value of the expansion parameter Q in the scattering calculation. +C. +Correlated truncation uncertainties +For functional quantities y(x) we employ a correlated approach that includes information at nearby values of x. +This approach is better for observables such as a differential cross section, which we know does not vary wildly from +values at nearby angles. It also starts from Eq. (27) and treats the coefficients cn(x) as independent draws from +an underlying Gaussian process. This Gaussian process encodes information about the correlation length ℓ, and the +qualities of the underlying distribution can be learned from the order-by-order results. This training is followed up by +testing procedures which seek to confirm the Gaussian process has been appropriately fit to the available results, and +if not, to diagnose potential issues. From a well-fit Gaussian process we can then extract truncation uncertainties for +the functional quantities. For more details and applications, see Refs. [36, 43]. +In the following examples, we examine proton scattering for 4He, 12C, and 16O at various projectile energies and +compare to the available experimental data. In each case, we show the convergence with respect to chiral order +1 One commonly applies a Similarity Renormalization Group (SRG) transformation to the NN potential in order to improve the conver- +gence of the many-body calculation. However, this leads to induced three-nucleon forces that are non-negligible; omitting those would +lead to a strong dependence on the SRG parameter. We therefore choose to not employ such a transformation here. For the binding +energies we use an exponential extrapolation to the complete basis, with associated uncertainties, see Ref. [71] for details. Radii converge +rather slowly in a harmonic oscillator basis, and they do not necessarily converge monotonically with increasing Nmax; furthermore, in +the scattering calculations we use densities obtained at fixed values of the harmonic oscillator parameters Nmax and ¯hΩ. We therefore +simply give in Table I our results for the point-proton radii of 12C and 16O at Nmax = 10, averaged over the range 16 ≤ ¯hΩ ≤ 28 MeV +(the same range as is used for the scattering calculations). The numerical uncertainty estimates for the radii listed in Table I correspond +to the spread over this ¯hΩ interval; this is a systematic uncertainty due to the Gaussian fall-off of harmonic oscillator basis functions, +and is therefore strongly correlated for the different chiral orders. However, the trend of a significant increase in the radii going from +LO to NLO, followed by a smaller increase going from NLO to N2LO, is robust, and correlates with the decrease in binding energies +going from LO to NLO to N2LO. Note that we did not include any chiral EFT corrections to the R2 operator; and the experimental +point-proton radii are extracted from the charge radius measured in electron scattering experiments, using standard proton and neutron +finite-size corrections, relativistic corrections, and meson-exchange corrections. + +9 +and the resulting decrease in the size of the chiral truncation uncertainties, as well as discuss any associated physics +insights. To avoid concerns about the expansion parameter increasing at larger angles, we mostly restrict our analysis +to forward angles where we expect the expansion parameter to be independent of the scattering angle. +For proton scattering on 4He, we see good agreement with experiment for the differential cross sections (Fig. 4) at +lower projectile energies. Below 100 MeV, most data points fall within the 2σ uncertainty band, and at 100 MeV a +majority of the data points are within the 1σ band. At the highest energy of 200 MeV, the chosen interaction seems +unable to reproduce the experimental data, though this is not uncommon for scattering from 4He. +The analyzing powers for proton scattering on 4He (Fig. 5) is more complicated. For the lower energies of 65 and +71 MeV, the experimental data shows a near zero value, regardless of scattering angle. In the scattering of a spin-1/2 +particle from a spin-0 nucleus, this indicates that there is no spin-orbit force at play. This behavior is only reproduced +by the LO result, for which the chiral NN interaction only contains the one-pion exchange and contact terms, which +do not produce a spin-orbit force. At NLO the two-pion exchange diagrams are responsible for reproducing the NN +p-waves and thus provide a spin-orbit force that leads to a non-zero value for the analyzing power in NA scattering. +At N2LO there are no new terms in the two-nucleon sector, and thus Ay does not change its shape at that chiral order. +Therefore, one needs to conclude that in this case other physics which goes beyond the leading order NA effective +interaction may be needed to describe the analyzing power. +For the higher energy of 200 MeV, all of the experimental data points are within the 2σ uncertainty band, though +there is a slight offset in the shape. In all cases, the analyzing power is more difficult to reproduce using this interaction, +though other interactions have done better [39, 41] +For proton scattering from 12C, the differential cross sections (Fig. 6) are reliably reproduced by the central value of +the N2LO calculations up to 100 MeV laboratory kinetic energy, and systematically over-predict at higher energies. As +the projectile energy increases, the expansion parameter increases and as a result uncertainty bands become larger. +This is most noticeable at 160 MeV: the experimental data is within the 1σ band, but the size of that band, as +well as the 2σ band, are so large that they are not practically useful. The gray bars in the cross section panels for +N2LO indicate the momentum transfer up to where we expect the expansion parameter to be dominated by the c.m. +momentum pNA. Once the momentum transfer exceeds the value given by the bar, the uncertainty is dominated by +the momentum transfer q, and is thus underrepresented by the method we use. Note that the vertical bar is at the +same scattering angle θc.m., but different momentum transfer q, as function of the projectile energy since pNA is a +function of the projectile energy as given in Eq. (29). Looking at the lower energies, the increasing agreement with +experiment in the first peak and minimum as higher orders in the chiral NN interaction are included gives the correct +trend. Minima in the differential cross section correlate with the size of the target nucleus. It is well well known [31], +and also evident from Table I, that the nuclear binding energy calculated with the LO of the chiral NN interaction is +way too large and correspondingly the radius much too small. Only when going to NLO and N2LO the binding energy +as well as the radius move into the vicinity of their experimental values. This finding from structure calculations is +corroborated by the calculations in Fig. 6, where with increasing chiral order the calculated first diffraction minimum +moves towards smaller momentum transfers indicating a larger nuclear size. +The analyzing powers for proton scattering on 12C are at 65 MeV also almost zero for small momentum transfers +and rise at q = 1.2 fm−1 to its maximum value of +1. +This is captured by the NLO calculation where spin- +contributions occur in the NN interaction (Fig. 7). For 65 MeV, the experimental data is mostly within the 2σ band +until approximately θc.m. = 60◦, where we expect the expansion parameter to being increasing and the uncertainty +bands to thus be underestimates. For 122 MeV, the very forward direction is inside the 1σ band, but the overall +shape of the experimental data is not well captured by this interaction. +For proton scattering from 16O, the differential cross sections (Fig. 8) are similar to the 12C case. Namely, the lower +energies do reasonably well at describing the data within the 2σ bands, but as the projectile energy increases the +uncertainty bands increase to unhelpful sizes. At the lowest energy of 65 MeV, we see a better and better reproduction +of the first minimum in the differential cross section as the chiral order increases. Again, this first minimum is known +to be related to the size of the nucleus, so this is an important feature to reproduce from both a structure, see Table I, +and reaction perspective. +The analyzing powers for proton scattering on 16O (Fig. 9) are again similar to the 12C case. At lower energies +(65 and 100 MeV), we again see a good reproduction to within 1σ or 2σ of the forward direction data, but beyond +θc.m. = 60◦, the experimental data is outside the uncertainty bands. At the higher energy of 135 MeV, many of the +experimental data are within the uncertainty bands but for a nucleus of this size, the expansion parameter has already +increased such that the resulting uncertainty bands are unhelpfully large. +As stated toward the beginning of the section we omit three-nucleon forces for consistency with the leading-order +optical potential which only treats two active nucleons. Those three-nucleon forces already appear at N2LO in the +chiral expansion, however, including them consistently in the structure as well as reaction calculation requires going +beyond the leading-order optical potential and is beyond the scope of this work. For the sake of investigating truncation +errors in the chiral NN force, one may carry out inconsistent calculation in the sense that the structure part of the + +10 +calculation is kept fixed at N2LO, and in the reaction part higher orders in the NN force are used. Proceeding in +this fashion is sensible, since the scattering calculation is more sensitive to the NN force compared to the structure +calculation, provided this structure calculation gives a reasonable description of the ground state one-body density. +To show how the chiral truncation error develops when higher chiral orders in the NN interaction are introduced, +we show in Fig. 10 proton scattering from 16O at 100 MeV projectile energy, where the higher chiral orders are only +employed in the scattering part through the corresponding Wolfenstein amplitudes. In both, the differential cross +section as well as the analyzing power the two most right panels depicting the inconsistent calculation show that +the uncertainty bands become smaller when higher chiral orders in the NN interaction are included. However, these +uncertainty bands are not necessarily realistic due to missing higher-body effects, which include higher orders in the +chiral force as well as higher orders in the multiple scattering expansion. Therefore, we can not draw firm conclusions +from the fact that data are outside the uncertainty estimates. Nevertheless, it is obvious that the decrease in the +uncertainties in the chiral truncation is rather slow due to the large expansion parameter. Furthermore, the medians +of the calculations shown in Figs. 8 and 9 do not change when higher chiral orders are considered in Fig. 10, which +further indicates that the smaller error bands of the higher order chiral truncations may be artificial. +D. +Analysis of Posteriors +Even while restricting our analysis to a region where we expect the expansion parameter to be constant, we can +still observe effects on the uncertainty bands if the expansion parameter is large, as noted in many of the results at +larger projectile energies. In fact, this behavior will place limits on the size of nucleus that can be considered with +this approach, since pNA as defined by Eq. (29) will continue to increase as A increases, yielding Q > 1 eventually. +While this situation is not ideal, we nonetheless find support for it in our analysis after examining the posteriors for +Q, in accordance with Ref. [36, 43]. +In Fig. 11, we calculated posteriors for the differential cross sections in proton scattering from 4He, 12C, and 16O +at the energies previously discussed. From these, we can extract a single best guess for the value of Q based on +the order-by-order calculations and compare that to the expectation for Q based on Eq. (30). For 16O, the largest +nucleus considered, we see generally good agreement between the expected value of Q and the best guess value from +the posteriors (Fig.11c). However, as the nucleus decreases in size and as the laboratory energy decreases, some +differences begin to emerge between the two values. In Fig. 11b for 12C, the comparisons are roughly similar to the +16O case, but for the 4He analysis (Fig. 11a), the differences are more pronounced, especially for the lower laboratory +energies. A similar analysis of neutron scattering on 12C did not show any significant differences between the two +values [43], which implies 4He may be the outlier in this approach. This analysis may imply scattering from 4He +with projectiles at lower energies could be analyzed with a smaller expansion parameter Q, though the higher energy +results still favor the larger expansion parameter. As the smallest nucleus considered here, it may also point to the +few-body character of 4He, which has not historically been well captured in an optical potential approach. +V. +OUTLOOK +Procedures that quantify the theoretical uncertainties associated with the underlying chiral EFT NN interaction +are by now well established for the NN and nucleon-deuteron systems as well as nuclear structure calculations, while +the systematic study of chiral truncation uncertainty is not as widely used in ab initio effective interaction employed +to describe the scattering of protons or neutrons from nuclei. Contributing factors for this relatively slow development +include that when considering a multiple scattering approach to deriving this effective NA interaction in an ab initio +fashion only recent progress in calculating the leading-order term in the multiple scattering approach has allowed to +treat the NN interaction on the same footing in the structure and reaction part [41] by considering the spin of the +struck target nucleon. Though calculations showed that the latter does not contribute significantly to observables when +considering scattering from nuclei with a 0+ ground state, one nevertheless needs a consistent ab initio implementation +of the leading-order term of the effective NA interaction in order to study the theoretical uncertainties imprinted on +NA observables by the chiral EFT NN interaction. +In this work we carry out a systematic study of chiral truncation uncertainties of the EKM chiral interaction on the +ab initio effective NA interaction calculated in leading order of the spectator expansion for 4He, 12C, and 16O. We find +that this interaction allows for a good description of experiment at energies around 100 MeV projectile kinetic energy +and slightly lower, provided we focus on regions of momentum transfer where the analysis of the EFT truncation +uncertainty is valid. When considering the lower energy of 65 MeV, the agreement with data starts to deteriorate. +This is an indication that errors other than the truncation error in the chiral interaction should come into play, +specifically errors that result from the spectator expansion itself. Theoretical consideration of the next-to-leading- + +11 +order term in the spectator expansion are described in some detail in this work in order to lay out necessary theoretical +and computational developments for this nontrivial endeavor. At at the next-to-leading order three-nucleon forces +will naturally enter the effective interaction. At present this step has only been attempted in approximative fashions, +namely by approximating the next-to-leading order in the propagator expansion via a nuclear mean field force [54] or +by introducing an effective, density dependent NN potential in the scattering part of the calculation [46]. Since we +are not considering next-to-leading order terms in the spectator expansion, we restrict our analysis to N2LO in the +chiral interaction and only consider two-nucleon forces. In this case the choice of the EKM interaction with a semi- +local coordinate space regulator of 1.0 fm is advantageous [38], since this specific interaction gives a slightly better +description of the ground state energies in the upper p-shell compared to other more recent chiral EFT interactions +when using two-nucleon interactions only. +In our study the chiral truncation errors at energies larger than 100 MeV increase considerably and the agreement +with experiment deteriorates. The increase in the chiral truncation error can simply be traced back to the expansion +parameter in our approach is getting too large. The deterioration of the agreement with experiment when going to +higher energies is more difficult to answer. One conclusion may be that the specific EKM chiral interaction employed +here in using the leading-order in the spectator expansion is not well suited to describe proton-nucleus scattering +observables for 4He, 12C, and 16O at higher energies. For the chiral NN interaction from Ref. [73] this is not the case +as shown in Refs. [40, 41]. Therefore one will have to investigate what features of a chiral NN interaction are most +relevant for a description of NA scattering observables for light nuclei. +To put this in perspective, let us reconsider the basic ideas of the spectator expansion. By design, the leading-order +term should be dominant at energies 150 MeV projectile kinetic energy and higher, since the reaction time of the +projectile with nucleons inside the nucleus is short, and thus an ‘impulse approximation’ is in general very good. +However, we do not want to consider here projectile energies larger than 400 MeV, where a relativistic treatment +e.g. via the Dirac equation may be preferred [74, 75]. Thus at energies around 200 MeV the leading order term by +design should give a reasonably good description of NA scattering data. This has been the case in the microscopic +calculations of the 1990s (see e.g. [9–14]) and a set of recent calculations with specific chiral NN interactions [40, 41, 46]. +Attempts to go beyond the leading order by incorporating 3NFs in a density dependent fashion into the many-body +propagator [46] indicate that effects at 200 MeV are only visible at higher momentum transfer. 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We therefore simply give in Table I our results for the point- +proton radii of 12C and 16O at Nmax = 10, averaged over the range 16 ≤ ¯hΩ ≤ 28 MeV (the same range as is used +for the scattering calculations). The numerical uncertainty estimates for the radii listed in Table I correspond to the +spread over this ¯hΩ interval; this is a systematic uncertainty due to the Gaussian fall-off of harmonic oscillator basis +functions, and is therefore strongly correlated for the different chiral orders. However, the trend of a significant increase +in the radii going from LO to NLO, followed by a smaller increase going from NLO to N2LO, is robust, and correlates +with the decrease in binding energies going from LO to NLO to N2LO. 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Stairs, Canadian Journal of Physics 48, 2629 (1970), https://doi.org/10.1139/p70-326. +[79] G. A. Moss et al., Phys. Rev. C 21, 1932 (1980). +[80] M. Ieiri, H. Sakaguchi, M. Nakamura, H. Sakamoto, H. Ogawa, M. Yosol, T. Ichihara, N. Isshiki, Y. Takeuchi, H. Togawa, +T. Tsutsumi, S. Hirata, T. Nakano, S. Kobayashi, T. Noro, and H. Ikegami, Nuclear Instruments and Methods in Physics +Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 257, 253 (1987). +[81] S. Kato, K. Okada, M. Kondo, K. Hosono, T. Saito, N. Matsuoka, K. Hatanaka, T. Noro, S. Nagamachi, H. Shimizu, +K. Ogino, Y. Kadota, S. Matsuki, and M. Wakai, Phys. Rev. C 31, 1616 (1985). +[82] K. Strauch and F. Titus, Phys. Rev. 103, 200 (1956). +[83] G. Gerstein, J. Niederer, and K. Strauch, Phys. Rev. 108, 427 (1957). +[84] H. O. Meyer, P. Schwandt, W. W. Jacobs, and J. R. Hall, Phys. Rev. C 27, 459 (1983). +[85] M. Ieiri, H. Sakaguchi, M. Nakamura, H. Sakamoto, H. Ogawa, M. Yosol, T. Ichihara, N. Isshiki, Y. Takeuchi, H. Togawa, +T. Tsutsumi, S. Hirata, T. Nakano, S. Kobayashi, T. Noro, and H. Ikegami, Nucl. Instrum. Methods Phys. Res. A 257, +253 (1987). +[86] H. Sakaguchi, M. Nakamura, K. Hatanaka, A. Goto, T. Noro, F. Ohtani, H. Sakamoto, and S. Kobayashi, Phys. Lett. B +89, 40 (1979). +[87] H. Seifert, Energy Dependence of the Effective Interaction for Nucleon-Nucleus Scattering, Ph.D. thesis, University of +Maryland (1990). +[88] J. J. Kelly, W. Bertozzi, T. N. Buti, J. M. Finn, F. W. Hersman, C. Hyde-Wright, M. V. Hynes, M. A. Kovash, B. Murdock, +B. E. Norum, B. Pugh, F. N. Rad, A. D. Bacher, G. T. Emery, C. C. Foster, W. P. Jones, D. W. Miller, B. L. Berman, +W. G. Love, J. A. Carr, and F. Petrovich, Phys. Rev. C 39, 1222 (1989). + +14 +[89] J. J. Kelly, J. M. Finn, W. Bertozzi, T. N. Buti, F. W. Hersman, C. Hyde-Wright, M. V. Hynes, M. A. Kovash, B. Murdock, +P. Ulmer, A. D. Bacher, G. T. Emery, C. C. Foster, W. P. Jones, D. W. Miller, and B. L. Berman, Phys. Rev. C 41, 2504 +(1990). +TABLES +4He +12C +16O +Binding energy (MeV) +LO +45.45(0.01) +137.(1.) +224.(2.) +NLO 28.53(0.01)(3.5) +97.(3.)(9.) +156.(5.)(14.) +N2LO 28.11(0.01)(0.9) +94.(4.)(3.) +149.(5.)(4.) +expt +28.30 +92.16 +127.62 +Point-proton radius (fm) +LO +1.08(0.02) +1.85(0.17) +1.8(0.2) +NLO +1.40(0.02)(0.08) +2.04(0.16)(0.09) +2.05(0.16)(0.10) +N2LO +1.42(0.02)(0.02) +2.12(0.15)(0.03) +2.11(0.15)(0.03) +expt +1.46 +2.32 +2.58 +TABLE I. Ground state binding energies (top) and point-proton RMS radii (bottom) of 4He, 12C, and 16O with LO, NLO, and +N2LO LENPIC SCS NN potentials. Both our estimated numerical uncertainties (first set of uncertainties) and chiral truncation +uncertainty estimates (second set of uncertainties, not evaluated for LO) are given. + +15 +FIG. 1. The angular distribution of the differential cross section divided by the Rutherford cross section (upper panel) and the +analyzing power (Ay) for elastic proton scattering from 8He at 71 MeV laboratory kinetic energy as function of the momentum +transfer q and the c. m. angle calculated with the LENPIC SCS chiral interaction [19] with a cutoff R = 1 fm. The calculations +are based on nonlocal densities using ¯hΩ = 14 MeV at Nmax = 14. The solid (red) line stands for using the Watson optical +potential while the black (dashed) line represents the KMT prescription. + +Oc.m. [deg] +20 +40 +60 +80 +100 +120 +103 +102 +101 +8He +100 +71 MeV +10-1 +1.0 +0.5 +0.0 +KMT +-0.5 +Watson +-1.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q [fm-1]16 +FIG. 2. The expansion parameter Q, defined by Eq. (30) where Λb = 600 MeV, as a function of the center-of-mass angle θc.m. +for a range of lab projectile energies Elab. In this case of nucleon-nucleus (NA) elastic scattering, the transition between when +the expansion parameter is dominated by the center-of-mass momentum and the momentum transfer can easily be identified. + +1.0 +0.8 +0.6 +O +0.4 +A = 4, Eiab = 65 MeV +0.2 +A = 4,Eiab = 71 MeV +A = 4,Elab = 100 MeV +A = 4, Eiab = 200 MeV +0.0 +20 +40 +60 +80 +100 +120 +140 +160 +0 +180 +Oc.m. +[deg]17 +FIG. 3. Reaction cross section for proton scattering on (a) 4He at 65 MeV and (b) 16O at 100 MeV, both at N2LO as a function +of Nmax. The error bars show a 68% credible interval (CI) from using a pointwise error estimation with the LO, NLO, and +N2LO results. The shaded regions show variations with respect to the harmonic oscillator parameter ¯hΩ. The values of the +expansion parameters used were Q = 0.47 for4He at 65 MeV and Q = 0.69 for 16O at 100 MeV. Note the different scales in (a) +and (b). + +160 +4He(p,p)4He, 65 MeV +(a) +155 +150 +145 +(mb) +140 +135 +6 +130 +125 +hΩ = 16 - 28 MeV +120 +68% CI +115 +8 +10 +12 +14 +16 +18 +20 +Nmax600 +160(p,p)160, 100 MeV +(b) +500 +400 +(mb) +300 +200 +100 +hΩ = 16 - 28 MeV +68% CI +0 +2 +4 +6 +8 +10 +Nmax18 +FIG. 4. Differential cross section divided by Rutherford for proton scattering on 4He at (first row) 65 MeV, (second row) 71 +MeV, (third row) 100 MeV, and (fourth row) 200 MeV for LO (left column), NLO (middle column), and N2LO (right column) +with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands. Black dots are experimental data from Refs. [76] +(65 MeV), [77] (71 MeV), [78] (100 MeV), and [79] (200 MeV). + +e.m. [deg] +Oc.m. [deg] +Oc.m. +[deg] +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +500 +N2LO +400 +LO +NLO +65 MeV +/Ruth +300 +200 +100 +0 +L +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +500 +N2LO +400 +LO +NLO +71 MeV +300 +200 +100 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +500 +N2LO +400 +Ruth +LO +NLO +100 MeV +300 +200 +I ++* +100 ++ +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +400 +N2LO +LO +NLO +200 MeV +200 +0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1]19 +FIG. 5. Analyzing power for proton scattering on 4He at (first row) 65 MeV, (second row) 71 MeV, and (third row) 200 MeV) +for LO (left column), NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter +bands) error bands. Black dots are experimental data from Refs. [76] (65 MeV), [77] (71 MeV), and [79] (200 MeV). + +Oc.m. +[deg] +Oc.m. +[deg] +Oc.m. +[deg] +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +1.0 +0.5 +0.0 +-0.5 +N2LO +上 +LO +NLO +65 MeV +-1.0 +L +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +1.0 +0.5 +A +0.0 +-0.5 +N2LO +LO +NLO +71 MeV +-1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +1.0 +产 +0.5 +A' +0.0 +-0.5 +N2LO +LO +NLO +200 MeV +I +1 +-1.0 +. +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1] +q [fm-1]20 +FIG. 6. +Differential cross section divided by Rutherford for proton scattering on 12C at (first row) 65 MeV, (second row) 100 +MeV, (third row) 122 MeV, and (fourth row) 160 MeV for LO (left column), NLO (middle column), and N2LO (right column) +with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands. Black dots/purple triangles are experimental data +from Refs. [80] (65 MeV, black dots), [81] (65 MeV, purple triangles), [82] (96 MeV, purple triangles), [83] (99 MeV, black +dots), and [84] (122 MeV and 160 MeV). Figure taken from Ref.[43]. + +Oe.m. [deg] +Gec.m. [deg] +Oe.m. [deg] +20 +40 +60 +80 +20 +40 +60 +80 +20 +40 +60 +80 +60 +N2LO +LO +NLO +65 MeV +40 +20 +0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +20 +40 +60 +20 +40 +60 +20 +40 +60 +60 +N2LO +LO +NLO +100 MeV +40 +20 +0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +60 +N2LO +LO +NLO +40 +122 MeV +20 +7 +*3 ++ +0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +60 +T +N2LO +LO +NLO +160 MeV +40 +20 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm- +11 +q [fm-1] +q [fm-21 +FIG. 7. Analyzing power for proton scattering on 12C at (first row) 65 MeV and (second row) 122 MeV for LO (left column), +NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands. +Black dots are experimental data from Refs. [85] (65 MeV) and [84] (122 MeV). Figure taken from Ref. [43]. + +Oc.m. [ +[deg] +Oc.m. [deg] +Oc.m. [deg] +20 +20 +40 +40 +60 +80 +60 +80 +20 +40 +60 +80 +1.0 +T ++ +T +0.5 ++ +0.0 +N2LO +LO +NLO +65 MeV +-0.5 +-1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +1.0 +* +I +* +0.5 +章 +0.0 +N2LO +LO +NLO +122 MeV +0.5 +-1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1] +q [fm-1]22 +FIG. 8. Differential cross section divided by Rutherford for proton scattering on 16O at (first row) 65 MeV, (second row) 100 +MeV, (third row) 135 MeV, and (fourth row) 180 MeV for LO (left column), NLO (middle column), and N2LO (right column) +with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands. Black dots are experimental data from Refs. [86] +(65 MeV), [87] (100 MeV), [88] (135 MeV), and [89] (180 MeV). Figure taken from Ref. [43]. + +Oc.m. [deg] +Oc.m. [deg] +Oc.m. [deg] +20 +40 +60 +80 +20 +40 +60 +80 +20 +40 +60 +80 +40 +LO +NLO +N2LO +65 MeV +20 +0 ++ ++ ++ +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +20 +40 +60 +20 +40 +60 +20 +40 +60 +60 +LO +N2LO +NLO +100 MeV +40 +20 ++ +* ++ ++ ++ +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +60 +LO +NLO +N2LO +/Ruth +135 MeV +40 ++ ++++ ++ ++++ +0 ++ ++ +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +60 +LO +NLO +N2LO +180 MeV +40 +20 ++ ++ ++ ++ ++ ++ +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1] +q [fm-1]23 +FIG. 9. Analyzing power for proton scattering on 16O at (first row) 65 MeV, (second row) 100 MeV, and (third row) 135 MeV +for LO (left column), NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter +bands) error bands. Black dots are experimental data from Refs. [86] (65 MeV), [87] (100 MeV), and [88] (135 MeV). Figure +taken from Ref. [43]. +FIG. 10. Differential cross section divided by the Rutherford cross section (top) and analyzing power (bottom) for proton +scattering from 16O at 100 MeV. The first 3 columns are the same as the second rows of Figs. 8 and 9. The additional two +rightmost panels are inconsistent calculations with use up to N2LO in the structure calculations and up to N3LO (fourth +column) or N4LO (fifth column) in the reaction calculation. Due to the inconsistency of the calculation the uncertainty bands +are not fully realistic. The data are the same as cited in Figs. 8 and 9. + +Oc.m. [deg] +Oc.m. [deg] +Oc.m. [deg] +20 +40 +60 +80 +20 +40 +60 +80 +20 +40 +60 +80 +1.0 +0.5 +0.0 +N2LO +LO +NLO +65 MeV +-0.5 +-1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +20 +60 +20 +40 +60 +20 +40 +60 +1.0 +0.5 +E3 +A +0.0 +N2LO +-0.5 +LO +NLO +100 MeV +-1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +1.0 ++ ++ ++ +0.5 +A +0.0 +N2LO +LO +NLO +-0.5 +135 MeV +-1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1] +q [fm-1]e.m. [deg] +Oe.m. [deg] +e.m. [deg] +Cc.m. [deg] +De.m. [deg] +40 +60 +20 +60 +20 +40 +60 +20 +40 +60 +20 +20 +40 +40 +60 +60 +LO +NLO +N2LO +N3LO +N4LO +（鄂）/ +20/ +++ +*+++ +++ +0E +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-=1] +q [fm-"] +g [fm-1] +q [fm=1]Gc.m. [deg] +Oe.m. [deg] +Se.m. [deg] +Ge.m. [deg] +Cc.m. [deg] +20 +40 +60 +20 +40 +60 +20 +40 +60 +20 +40 +60 +20 +60 +1.0 +0.5 +A +0.0 +0.5 +LO +NLO +N2LO +N3LO +N4LO +1.0 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +q [fm-1] +q [fm-1] +q [fm=1] +q [fm=1] +q [fm-1]24 +FIG. 11. Posterior plots for the expansion parameter Q given the differential cross sections for proton scattering on (a) 4He, +(b) 12C, and (c) 16O. + +Q = PNA/Ab +a +Best guess +65 MeV +0 +1 +71 MeV +0 +)10) +4He(p, p)4He +100 MeV +0 +1. +200 MeV +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q(b) +65 MeV +0 +1 +100 MeV +0 +)10) +12C(p, p)12C +122 MeV +0 +1. +Q = PNA/Ab +Best guess +160 MeV +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q(c) +65 MeV +0 +1 +100 MeV +0 +)10) +160(p, p)160 +135 MeV +0 +1 +Q = PNA/Ab +Best guess +180 MeV +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q \ No newline at end of file diff --git a/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf b/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..40fd2834e71f275631f702d5410692dc4add1e41 --- /dev/null +++ b/TtE4T4oBgHgl3EQfLwyC/content/2301.04941v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8b7586246ff9dba2c515afc674c3ce9f36c4b4a7329aa65b318a324374f7fd9b +size 145858 diff --git a/TtE4T4oBgHgl3EQfLwyC/vector_store/index.faiss b/TtE4T4oBgHgl3EQfLwyC/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..427d47376dbb1e1368d13eb2d04270ca5cbe8bd7 --- /dev/null +++ b/TtE4T4oBgHgl3EQfLwyC/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d3f311d17af2577b00caa2d6b562cf4d9e7a5a4b9f7799df7a609c9f94c4fca4 +size 1310765 diff --git a/TtE4T4oBgHgl3EQfLwyC/vector_store/index.pkl b/TtE4T4oBgHgl3EQfLwyC/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e7514c14786c66a5ad8f9a2e604de9eb63c580e7 --- /dev/null +++ b/TtE4T4oBgHgl3EQfLwyC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3e6b9236feb9155314902b047151d976731f558b887cb4b15e1bcadf97ea7abf +size 52589 diff --git a/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf b/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..56d7fc2b94880402dbb7e4cb40c9e28190384dc8 --- /dev/null +++ b/U9E2T4oBgHgl3EQftwgF/content/2301.04072v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2afbe5f67b298e960c154bbc9e9e64afbffb4bc74588b239af0188dd34cd4743 +size 2279743 diff --git a/U9E2T4oBgHgl3EQftwgF/vector_store/index.faiss b/U9E2T4oBgHgl3EQftwgF/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4d53d60c161c2006f93f82445740fa559e32b4e8 --- /dev/null +++ b/U9E2T4oBgHgl3EQftwgF/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9821b60254e473843ed06698fcaab2a84980eee08792151a287228e349e28b86 +size 2752557 diff --git a/U9E2T4oBgHgl3EQftwgF/vector_store/index.pkl b/U9E2T4oBgHgl3EQftwgF/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1381eeab6dc74d0b42dacc2e13ae3c6cf9b61a07 --- /dev/null +++ b/U9E2T4oBgHgl3EQftwgF/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:45720c34239fb2b71e2df619d3924e448fbb2e0c982cbcd7f4f3f2f0f5e1a714 +size 104676 diff --git a/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/2301.08312v1.pdf.txt b/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/2301.08312v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fddba5578b8baa97fe5e690185e8b6f5ce835b0 --- /dev/null +++ b/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/2301.08312v1.pdf.txt @@ -0,0 +1,867 @@ +arXiv:2301.08312v1 [math.NT] 19 Jan 2023 +Computing torsion for plane quartics +without using height bounds +Raymond van Bommel∗ +23 January 2023 +Abstract. We describe an algorithm that provably computes the rational torsion +subgroup of the Jacobian of a curve without relying on height bounds. Instead, it +relies on computing torsion points over small number fields. Both complex analytic +and Chinese remainder theorem based methods are used to find such torsion points. +The method has been implemented in Magma and used to provably compute the +rational torsion subgroup for more than 98% of Jacobians of curves in a dataset due +to Sutherland consisting of 82240 plane quartic curves. +1 +Introduction +In the 1920s, Mordell and Weil proved that for abelian varieties over a number field K +the group of rational points is finitely generated, [Mor22, Weil29]. In particular, +the rational torsion subgroup is finite. The torsion conjecture asserts that there +are only finitely many possible torsion groups, when the dimension of the abelian +variety and the degree [K : Q] are fixed. The conjecture has been proved in the +case of elliptic curves, first over Q by Mazur, and finally for all number fields by +Merel, [Maz78, Mer96]. The exact determination of which groups can occur, and +which of them occur infinitely often, is still an active area of research. In higher +dimensions, even in the case of abelian surfaces, no upper bound is known for the +rational torsion subgroup of an abelian variety over Q. +The torsion subgroup is also of importance for the Birch and Swinnerton-Dyer con- +jecture, [BSD65]. The action of the absolute Galois group of Q on the torsion plays +∗Raymond van Bommel has been supported by the Simons Collaboration on Arithmetic Geo- +metry, Number Theory, and Computation (Simons Foundation grant 550033). +Keywords: Abelian varieties, Galois representations, Birch-Swinnerton-Dyer conjecture, Jac- +obians, Curves +Mathematics Subject Classification (2020): 11G10, 11F80, 11G40, 11G30, 14H40 +1 + +an essential role in the definition of the Tate module and subsequently the L-function +of an abelian variety over a number field. The order of vanishing of this conjecturally +analytic function at 1, is asserted to equal the rank of the free part of the group +of rational points. Moreover, a formula links the leading Taylor coefficient of this +L-function to several arithmetic invariants, among which the order of the rational +group of torsion points. +In this present paper, we consider the question of explicitly computing the rational +torsion subgroup in case the abelian variety is the Jacobian of a curve defined over Q. +Recently, such a computation has been done by M¨uller and Reitsma for hyperelliptic +curves of genus 3, [M¨uRe22]. For small genus, the torsion is typically computed by +doing some form of an exhaustive search. The N´eron-Tate height �h, or canonical +height, of torsion points is known to be 0. Then one chooses a na¨ıve height h, finds a +bound |h − �h| < c, and subsequently uses the fact that the na¨ıve height of a torsion +point is at most c to find all of them, [Sto02, Sto17]. Such height bounds are known +and relatively small for genus 1, 2, and 3, but especially in genus 3 and higher the +enumeration of all rational points up to these height bounds can still be challenging. +Therefore, we would like to advocate an alternative approach. +In practice, it seems that torsion points actually have a way smaller na¨ıve height +than the bound given by the height bounds, and are not too hard to find. However, +the problem then still remains to prove that one has found the complete rational +torsion group. For this purpose, one could use the following lemma whose proof can +be found in [Katz81, Appendix]. +Lemma 1. Let K be a number field, let p be a prime of OK over the prime number +p, and let A be an abelian variety over K. Suppose that A has good reduction at p +and that Ap is its reduction. Moreover, suppose that p > e(p/p) + 1, where e(p/p) +is the ramification index of p over p. Then the natural reduction map +A(K)[n] −→ Ap(OK/p) +is injective for any integer n such that p ∤ n. +In many cases, it does not suffice to only use K = Q in this lemma. For example, +there could be nonrational 2-torsion points P2, P3, and P6 such that Pj is defined +over Q(√j) for j = 2, 3, 6. In this case, for any prime p of good reduction, at least +one of the three points will reduce to a point defined over Fp, causing the reduction +map to never be surjective on the Q-rational 2-torsion points. In this case we say +that the abelian variety has a fake torsion point. +The solution that we propose is to also search for torsion points defined over number +fields of small degree to account for the nonsurjectivity of the reduction map. In +Section 2, we discuss the background needed for this approach: the Weil pairing, +Weil polynomials, the Newton-Raphson method, and different ways to do Jacobian +arithmetic. In Section 3, we study the phenomenon of fake torsion points. The core +2 + +section of this paper in which we explain our actual methods to find torsion points +over number fields is Section 4. In the final section 5, we talk about the computation +of the torsion groups in a dataset consisting of 82240 plane quartic curves, [Suth19]. +The implementation of our method in Magma can be found at [code]. +Notation. Throughout this text, C denotes a smooth projective plane quartic curve +over Q, the symbol p denotes a prime number, J is the Jacobian of C, and Cp and +Jp are the reduction of C and J, respectively, modulo p, when p is a prime of good +reduction. +Acknowledgements. The author wishes to thank Edgar Costa, Bjorn Poonen, +David Roe, and Andrew Sutherland for useful discussions that helped to improve +the method, and for their help running the parallel computation on the servers of +the Simons Collaboration at Massachusetts Institute of Technology. +2 +Preliminaries +2.1 +Weil pairing +For abelian varieties A over a field K, there is a pairing +w: A(K)[n] × A∨(K)[n] → µn +� +K +� +, +called the Weil pairing, for any integer n prime to the characteristic of K. +In +particular, in the case of a Jacobian J of a curve, when the theta divisor on J +induces a principal polarisation J → J∨, we get a pairing +w: J(K)[n] × J(K)[n] → µn +� +K +� +. +This pairing, which we will also call the Weil pairing, is symplectic, i.e. alternating +and nondegenerate. +Now consider the case K is a number field and p is a prime of residue characteristic +p ∤ n satisfying the conditions of Lemma 1, then the Weil pairing is compatible with +the reduction map modulo p, i.e. the following diagram is commutative. +J(K)[n] × J(K)[n] +� +� +µn +� +K +� +� +Jp(Fp)[n] × Jp(Fp)[n] +� µn +� +Fp +� +3 + +Moreover, the absolute Galois group GK := Gal +� +K/K +� +has to respect the Weil +pairing, i.e. +σ(w(x, y)) = w(σ(x), σ(y)) +for all x, y ∈ J +� +K +� +[n] and all σ ∈ GK. In particular the action of GK must factor +through the general symplectic group GSp(J +� +K +� +[n], w). In the case n = ℓ is prime, +this group can be identified with the classical general symplectic group GSp(2g, Fℓ), +where g is the dimension of J. +2.2 +Weil polynomials +Let A be an abelian variety over Q, and let p be an odd prime of good reduction. +Then its reduction Ap is an abelian variety over Fp and for any prime ℓ ̸= p, we +can consider the Tate module Vℓ := Qℓ ⊗Zℓ limn Ap[ℓn](Fp). The Frobenius element +in Gal(Fp/Fp) acts on this Qℓ-vector space of dimension 2dim(A). Its characteristic +polynomial PAp, the Weil polynomial, has coefficients in Z, is independent of the +choice of ℓ, and has the property that #Ap = PAp(1). +When J = Jac(C) is the Jacobian of a curve and p is a prime of good reduction +of the curve, the polynomial PJp can be computed by computing the characteristic +polynomial of Frobenius on the ´etale cohomology group H1 +´et(C, Qℓ). It is feasible +to compute PJp mod p for p of size about 106 in several minutes using algorithms +described in [Cos15]. +2.3 +Newton-Raphson method and precision +For part of our computation, we will use complex valued numerical computations in +order to try to find algebraic torsion points in J(Q). For this reason, we will briefly +recall the numerical methods that we use, their stability, speed of convergence, and +the loss of precision that might occur. +The main numerical method that we use is the Newton-Raphson method. +The +method attempts to find a zero for a holomorphic function f : C → C by starting +with some initial guess x0 ∈ C for the root and iteratively computing the next +approximation xn+1 = xn − f(xn) +f′(xn). If x0 is close enough to some simple root r of f, +then the sequence (xn) will converge quadratically fast to r. +However, when r is a root of multiplicity at least 2 (or in practice also when f has +two simple roots very close to one another) problems may arise. As xn gets closer to +r, the denominator f ′(xn) will get close to 0, requiring us to use a lot more precision +to reliably compute the fraction f(xn) +f′(xn). In principle, the method would still converge, +but the following example explains why it is inevitable that we lose precision. +4 + +Example 2. Suppose we are looking for a root of x2 − 2x + 1 close to the starting +point x0 = 1.1, and we are doing computations with 1000 digits of precision. Then +we might end up finding the approximate root �r = 1−10−500. As r2−2r+1 = 10−1000, +this is a root within the precision of our computation, even though the actual root +r = 1 lies at distance 10−500. We lost half of our digits of precision. +If we now continue using �r instead of r and look for a root of x2 − 2x + �r, then we +could end up with the approximate root 1 + 10−250 instead of the intended solution +1. So the loss of digits in the process can accumulate. +The Newton-Raphson can also be used for multivariate functions Cn → Cn. One +has to replace f ′(xn) by the Jacobian matrix J of the function evaluated at xn. +Here, problems arise when J(r) has an eigenvalue of zero, then the computation of +the inverse J(xn)−1 might become numerically unstable. +2.4 +Jacobian arithmetic +For hyperelliptic curves, points on the Jacobian are typically represented using Mum- +ford coordinates, [Can87], which gives a unique way to represent each curve. For +general curves Khuri-Makdisi, [Khu04, Khu07, Khu18], developed ways to represent +points on the Jacobian and do arithmetic. In [FOR08] Flon, Oyono, and Ritzenthaler +describe a method specifically tailored to nonhyperelliptic genus 3 curves. +Many of these methods have been designed with the goal of implementing fast +arithmetic over finite fields, which has potential applications in cryptography. Often +these methods make assumptions on the curve that are easy to satisfy over finite +fields, but not over number fields. For the purpose of our computation, we used two +different methods to representing points on the Jacobian and do arithmetic. Even +though these methods are most likely not state of the art in terms of their efficiency, +we will describe them to inform the reader about the representations we used. +2.4.1 +Over exact fields +Our goal will be to reconstruct points on the Jacobian from their reductions modulo +different primes p. For this reason, we would like represent points on the Jacobian +in such a way that it has the following properties: +(α) the representation of each point is unique; +(β) for a prime p of good reduction, and any point P ∈ J(Q), there is a way +to reduce the representation of P modulo p, and for all but finitely many of +the primes p this reduced representation is the unique representation for the +reduction P ∈ Jp(Fp) of P modulo p. +5 + +If C does not have a Q-rational divisor of degree 1, one can probably find a way to +use the linear algebra methods as described in [Khu04]. However, because 99% of +the curves in [Suth19] that we are considering have a Q-rational point, we decided +to only implement the algorithm in this case. +So we suppose that C has a Q-rational point P. Then for any divisor D on C, there +is a nonpositive integer m such that h0(D − mP) = 1. If there are multiple such +m, we take the largest one. Then looking at the divisor of any nonzero function +f ∈ H0(D − mP), we find a way to represent D as mP + E, where E is an effective +divisor. This representation is unique by construction. +Lemma 3. This representation has property (β). +Proof. Let N be the product of the primes of bad reduction of C. Then C has +a smooth model C over Spec(Z[1/N]) and we can take the closures D and P in- +side C of D and P, respectively. We consider the sheafs F := OC(D − mP) and +G := OC(D − (m + 1)P). We have that F(CQ) and G(CQ) are Q-vector spaces of +dimensions 1 and 0, respectively. In particular, this implies that for all but finitely +many p the Fp-vector spaces F(CFp) and G(CFp) have dimensions 1 and 0, respect- +ively, see [Har77, Theorem III.12.8, p. 288] for example. For these primes p, the +representation of D mod p has the same m and uses E mod p. +To add two points m1P + E1 and m2P + E2, we use Magma’s built in Gr¨obner basis +function to compute the Riemann-Roch spaces H0(E1 + E2 − mP). This is most +likely not the most efficient way to do this, but as this part of the computation +was not a bottleneck for the whole computation, we did not put any effort into +optimising this. +2.4.2 +Over the complex numbers +For most divisor classes in J(C), the representation described in the previous sub- +section would be of the shape −3P +E with E effective of degree 3. Our (potential) +torsion points, being special points on the Jacobian, quite regularly have a repre- +sentation with m > −3. When we are doing numerical computations on J(C), this +often causes numerical instability for our algorithms. Luckily, there is an abundance +of points on J(C) and therefore we can use the following alternative presentation +for elements of J(C). +We represented them as Q1 + Q2 + Q3 − P1 − P2 − P3, where P1, P2, P3 ∈ C(C) are +three randomly generated points that are chosen in advance and Q1, Q2, Q3 ∈ C(C) +is a triple of points depending on the divisor class. Because the Pi are randomly +generated, we have a practical guarantee that the set {Q1, Q2, Q3} will be unique +for any divisor class that we encounter in our computation. +6 + +Proposition 4. Let D ∈ J(C) be any divisor class. Then for a general choice of +P1, P2, P3 ∈ C(C), the class D has a unique representation Q1+Q2+Q3−P1−P2−P3, +where two representations are called the same if the Qi are the same up to reordering. +Proof. If D has two such representations, this implies that h0(D+P1+P2+P3) > 1. +The dimension h0(D+P1+P2+P3) is upper semicontinuous as function in P1, P2, P3 +by [Har77, Theorem III.12.8, p. 288]. If D + P1 + P2 + P3 ∼ 3Q, where Q ∈ C(C) +is a non-Weierstraß point, then this dimension is equal to 1. In particular, for all +but a codimension 1 set of (P1, P2, P3) ∈ C(C)3, the dimension must equal 1 and +the representation must be unique. +The representation also has the following useful property. +Proposition 5. Let D ∈ J(C) be a nonzero divisor class. Then for a general choice +of P1, P2, P3 ∈ C(C), the unique representation Q1 + Q2 + Q3 − P1 − P2 − P3 for D +has the property that {Q1, Q2, Q3} ∩ {P1, P2, P3} = ∅. +Proof. It suffices to show that for a general choice of P1, P2 ∈ C(C), the class D +is not equivalent to Q1 + Q2 − P1 − P2 for any Q1, Q2 ∈ C(C). Equivalently, we +like to show that h0(D + P1 + P2) = 0 generically. Suppose that this is not the +case, then this dimension must be at least 1 for any choice of P1 and P2 by the +semicontinuity in [Har77, Theorem III.12.8, p. 288]. In particular, for any distinct +P1, P2, P3 ∈ C(C), we now have three ways of representing D: +Q1 + Q2 + P3 − +� +i +Pi, +R1 + P2 + R3 − +� +i +Pi, +and +P1 + S2 + S3 − +� +i +Pi, +for certain Q1, Q2, R1, R3, S2, S3 ∈ C(C). By the uniqueness of the representation, +we now must have that {Q1, Q2, P3} = {R1, P2, R3} = {P1, S2, S3} = {P1, P2, P3}. +In particular, D = 0, which is a contradiction. +To add two points, we use the following algorithm, which is a modified version of +the algorithm in [FOR08]. +Algorithm 6. Input: two triples of points Q1, Q2, Q3 and R1, R2, R3 representing +points Q = � +i Qi − � +i Pi and R = � +i Ri − � +i Pi on J(C). +Output: a triple of points S1, S2, S3 representing the point Q + R = � +i Si − � +i Pi. +Step 1. Pick (another) random point B ∈ C(C). +Step 2. Find the line ℓ through P1 and P2, and compute the residual intersection +A of this line with C, i.e. A is an effective divisor of degree 2 such that C +intersects ℓ in P1 + P2 + A. +7 + +Step 3. Find the cubic c through Q1, Q2, Q3, R1, R2, R3, A, and B, and compute the +residual intersection E of this cubic with C, i.e. E is an effective divisor of +degree 3 such that C intersects c in � +i Qi + � +i Ri + A + B + E. +Step 4. Find the conic n through B, P3, and E and compute the residual intersec- +tion S of this conic with C, i.e. S is an effective divisor of degree 3 such +that C intersects n in B + P3 + E + S. +Step 5. Output the three points S1, S2, and S3 of which S consists. +Proposition 7. The output of Algorithm 6 is correct. +Proof. Consider the rational function +c +ℓn. By construction, its associated principal +divisor is +� c +ℓn +� += +� +i +Qi + +� +i +Ri + A + B + E − P1 − P2 − A − B − P3 − E − S += +� +i +Qi + +� +i +Ri − +� +i +Pi − +� +i +Si. +In particular, we see that � +i Si−� +i Pi is equivalent to � +i Qi+� +i Ri−2 � +i Pi. +Remark 8. To find the intersection of a line/conic/cubic with f numerically, using +the root finding algorithms described in subsection 2.3, it is beneficial to not have +any points of intersection with multiplicity higher than 1. In general, we expect the +divisors P1 + P2 + A and � +i Qi + � +i Ri + A + B + E to not have any double points. +This causes the computation of A and E in Step 2 and Step 3 to be numerically +stable and fast without any difficulty. In Step 5, there could be one double point in +the divisor B + P3 + E + S. The divisor S could namely contain P3, but according +to Proposition 5, this only happens in the case P + Q = 0. In all other cases, there +is generally no double point and our algorithm to compute S will be numerically +stable and fast. +Another way that we will use to represent points in J(C) is by the means of an +element in a complex torus C3/Λ. The computation of a period lattice Λ and an +Abel-Jacobi map ι: J(C) → C3/Λ mapping Q1 + Q2 + Q3 − P1 − P2 − P3 to a cor- +responding point in the complex torus has been implemented in Magma by Neurohr, +see also [Neu18]. We will also write ι(Q1, Q2, Q3) for ι(Q1 +Q2 +Q3 −P1 −P2 −P3). +In order to go back from a point in C3/Λ to a divisor class, we use the following +algorithm to invert the Abel-Jacobi map. +Algorithm 9. Input: an element x ∈ C3/Λ. +Output: a triple of points Q1, Q2, Q3 ∈ C(C) such that ι(Q1, Q2, Q3) is close to x. +Step 1. Pick some integer n. We found that n = 14 worked good in practice for +our examples. +8 + +Step 2. Use Newton-Raphson (see subsection 2.3) with starting point (P1, P2, P3) +to numerically approximate a solution to ι(Q1,n, Q2,n, Q3,n) = +1 +2n · x. +Step 3. Add Q1,n+Q2,n+Q3,n−� +i Pi to itself using Algorithm 6. The output of this +addition is an approximate solution to ι(Q1,n−1, Q2,n−1, Q3,n−1) = +1 +2n−1 · x. +We then use Newton-Raphson to increase the precision of this solution +(Q1,n−1, Q2,n−1, Q3,n−1). Decrease n by 1 and repeat this step until n = 0. +Step 4. Use Newton-Raphson to refine (Q1,0, Q2,0, Q3,0) to the desired precision and +output the triple. +The reason for choosing an n and dividing by 2n first, is to make sure that the +starting point (P1, P2, P3) in Step 2 is close enough to the solution for the Newton- +Raphson method to actually converge. +3 +Fake torsion points +Let P ∈ J(Q) be a point and let ℓ be a prime number. We define +Dℓ(P) = {Q ∈ J(Q) | ℓ · Q = P} +and +Dℓ,p(P) = {Q ∈ Jp(Fp) | ℓ · Q = P}, +for every odd prime p ̸= ℓ of good reduction. This is a torsor under the action of +J[ℓ](Q) or Jp[ℓ](Fp), respectively. We already saw in the introduction that it could +happen that the set of Q-rational points in Dℓ(P) is smaller than any of the sets +of Fp-rational points in Dℓ,p(P). In case this happen, we say that P has a fake +ℓ-divisor. +In order to understand this phenomenon better, one considers the action of the +absolute Galois group Gal(Q/Q) on Dℓ(P). Because the action of Gal(Q/Q) has +to respect the symplectic form on J[ℓ], the action factors through a subgroup H of +the affine general symplectic group AGSp(2g, Fℓ). In the case P = 0, we actually +have that H is a subgroup of the smaller group GSp(2g, Fℓ). In the case P ̸= 0, +we actually have that H also lies in a smaller subgroup of AGSp(2g, Fℓ), as H must +commute with the multiplication by n if gcd(n, ℓ) = 1. +For each odd prime p ̸= ℓ of good reduction, there is a conjugacy class Frobp of +H which describes how Frobenius acts on Dℓ,p(P). Using these, we can exactly +determine for which H the point P has a fake ℓ-divisor. +Proposition 10. The point P has a fake ℓ-divisor if and only if for every element +h ∈ H we have +Dℓ(P) ⊃ Stab(h) ⊋ Stab(H) := {x ∈ Dℓ(P) | ∀h ∈ H : h(x) = x}. +9 + +Proof. The set Stab(H) is exactly the set of Q-rational points in Dℓ(P). For each +odd prime p ̸= ℓ of good reduction, the set of points in Dℓ(P) reducing to an Fp- +rational point in Dℓ,p(P) is exactly Stab(h) for some h ∈ Frobp. Because of the +Chebotarev density theorem, every conjugacy class will occur as Frobp for some odd +prime p ̸= ℓ, which concludes the proof of the proposition. +Looking at the group H, one cannot only determine whether there is a fake torsion +point, but also the degrees of the actual torsion points. By enumerating all the +appropriate subgroups of AGSp(2g, Fℓ), we get the following result that shows that +in certain cases the nonexistence of rational ℓ-divisors of P can be explained by +points of degree at most 12. +Proposition 11. Suppose either ℓ = 2, or both ℓ = 3 and P = 0. Then there exist +points Q1, . . . , Qk ∈ Dℓ(P) such that [Q(Qi) : Q] ⩽ 12 and a prime number p with +the following properties. If P ̸= 0, then Dℓ,p(P) = {Q1 mod p, . . . , Qk mod p}. If +P = 0, then Dℓ,p(P) = ⟨Q1 mod p, . . . , Qk mod p⟩. +Proof. This is a big group theoretic computation, enumerating all the appropriate +subgroups of GSp(6, Fℓ) or AGSp(6, Fℓ), and figuring out the degrees of the fake +torsion points needed. The code can be found at [code, extra/subgroups.m]. +4 +Methods +In this section, we explain the main result of this paper: two methods to find torsion +points over number fields. For the first method, we use the Chinese remainder the- +orem, taking torsion points modulo pi for different primes pi and trying to combine +them into one torsion point over a number field. For the second method, we use a +complex analytic approach, computing a complex approximation of torsion points +up to high enough precision to reconstruct them algebraically. One could also imag- +ine a third method, where one uses Hensel lifting to try to construct torsion points +using methods from [Mas20], but this approach has not been implemented as of +now. +4.1 +Algebraic reconstruction +Given a rational number α = r +s and its residue class modulo N for some suitable +N ≫ max(r2, s2), one could wonder if it is possible to construct α from this residue +class. This question has been answered positively in [Wang81, WGD92] with a fast +algorithm using the Euclidean algorithm. +10 + +In this section, we will consider an algebraic number α ∈ Q, its associated number +field K = Q(α) and prime ideals p1, . . . , pk, such that vpi(α) ≥ 0 for i = 1, . . . , k. +Then we can reduce α modulo each pi and we get finite field elements αi ∈ Fpi. The +question one can ask now is: can we reconstruct α from the αi? We will describe an +algorithm that attempts to do this. Even though it is still practical for our purpose, +the algorithm is definitely not as efficient as the rational reconstruction algorithm +mentioned before. +For each i, let pi be the residue class field characteristic of pi, and let fi ∈ Z[x] be +a lift of the minimum polynomial of αi over Fpi. Then we can consider the ideal +Ii = (fi, pi) of Z[x]. The minimum polynomial f of α is an element of Ii for each i +and hence of the intersection I := � +i Ii. The idea of our approach is to find a small +element in I. +Algorithm 12. Input: prime numbers pi and polynomials fi as described above. +Output: candidate minimum polynomial f for α. +Step 1. Compute a Gr¨obner basis G for the ideal I = � +i(fi, pi) ⊂ Z[x]. +Step 2. Set d := 1, the degree for the candidate polynomial f that we are currently +considering. +Step 3. For each g ∈ G, compute Bd +g := {xi · g | i ∈ Z≥0 such that deg(xi · g) ≤ d}. +Let Bd := � +g Bd +g and Λd ⊂ R{x0, . . . , xd} be the lattice generated by Bd. +Step 4. Find a short vector f ∈ Λd. Compute |f|, the maximum of the absolute +values of the coefficients of f. +Step 5. If (2|f|)d+1 is significantly smaller than lcm({pi}) and f ̸≡ 0 mod pi for +any i, then return f, otherwise set d := d + 1 and return to Step 3. +For Step 4 of the algorithm one could use any algorithm to find short vectors. In +our implementation we used the LLL algorithm by Lenstra, Lenstra, and Lov´asz, +see [LLL82]. In Step 5, we do a heuristic check to see if the polynomial f that +we are currently considering is small enough. For this purpose, we compare the +number of polynomials of the same degree with coefficients of equal or smaller size +with the product of the primes p over which we have information about f mod p. If +the latter is much greater than the former, this suggests that the polynomial that +we are currently considering might be the correct one. +Example 13. Suppose that k = 2, p1 = 1009, p2 = 1019, f1 = x − 55 and +f2 = x − 241. Then we find G = {x + 635615, 1028171}. For d = 1, the shortest +vector that we can find is x − 392556, which is too big to pass the test in Step 5. +For d = 2, we find the short vector x2 + 2, which we will output as f. +11 + +4.2 +Finding torsion points: the CRT method +In this section, we will describe how to find torsion points using the Chinese re- +mainder theorem. We assume that ℓ is prime and that we have some ℓ-power torsion +points Q ∈ J(Q). Our goal is to find points R ̸= 0 such that ℓR = Q. In this section, +all points will be represented using the representation described in subsection 2.4.1. +We give an outline of the method. +Algorithm 14. Input: a prime number ℓ, a subgroup of known torsion points +K ⊂ J[ℓ](Q), and a point Q ∈ J(Q) as described above. +Output: a (possibly empty) list of nonzero points R ∈ J(Q) such that ℓR = Q. +Step 1. Pick some medium size (≈ 106) auxiliary prime numbers p1, . . . , pk, such +that C has good reduction at these primes. +Step 2. For each pi, compute the Weil polynomial Ppi modulo pi of the reduction +Jpi as described in subsection 2.2. Using inequalities for the coefficients +of Ppi found in [Hal10], construct a finite set B containing all the possible +values of Ni := #Jpi(Fpi) = Ppi(1). +Step 3. Take a random point S ∈ Jpi(Fpi) and use a baby step giant step approach +to identify all b ∈ B such that b · S = 0. Discard all other elements of B. +Repeat this step until #B = 1, which must mean that B = {Ni}. +Step 4. For each pi, decompose Ni as ℓei · qi, where qi has no factors ℓ. Then +generate a bunch of random points S in Jpi(Fpi) and compute qi · S, +which is an element of Jpi[ℓ∞]. Keep finding new points, until there are +enough points to generate the ℓ-power torsion of Jpi(Fpi). +Step 5. For each pi, find the set Dpi of points Ri ∈ Jpi(Fpi) such that ℓRi = Q +mod pi, and compute the image Kpi of K inside Jpi(Fpi). Discard some +of the primes pi for which the set Dpi is relatively large. +Step 6. For each finite set I ⊂ {1, . . . , k} for which DI := � +i∈I Dpi/Kpi not too +large, enumerate all elements (Ri)i∈I of DI and execute the next three +steps for each such element. After finishing that, continue to Step 10. +Step 7. For each i ∈ I and V ∈ K compute a representation +Ri + V = mi,vP + +−mi,V +� +m=1 +Ri,V,m, +where +Ri,V,m ∈ Cp(Fp) +as in subsection 2.4.1. If the multisets {mi,V : V ∈ K} are not all equal for +the different i ∈ I, disregard this element of DI. Otherwise, compute the +polynomials Px,i = � +m,R(T − x(Ri,V,m)) and Py,i = � +m,R(T − y(Ri,V,m)) +inside Fpi[T]. +12 + +Step 8. Use algebraic reconstruction, as described in subsection 4.1, to try to lift +the matching coefficients of the Px,i and Py,i for the different i ∈ I to +elements of a number field. If the coefficients lift, and we get polynomials +Px, Py ∈ Q[T], apply the next step to them. +Step 9. For all possible combinations of the roots of Px and Py see which ones +give points on C(Q). Then try all combinations of m of these points to +see if we can find an R ∈ J(Q) such that ℓR = Q. Use the Jacobian +arithmetic described in subsection 2.4.1 to verify this. +Step 10. After finishing the loop described in Step 5, output all R with ℓR = Q +that we found in Step 9 during the computation. +Steps 1 through 4 are precomputation steps that only need to be done once for +each curve. In most cases, the CRT method was the faster method to find torsion +points over number fields. The biggest bottleneck of the method is the combinatorial +explosion that can take place in Steps 6 through 9; the sets DJ can become very big +in cases where there is a lot of fake torsion. +4.3 +Finding torsion points: the analytic method +The following analytic method to find torsion points has the advantage that there +will be no combinatorial explosion of trying to combine torsion points modulo differ- +ent primes into a torsion point over a number field. The downside is that we cannot +utilise the fact that Jpi(Fpi)[ℓn] is typically much smaller than J(C)[ℓn]. Recall that +we assumed the existence of a point P ∈ C(Q) and that we picked such a point at +the start. +Algorithm 15. Input: a prime number ℓ, a subgroup of known torsion points +K ⊂ J[ℓ](Q), and a point Q ∈ J(Q) as described above. +Output: a (possible empty) list of nonzero points R ∈ J(Q) such that ℓR = Q. +Step 1. Choose some P1, P2, P3 ∈ C(C) as in subsection 2.4.2. Now write Q as +Q1+Q2+Q3−P1−P2−P3. Compute an Abel-Jacobi map ι: J(C) → C/Λ +and compute the image ι(Q). +Step 2. Pick an element x in each class in +� 1 +ℓι(Q) + 1 +ℓΛ +� +/ι(K) and apply the fol- +lowing three steps for each element. +Step 3. Use Algorithm 9 to find a point R ∈ J(C) such that ι(R) is close to x. Use +a modified version of Algorithm 6 to write R as R1 + R2 + R3 − 3P for +some R1, R2, R3 ∈ C(C). +Step 4. Compute Mumford coordinates for R, i.e. compute the product polynomial +Px := � +i(T − x(Ri)) in C[T] and a polynomial Py of degree 2 such that +Py(x(Ri)) = y(Ri). +13 + +Step 5. Use a short lattice vector algorithm to try to find algebraic relations for the +coefficients of Px and Py. If this succeeds, reconstruct the corresponding +point in J(Q), which we also call R by abuse of notation. +Step 6. After finishing the loop described in Step 2, output all R with ℓR = Q that +we found in Step 5 during the computation. +In practice, to recognise torsion points over number fields, we need several hundreds +of digits of precision. This together with the sheer number of potential points we +need to try (typically ℓ6) makes the method slow in practice and only practical for +ℓ = 2 or ℓ = 3. +5 +Results +The algorithm has been implemented by the author in Magma and is publicly available +at [code]. It has been run on a dataset consisting of 82240 plane quartic curves found +by Andrew Sutherland, see [Suth19]. As a result, the rational torsion subgroup has +been computed successfully for 81357 of the Jacobians of these curves. The total +runtime for this computation was approximately 8 core months and has been done +in parallel on a machine of the Simons Collaboration at Massachusetts Institute of +Technology. For each computed torsion group a proof has been stored in the form +of a list of primes, and a list torsion points over Q and over some number fields +which together can be used to prove the completeness of the list of rational torsion +points using Lemma 1. These proofs can be verified significantly faster than it took +to construct them and are stored in the file [code, extra/proofs.tar.xz]. In Table +5, you can see the 64 different orders of the torsion groups that we found and how +often each of them occurred. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +58702 +8855 +5101 +2695 +1106 +1435 +616 +697 +452 +214 +51 +382 +42 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +130 +78 +129 +7 +127 +30 +51 +55 +17 +2 +86 +14 +14 +19 +31 +1 +24 +31 +32 +33 +35 +36 +38 +39 +40 +41 +42 +44 +45 +48 +49 +50 +51 +52 +3 +20 +12 +5 +33 +4 +7 +15 +1 +15 +2 +7 +14 +2 +2 +2 +2 +54 +56 +57 +60 +62 +64 +65 +66 +70 +72 +75 +80 +84 +96 +98 +105 +160 +2 +4 +4 +8 +1 +6 +1 +3 +3 +5 +2 +2 +1 +3 +1 +1 +1 +Table 1: Order statistics for torsion groups +For the majority of the 883 missing torsion groups, the reason that we could not +compute them was the failure to find a rational point on the plane quartic curve. For +14 + +some of these curves, we could verify the nonexistence of rational points by proving +that there are no points over some local field. For the remaining curves, which might +give rise to counterexamples for the Hasse principle, we did not attempt to verify +the nonexistence of rational points. +To conclude this section we exhibit an example where we managed to find a torsion +point over a degree 12 number field in order to certify the correctness of the computed +rational torsion subgroup. +Example 16. Consider the smooth plane quartic C : f = 0 with +f = x3y−xy3+y4+x3z+2x2yz+2xy2z−y3z+x2z2+2xyz2+y2z2−2xz3−yz3+z4. +Its Jacobian J modulo 11 has 1772 points, and J modulo 67 has 274944 points. +As the primes 11 and 67 are both primes of good reduction, this implies that the +torsion subgroup of J can have at most order gcd(1772, 274944) = 4. +Besides 0, we find a second rational torsion point +T2 = +� 1 +19(−10θ + 4) : −1 : 1 +� ++ +� 1 +19(−10¯θ + 4) : −1 : 1 +� +− 2 · (1 : 0 : 0), +where θ and ¯θ are zeros of x2 − 27 +10x − 1713 +100 . We easily find that there are no other +points of order 2. After looking at a lot of different primes and seeing that T2 has +a 2-divisor modulo each of these primes, we suspect that T2 might have a (fake) +2-divisor. +After about an hour of computation time, our program finds a torsion point T4 over +a degree 12 number field K defined by adjoining to Q a root of +x12 − 5x10 − 2x9 − 20x8 − 20x7 + 7x6 − 50x5 + 26x4 − 40x3 − 58x2 − 24x − 15. +This point satisfies 2T4 = T2. As the prime 67 splits into four primes of residue +degrees 67, 67, 672, and 678 in the ring of integers of K, the point T4 explains two of +the 2-divisors of T2 modulo 67. As there are only two 2-divisors of T2 in J mod 67, +we conclude that T2 doesn’t have a 2-divisor over Q, and {0, T2} is the full torsion +subgroup of J. +References +[BSD65] +B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II. J. Reine +Angew. Math. 218 (1965), 79–108. 1 +[code] +R. van Bommel, genus3torsion. Magma code, available at: https://git +hub.com/rbommel/genus3torsion 3, 10, 14 +[Can87] +David G. Cantor, Computing in the Jacobian of a hyperelliptic curve. +Math. Comp. 48 (1987), no. 177, 95–101. 5 +15 + +[Cos15] +Edgar Costa, Effective computations of Hasse–Weil zeta functions. Thesis +(Ph.D.)–New York University. 2015. ISBN: 978-1321-95392-3. ProQuest +LLC. 4 +[FOR08] +St´ephane Flon, Roger Oyono, Christophe Ritzenthaler, Fast addition on +non-hyperelliptic genus 3 curves. Algebraic geometry and its applications, +1–28, Ser. Number Theory Appl., 5, Word Sci. Publ., Hackensack, NJ, +2008. 5, 7 +[Hal10] +Safia Haloui, The characteristic polynomials of abelian varieties of dimen- +sions 3 over finite fields. J. Number Theory 130 (2010), no. 12, 2745–2752. +12 +[Har77] +Robin Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, +No. 52. Springer-Verlag, New York-Heidelberg, 1977. 6, 7 +[Khu04] +Kamal Khuri-Makdisi, Linear algebra algorithms for divisors on an al- +gebraic curve. Math. Comp. 73 (2004), no. 245, 333-357. 5, 6 +[Khu07] +Kamal Khuri-Makdisi, Asymptotically fast group operations on Jacobi- +ans of general curves. Math. Comp. 76 (2007), no. 260, 2213–2239. 5 +[Khu18] +Kamal Khuri-Makdisi, On Jacobian group arithmetic for typical divisors +on curves. Res. Number Theory 4 (2018), no. 1, Paper No. 3, 29 pp. 5 +[LLL82] +A. K. Lenstra, H. W. Lenstra Jr., L. Lov´asz, Factoring polynomials with +rational coefficients. Math. Ann. 261 (1982), no. 4, 515–534. 11 +[Katz81] +Nicholas M. Katz, Galois properties of torsion points on abelian varieties. +Invent. Math. 62 (1981), no. 3, 481–502. 2 +[Mas20] +Nicolas Mascot, Hensel-lifting torsion points on Jacobians and Galois +representations. Math. Comp. 89 (2020), no. 323, 1417–1455. 10 +[Maz78] +B. Mazur, Modular curves and the Eisenstein ideal. With an appendix +by Mazur and M. Rapoport. Inst. Hautes ´Etudes Sci. Publ. Math. No. 47 +(1977), 33–186 (1978). 1 +[Mer96] +Lo¨ıc Merel, Bornes pour la torsion des courbes elliptiques sur les corps +de nombres. Invent. Math. 124 (1996), no. 1–3, 437–449. 1 +[Mor22] +L. J. Mordell, On the rational solutions of the indeterminate equations of +the third and fourth degrees. Cambr. Phil. Soc. Proc. 21 (1922), 179–192. +1 +[M¨uRe22] J. Steffen M¨uller, Berno Reitsma, Computing torsion subgroups of Jac- +obians of hyperelliptic curves of genus 3. Preprint. arXiv:2211.03372. 2 +[Neu18] +Christian Neurohr, Efficient integration on Riemann surfaces & applica- +tions. PhD thesis (2018), https://oops.uni-oldenburg.de/3607/1/ne +ueff18.pdf. 8 +[Sto02] +Michael Stoll, On the height constant of genus two. II. Acta Arith. 104 +(2002), no. 2, 165–182. 2 +[Sto17] +Micael Stoll, An explicit theory of heights for hyperelliptic Jacobians of +genus three. Algorithmic and experimental methods in algebra, geometry, +and number theory, 665–715, Springer, Cham, 2017. 2 +[Suth19] +A. V. Sutherland, A database of nonhyperelliptic genus 3 curves over Q. +Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), Open +Book Series 2 (2019), 443–459. 3, 6, 14 +16 + +[Wang81] Paul S. Wang, A p-adic algorithm for univariate partial fractions. Sym- +bolic and algebraic computation, Proc. AMC Symp., Snowbird/Utah +1981, 212–217. 10 +[Weil29] +Andr´e Weil, L’arithm´etique sur les courbes alg´ebriques. Acta Math. 52, +no. 1, 281–315. 1 +[WGD92] Paul S. Wang, M. J. T. Guy, James H. Davenport. P-adic reconstruction +of rational numbers, SIGSAM Bull. 16 (1982), no. 2, 2–3. 10 +17 + diff --git a/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/load_file.txt b/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0bc05be33b736a23575ae6bf89d3c21100be02af --- /dev/null +++ b/U9E_T4oBgHgl3EQfxhwl/content/tmp_files/load_file.txt @@ -0,0 +1,682 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf,len=681 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='08312v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='NT] 19 Jan 2023 Computing torsion for plane quartics without using height bounds Raymond van Bommel∗ 23 January 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We describe an algorithm that provably computes the rational torsion subgroup of the Jacobian of a curve without relying on height bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Instead, it relies on computing torsion points over small number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Both complex analytic and Chinese remainder theorem based methods are used to find such torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The method has been implemented in Magma and used to provably compute the rational torsion subgroup for more than 98% of Jacobians of curves in a dataset due to Sutherland consisting of 82240 plane quartic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 Introduction In the 1920s, Mordell and Weil proved that for abelian varieties over a number field K the group of rational points is finitely generated, [Mor22, Weil29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, the rational torsion subgroup is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The torsion conjecture asserts that there are only finitely many possible torsion groups, when the dimension of the abelian variety and the degree [K : Q] are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The conjecture has been proved in the case of elliptic curves, first over Q by Mazur, and finally for all number fields by Merel, [Maz78, Mer96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The exact determination of which groups can occur, and which of them occur infinitely often, is still an active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In higher dimensions, even in the case of abelian surfaces, no upper bound is known for the rational torsion subgroup of an abelian variety over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The torsion subgroup is also of importance for the Birch and Swinnerton-Dyer con- jecture, [BSD65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The action of the absolute Galois group of Q on the torsion plays ∗Raymond van Bommel has been supported by the Simons Collaboration on Arithmetic Geo- metry, Number Theory, and Computation (Simons Foundation grant 550033).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Keywords: Abelian varieties, Galois representations, Birch-Swinnerton-Dyer conjecture, Jac- obians, Curves Mathematics Subject Classification (2020): 11G10, 11F80, 11G40, 11G30, 14H40 1 an essential role in the definition of the Tate module and subsequently the L-function of an abelian variety over a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The order of vanishing of this conjecturally analytic function at 1, is asserted to equal the rank of the free part of the group of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Moreover, a formula links the leading Taylor coefficient of this L-function to several arithmetic invariants, among which the order of the rational group of torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In this present paper, we consider the question of explicitly computing the rational torsion subgroup in case the abelian variety is the Jacobian of a curve defined over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Recently, such a computation has been done by M¨uller and Reitsma for hyperelliptic curves of genus 3, [M¨uRe22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For small genus, the torsion is typically computed by doing some form of an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The N´eron-Tate height �h, or canonical height, of torsion points is known to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then one chooses a na¨ıve height h, finds a bound |h − �h| < c, and subsequently uses the fact that the na¨ıve height of a torsion point is at most c to find all of them, [Sto02, Sto17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Such height bounds are known and relatively small for genus 1, 2, and 3, but especially in genus 3 and higher the enumeration of all rational points up to these height bounds can still be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Therefore, we would like to advocate an alternative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In practice, it seems that torsion points actually have a way smaller na¨ıve height than the bound given by the height bounds, and are not too hard to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' However, the problem then still remains to prove that one has found the complete rational torsion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For this purpose, one could use the following lemma whose proof can be found in [Katz81, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Let K be a number field, let p be a prime of OK over the prime number p, and let A be an abelian variety over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Suppose that A has good reduction at p and that Ap is its reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Moreover, suppose that p > e(p/p) + 1, where e(p/p) is the ramification index of p over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then the natural reduction map A(K)[n] −→ Ap(OK/p) is injective for any integer n such that p ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In many cases, it does not suffice to only use K = Q in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For example, there could be nonrational 2-torsion points P2, P3, and P6 such that Pj is defined over Q(√j) for j = 2, 3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In this case, for any prime p of good reduction, at least one of the three points will reduce to a point defined over Fp, causing the reduction map to never be surjective on the Q-rational 2-torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In this case we say that the abelian variety has a fake torsion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The solution that we propose is to also search for torsion points defined over number fields of small degree to account for the nonsurjectivity of the reduction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In Section 2, we discuss the background needed for this approach: the Weil pairing, Weil polynomials, the Newton-Raphson method, and different ways to do Jacobian arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In Section 3, we study the phenomenon of fake torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The core 2 section of this paper in which we explain our actual methods to find torsion points over number fields is Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In the final section 5, we talk about the computation of the torsion groups in a dataset consisting of 82240 plane quartic curves, [Suth19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The implementation of our method in Magma can be found at [code].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Throughout this text, C denotes a smooth projective plane quartic curve over Q, the symbol p denotes a prime number, J is the Jacobian of C, and Cp and Jp are the reduction of C and J, respectively, modulo p, when p is a prime of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The author wishes to thank Edgar Costa, Bjorn Poonen, David Roe, and Andrew Sutherland for useful discussions that helped to improve the method, and for their help running the parallel computation on the servers of the Simons Collaboration at Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 Weil pairing For abelian varieties A over a field K, there is a pairing w: A(K)[n] × A∨(K)[n] → µn � K � , called the Weil pairing, for any integer n prime to the characteristic of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, in the case of a Jacobian J of a curve, when the theta divisor on J induces a principal polarisation J → J∨, we get a pairing w: J(K)[n] × J(K)[n] → µn � K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This pairing, which we will also call the Weil pairing, is symplectic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' alternating and nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Now consider the case K is a number field and p is a prime of residue characteristic p ∤ n satisfying the conditions of Lemma 1, then the Weil pairing is compatible with the reduction map modulo p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' the following diagram is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J(K)[n] × J(K)[n] � � µn � K � � Jp(Fp)[n] × Jp(Fp)[n] � µn � Fp � 3 Moreover, the absolute Galois group GK := Gal � K/K � has to respect the Weil pairing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' σ(w(x, y)) = w(σ(x), σ(y)) for all x, y ∈ J � K � [n] and all σ ∈ GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular the action of GK must factor through the general symplectic group GSp(J � K � [n], w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In the case n = ℓ is prime, this group can be identified with the classical general symplectic group GSp(2g, Fℓ), where g is the dimension of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 Weil polynomials Let A be an abelian variety over Q, and let p be an odd prime of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then its reduction Ap is an abelian variety over Fp and for any prime ℓ ̸= p, we can consider the Tate module Vℓ := Qℓ ⊗Zℓ limn Ap[ℓn](Fp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The Frobenius element in Gal(Fp/Fp) acts on this Qℓ-vector space of dimension 2dim(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Its characteristic polynomial PAp, the Weil polynomial, has coefficients in Z, is independent of the choice of ℓ, and has the property that #Ap = PAp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' When J = Jac(C) is the Jacobian of a curve and p is a prime of good reduction of the curve, the polynomial PJp can be computed by computing the characteristic polynomial of Frobenius on the ´etale cohomology group H1 ´et(C, Qℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' It is feasible to compute PJp mod p for p of size about 106 in several minutes using algorithms described in [Cos15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='3 Newton-Raphson method and precision For part of our computation, we will use complex valued numerical computations in order to try to find algebraic torsion points in J(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For this reason, we will briefly recall the numerical methods that we use, their stability, speed of convergence, and the loss of precision that might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The main numerical method that we use is the Newton-Raphson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The method attempts to find a zero for a holomorphic function f : C → C by starting with some initial guess x0 ∈ C for the root and iteratively computing the next approximation xn+1 = xn − f(xn) f′(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If x0 is close enough to some simple root r of f, then the sequence (xn) will converge quadratically fast to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' However, when r is a root of multiplicity at least 2 (or in practice also when f has two simple roots very close to one another) problems may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As xn gets closer to r, the denominator f ′(xn) will get close to 0, requiring us to use a lot more precision to reliably compute the fraction f(xn) f′(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In principle, the method would still converge, but the following example explains why it is inevitable that we lose precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Suppose we are looking for a root of x2 − 2x + 1 close to the starting point x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1, and we are doing computations with 1000 digits of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then we might end up finding the approximate root �r = 1−10−500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As r2−2r+1 = 10−1000, this is a root within the precision of our computation, even though the actual root r = 1 lies at distance 10−500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We lost half of our digits of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If we now continue using �r instead of r and look for a root of x2 − 2x + �r, then we could end up with the approximate root 1 + 10−250 instead of the intended solution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' So the loss of digits in the process can accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The Newton-Raphson can also be used for multivariate functions Cn → Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' One has to replace f ′(xn) by the Jacobian matrix J of the function evaluated at xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Here, problems arise when J(r) has an eigenvalue of zero, then the computation of the inverse J(xn)−1 might become numerically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4 Jacobian arithmetic For hyperelliptic curves, points on the Jacobian are typically represented using Mum- ford coordinates, [Can87], which gives a unique way to represent each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For general curves Khuri-Makdisi, [Khu04, Khu07, Khu18], developed ways to represent points on the Jacobian and do arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In [FOR08] Flon, Oyono, and Ritzenthaler describe a method specifically tailored to nonhyperelliptic genus 3 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Many of these methods have been designed with the goal of implementing fast arithmetic over finite fields, which has potential applications in cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Often these methods make assumptions on the curve that are easy to satisfy over finite fields, but not over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For the purpose of our computation, we used two different methods to representing points on the Jacobian and do arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Even though these methods are most likely not state of the art in terms of their efficiency, we will describe them to inform the reader about the representations we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 Over exact fields Our goal will be to reconstruct points on the Jacobian from their reductions modulo different primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For this reason, we would like represent points on the Jacobian in such a way that it has the following properties: (α) the representation of each point is unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' (β) for a prime p of good reduction, and any point P ∈ J(Q), there is a way to reduce the representation of P modulo p, and for all but finitely many of the primes p this reduced representation is the unique representation for the reduction P ∈ Jp(Fp) of P modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5 If C does not have a Q-rational divisor of degree 1, one can probably find a way to use the linear algebra methods as described in [Khu04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' However, because 99% of the curves in [Suth19] that we are considering have a Q-rational point, we decided to only implement the algorithm in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' So we suppose that C has a Q-rational point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then for any divisor D on C, there is a nonpositive integer m such that h0(D − mP) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If there are multiple such m, we take the largest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then looking at the divisor of any nonzero function f ∈ H0(D − mP), we find a way to represent D as mP + E, where E is an effective divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This representation is unique by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This representation has property (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Let N be the product of the primes of bad reduction of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then C has a smooth model C over Spec(Z[1/N]) and we can take the closures D and P in- side C of D and P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We consider the sheafs F := OC(D − mP) and G := OC(D − (m + 1)P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We have that F(CQ) and G(CQ) are Q-vector spaces of dimensions 1 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, this implies that for all but finitely many p the Fp-vector spaces F(CFp) and G(CFp) have dimensions 1 and 0, respect- ively, see [Har77, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 288] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For these primes p, the representation of D mod p has the same m and uses E mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' To add two points m1P + E1 and m2P + E2, we use Magma’s built in Gr¨obner basis function to compute the Riemann-Roch spaces H0(E1 + E2 − mP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This is most likely not the most efficient way to do this, but as this part of the computation was not a bottleneck for the whole computation, we did not put any effort into optimising this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 Over the complex numbers For most divisor classes in J(C), the representation described in the previous sub- section would be of the shape −3P +E with E effective of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Our (potential) torsion points, being special points on the Jacobian, quite regularly have a repre- sentation with m > −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' When we are doing numerical computations on J(C), this often causes numerical instability for our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Luckily, there is an abundance of points on J(C) and therefore we can use the following alternative presentation for elements of J(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We represented them as Q1 + Q2 + Q3 − P1 − P2 − P3, where P1, P2, P3 ∈ C(C) are three randomly generated points that are chosen in advance and Q1, Q2, Q3 ∈ C(C) is a triple of points depending on the divisor class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Because the Pi are randomly generated, we have a practical guarantee that the set {Q1, Q2, Q3} will be unique for any divisor class that we encounter in our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 6 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Let D ∈ J(C) be any divisor class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then for a general choice of P1, P2, P3 ∈ C(C), the class D has a unique representation Q1+Q2+Q3−P1−P2−P3, where two representations are called the same if the Qi are the same up to reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If D has two such representations, this implies that h0(D+P1+P2+P3) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The dimension h0(D+P1+P2+P3) is upper semicontinuous as function in P1, P2, P3 by [Har77, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If D + P1 + P2 + P3 ∼ 3Q, where Q ∈ C(C) is a non-Weierstraß point, then this dimension is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, for all but a codimension 1 set of (P1, P2, P3) ∈ C(C)3, the dimension must equal 1 and the representation must be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The representation also has the following useful property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Let D ∈ J(C) be a nonzero divisor class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then for a general choice of P1, P2, P3 ∈ C(C), the unique representation Q1 + Q2 + Q3 − P1 − P2 − P3 for D has the property that {Q1, Q2, Q3} ∩ {P1, P2, P3} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' It suffices to show that for a general choice of P1, P2 ∈ C(C), the class D is not equivalent to Q1 + Q2 − P1 − P2 for any Q1, Q2 ∈ C(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Equivalently, we like to show that h0(D + P1 + P2) = 0 generically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Suppose that this is not the case, then this dimension must be at least 1 for any choice of P1 and P2 by the semicontinuity in [Har77, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, for any distinct P1, P2, P3 ∈ C(C), we now have three ways of representing D: Q1 + Q2 + P3 − � i Pi, R1 + P2 + R3 − � i Pi, and P1 + S2 + S3 − � i Pi, for certain Q1, Q2, R1, R3, S2, S3 ∈ C(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' By the uniqueness of the representation, we now must have that {Q1, Q2, P3} = {R1, P2, R3} = {P1, S2, S3} = {P1, P2, P3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, D = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' To add two points, we use the following algorithm, which is a modified version of the algorithm in [FOR08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Input: two triples of points Q1, Q2, Q3 and R1, R2, R3 representing points Q = � i Qi − � i Pi and R = � i Ri − � i Pi on J(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output: a triple of points S1, S2, S3 representing the point Q + R = � i Si − � i Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Pick (another) random point B ∈ C(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Find the line ℓ through P1 and P2, and compute the residual intersection A of this line with C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' A is an effective divisor of degree 2 such that C intersects ℓ in P1 + P2 + A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 7 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Find the cubic c through Q1, Q2, Q3, R1, R2, R3, A, and B, and compute the residual intersection E of this cubic with C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' E is an effective divisor of degree 3 such that C intersects c in � i Qi + � i Ri + A + B + E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Find the conic n through B, P3, and E and compute the residual intersec- tion S of this conic with C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' S is an effective divisor of degree 3 such that C intersects n in B + P3 + E + S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output the three points S1, S2, and S3 of which S consists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The output of Algorithm 6 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Consider the rational function c ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' By construction, its associated principal divisor is � c ℓn � = � i Qi + � i Ri + A + B + E − P1 − P2 − A − B − P3 − E − S = � i Qi + � i Ri − � i Pi − � i Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In particular, we see that � i Si−� i Pi is equivalent to � i Qi+� i Ri−2 � i Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' To find the intersection of a line/conic/cubic with f numerically, using the root finding algorithms described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='3, it is beneficial to not have any points of intersection with multiplicity higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In general, we expect the divisors P1 + P2 + A and � i Qi + � i Ri + A + B + E to not have any double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This causes the computation of A and E in Step 2 and Step 3 to be numerically stable and fast without any difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In Step 5, there could be one double point in the divisor B + P3 + E + S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The divisor S could namely contain P3, but according to Proposition 5, this only happens in the case P + Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In all other cases, there is generally no double point and our algorithm to compute S will be numerically stable and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Another way that we will use to represent points in J(C) is by the means of an element in a complex torus C3/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The computation of a period lattice Λ and an Abel-Jacobi map ι: J(C) → C3/Λ mapping Q1 + Q2 + Q3 − P1 − P2 − P3 to a cor- responding point in the complex torus has been implemented in Magma by Neurohr, see also [Neu18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We will also write ι(Q1, Q2, Q3) for ι(Q1 +Q2 +Q3 −P1 −P2 −P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In order to go back from a point in C3/Λ to a divisor class, we use the following algorithm to invert the Abel-Jacobi map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Input: an element x ∈ C3/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output: a triple of points Q1, Q2, Q3 ∈ C(C) such that ι(Q1, Q2, Q3) is close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Pick some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We found that n = 14 worked good in practice for our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 8 Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use Newton-Raphson (see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='3) with starting point (P1, P2, P3) to numerically approximate a solution to ι(Q1,n, Q2,n, Q3,n) = 1 2n · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Add Q1,n+Q2,n+Q3,n−� i Pi to itself using Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The output of this addition is an approximate solution to ι(Q1,n−1, Q2,n−1, Q3,n−1) = 1 2n−1 · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We then use Newton-Raphson to increase the precision of this solution (Q1,n−1, Q2,n−1, Q3,n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Decrease n by 1 and repeat this step until n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use Newton-Raphson to refine (Q1,0, Q2,0, Q3,0) to the desired precision and output the triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The reason for choosing an n and dividing by 2n first, is to make sure that the starting point (P1, P2, P3) in Step 2 is close enough to the solution for the Newton- Raphson method to actually converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 3 Fake torsion points Let P ∈ J(Q) be a point and let ℓ be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We define Dℓ(P) = {Q ∈ J(Q) | ℓ · Q = P} and Dℓ,p(P) = {Q ∈ Jp(Fp) | ℓ · Q = P}, for every odd prime p ̸= ℓ of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This is a torsor under the action of J[ℓ](Q) or Jp[ℓ](Fp), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We already saw in the introduction that it could happen that the set of Q-rational points in Dℓ(P) is smaller than any of the sets of Fp-rational points in Dℓ,p(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In case this happen, we say that P has a fake ℓ-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In order to understand this phenomenon better, one considers the action of the absolute Galois group Gal(Q/Q) on Dℓ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Because the action of Gal(Q/Q) has to respect the symplectic form on J[ℓ], the action factors through a subgroup H of the affine general symplectic group AGSp(2g, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In the case P = 0, we actually have that H is a subgroup of the smaller group GSp(2g, Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In the case P ̸= 0, we actually have that H also lies in a smaller subgroup of AGSp(2g, Fℓ), as H must commute with the multiplication by n if gcd(n, ℓ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each odd prime p ̸= ℓ of good reduction, there is a conjugacy class Frobp of H which describes how Frobenius acts on Dℓ,p(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Using these, we can exactly determine for which H the point P has a fake ℓ-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The point P has a fake ℓ-divisor if and only if for every element h ∈ H we have Dℓ(P) ⊃ Stab(h) ⊋ Stab(H) := {x ∈ Dℓ(P) | ∀h ∈ H : h(x) = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The set Stab(H) is exactly the set of Q-rational points in Dℓ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each odd prime p ̸= ℓ of good reduction, the set of points in Dℓ(P) reducing to an Fp- rational point in Dℓ,p(P) is exactly Stab(h) for some h ∈ Frobp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Because of the Chebotarev density theorem, every conjugacy class will occur as Frobp for some odd prime p ̸= ℓ, which concludes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Looking at the group H, one cannot only determine whether there is a fake torsion point, but also the degrees of the actual torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' By enumerating all the appropriate subgroups of AGSp(2g, Fℓ), we get the following result that shows that in certain cases the nonexistence of rational ℓ-divisors of P can be explained by points of degree at most 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Suppose either ℓ = 2, or both ℓ = 3 and P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then there exist points Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , Qk ∈ Dℓ(P) such that [Q(Qi) : Q] ⩽ 12 and a prime number p with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If P ̸= 0, then Dℓ,p(P) = {Q1 mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , Qk mod p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If P = 0, then Dℓ,p(P) = ⟨Q1 mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , Qk mod p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This is a big group theoretic computation, enumerating all the appropriate subgroups of GSp(6, Fℓ) or AGSp(6, Fℓ), and figuring out the degrees of the fake torsion points needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The code can be found at [code, extra/subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4 Methods In this section, we explain the main result of this paper: two methods to find torsion points over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For the first method, we use the Chinese remainder the- orem, taking torsion points modulo pi for different primes pi and trying to combine them into one torsion point over a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For the second method, we use a complex analytic approach, computing a complex approximation of torsion points up to high enough precision to reconstruct them algebraically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' One could also imag- ine a third method, where one uses Hensel lifting to try to construct torsion points using methods from [Mas20], but this approach has not been implemented as of now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 Algebraic reconstruction Given a rational number α = r s and its residue class modulo N for some suitable N ≫ max(r2, s2), one could wonder if it is possible to construct α from this residue class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This question has been answered positively in [Wang81, WGD92] with a fast algorithm using the Euclidean algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 10 In this section, we will consider an algebraic number α ∈ Q, its associated number field K = Q(α) and prime ideals p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , pk, such that vpi(α) ≥ 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then we can reduce α modulo each pi and we get finite field elements αi ∈ Fpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The question one can ask now is: can we reconstruct α from the αi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We will describe an algorithm that attempts to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Even though it is still practical for our purpose, the algorithm is definitely not as efficient as the rational reconstruction algorithm mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each i, let pi be the residue class field characteristic of pi, and let fi ∈ Z[x] be a lift of the minimum polynomial of αi over Fpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then we can consider the ideal Ii = (fi, pi) of Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The minimum polynomial f of α is an element of Ii for each i and hence of the intersection I := � i Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The idea of our approach is to find a small element in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algorithm 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Input: prime numbers pi and polynomials fi as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output: candidate minimum polynomial f for α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Compute a Gr¨obner basis G for the ideal I = � i(fi, pi) ⊂ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Set d := 1, the degree for the candidate polynomial f that we are currently considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each g ∈ G, compute Bd g := {xi · g | i ∈ Z≥0 such that deg(xi · g) ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Let Bd := � g Bd g and Λd ⊂ R{x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , xd} be the lattice generated by Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Find a short vector f ∈ Λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Compute |f|, the maximum of the absolute values of the coefficients of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If (2|f|)d+1 is significantly smaller than lcm({pi}) and f ̸≡ 0 mod pi for any i, then return f, otherwise set d := d + 1 and return to Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For Step 4 of the algorithm one could use any algorithm to find short vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In our implementation we used the LLL algorithm by Lenstra, Lenstra, and Lov´asz, see [LLL82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In Step 5, we do a heuristic check to see if the polynomial f that we are currently considering is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For this purpose, we compare the number of polynomials of the same degree with coefficients of equal or smaller size with the product of the primes p over which we have information about f mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If the latter is much greater than the former, this suggests that the polynomial that we are currently considering might be the correct one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Suppose that k = 2, p1 = 1009, p2 = 1019, f1 = x − 55 and f2 = x − 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then we find G = {x + 635615, 1028171}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For d = 1, the shortest vector that we can find is x − 392556, which is too big to pass the test in Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For d = 2, we find the short vector x2 + 2, which we will output as f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 Finding torsion points: the CRT method In this section, we will describe how to find torsion points using the Chinese re- mainder theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We assume that ℓ is prime and that we have some ℓ-power torsion points Q ∈ J(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Our goal is to find points R ̸= 0 such that ℓR = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In this section, all points will be represented using the representation described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We give an outline of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algorithm 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Input: a prime number ℓ, a subgroup of known torsion points K ⊂ J[ℓ](Q), and a point Q ∈ J(Q) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output: a (possibly empty) list of nonzero points R ∈ J(Q) such that ℓR = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Pick some medium size (≈ 106) auxiliary prime numbers p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , pk, such that C has good reduction at these primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each pi, compute the Weil polynomial Ppi modulo pi of the reduction Jpi as described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Using inequalities for the coefficients of Ppi found in [Hal10], construct a finite set B containing all the possible values of Ni := #Jpi(Fpi) = Ppi(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Take a random point S ∈ Jpi(Fpi) and use a baby step giant step approach to identify all b ∈ B such that b · S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Discard all other elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Repeat this step until #B = 1, which must mean that B = {Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each pi, decompose Ni as ℓei · qi, where qi has no factors ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then generate a bunch of random points S in Jpi(Fpi) and compute qi · S, which is an element of Jpi[ℓ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Keep finding new points, until there are enough points to generate the ℓ-power torsion of Jpi(Fpi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each pi, find the set Dpi of points Ri ∈ Jpi(Fpi) such that ℓRi = Q mod pi, and compute the image Kpi of K inside Jpi(Fpi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Discard some of the primes pi for which the set Dpi is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each finite set I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' , k} for which DI := � i∈I Dpi/Kpi not too large, enumerate all elements (Ri)i∈I of DI and execute the next three steps for each such element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' After finishing that, continue to Step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each i ∈ I and V ∈ K compute a representation Ri + V = mi,vP + −mi,V � m=1 Ri,V,m, where Ri,V,m ∈ Cp(Fp) as in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If the multisets {mi,V : V ∈ K} are not all equal for the different i ∈ I, disregard this element of DI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Otherwise, compute the polynomials Px,i = � m,R(T − x(Ri,V,m)) and Py,i = � m,R(T − y(Ri,V,m)) inside Fpi[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 12 Step 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use algebraic reconstruction, as described in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1, to try to lift the matching coefficients of the Px,i and Py,i for the different i ∈ I to elements of a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If the coefficients lift, and we get polynomials Px, Py ∈ Q[T], apply the next step to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For all possible combinations of the roots of Px and Py see which ones give points on C(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Then try all combinations of m of these points to see if we can find an R ∈ J(Q) such that ℓR = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use the Jacobian arithmetic described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 to verify this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' After finishing the loop described in Step 5, output all R with ℓR = Q that we found in Step 9 during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Steps 1 through 4 are precomputation steps that only need to be done once for each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In most cases, the CRT method was the faster method to find torsion points over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The biggest bottleneck of the method is the combinatorial explosion that can take place in Steps 6 through 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' the sets DJ can become very big in cases where there is a lot of fake torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='3 Finding torsion points: the analytic method The following analytic method to find torsion points has the advantage that there will be no combinatorial explosion of trying to combine torsion points modulo differ- ent primes into a torsion point over a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The downside is that we cannot utilise the fact that Jpi(Fpi)[ℓn] is typically much smaller than J(C)[ℓn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Recall that we assumed the existence of a point P ∈ C(Q) and that we picked such a point at the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algorithm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Input: a prime number ℓ, a subgroup of known torsion points K ⊂ J[ℓ](Q), and a point Q ∈ J(Q) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Output: a (possible empty) list of nonzero points R ∈ J(Q) such that ℓR = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Choose some P1, P2, P3 ∈ C(C) as in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Now write Q as Q1+Q2+Q3−P1−P2−P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Compute an Abel-Jacobi map ι: J(C) → C/Λ and compute the image ι(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Pick an element x in each class in � 1 ℓι(Q) + 1 ℓΛ � /ι(K) and apply the fol- lowing three steps for each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use Algorithm 9 to find a point R ∈ J(C) such that ι(R) is close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use a modified version of Algorithm 6 to write R as R1 + R2 + R3 − 3P for some R1, R2, R3 ∈ C(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Compute Mumford coordinates for R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' compute the product polynomial Px := � i(T − x(Ri)) in C[T] and a polynomial Py of degree 2 such that Py(x(Ri)) = y(Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 13 Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Use a short lattice vector algorithm to try to find algebraic relations for the coefficients of Px and Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' If this succeeds, reconstruct the corresponding point in J(Q), which we also call R by abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' After finishing the loop described in Step 2, output all R with ℓR = Q that we found in Step 5 during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In practice, to recognise torsion points over number fields, we need several hundreds of digits of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This together with the sheer number of potential points we need to try (typically ℓ6) makes the method slow in practice and only practical for ℓ = 2 or ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5 Results The algorithm has been implemented by the author in Magma and is publicly available at [code].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' It has been run on a dataset consisting of 82240 plane quartic curves found by Andrew Sutherland, see [Suth19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As a result, the rational torsion subgroup has been computed successfully for 81357 of the Jacobians of these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' The total runtime for this computation was approximately 8 core months and has been done in parallel on a machine of the Simons Collaboration at Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For each computed torsion group a proof has been stored in the form of a list of primes, and a list torsion points over Q and over some number fields which together can be used to prove the completeness of the list of rational torsion points using Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' These proofs can be verified significantly faster than it took to construct them and are stored in the file [code, extra/proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='xz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' In Table 5, you can see the 64 different orders of the torsion groups that we found and how often each of them occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='Table 1: Order statistics for torsion groups ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='For the majority of the 883 missing torsion groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' the reason that we could not compute them was the failure to find a rational point on the plane quartic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For 14 some of these curves, we could verify the nonexistence of rational points by proving that there are no points over some local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' For the remaining curves, which might give rise to counterexamples for the Hasse principle, we did not attempt to verify the nonexistence of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' To conclude this section we exhibit an example where we managed to find a torsion point over a degree 12 number field in order to certify the correctness of the computed rational torsion subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Consider the smooth plane quartic C : f = 0 with f = x3y−xy3+y4+x3z+2x2yz+2xy2z−y3z+x2z2+2xyz2+y2z2−2xz3−yz3+z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Its Jacobian J modulo 11 has 1772 points, and J modulo 67 has 274944 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As the primes 11 and 67 are both primes of good reduction, this implies that the torsion subgroup of J can have at most order gcd(1772, 274944) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Besides 0, we find a second rational torsion point T2 = � 1 19(−10θ + 4) : −1 : 1 � + � 1 19(−10¯θ + 4) : −1 : 1 � − 2 · (1 : 0 : 0), where θ and ¯θ are zeros of x2 − 27 10x − 1713 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' We easily find that there are no other points of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' After looking at a lot of different primes and seeing that T2 has a 2-divisor modulo each of these primes, we suspect that T2 might have a (fake) 2-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' After about an hour of computation time, our program finds a torsion point T4 over a degree 12 number field K defined by adjoining to Q a root of x12 − 5x10 − 2x9 − 20x8 − 20x7 + 7x6 − 50x5 + 26x4 − 40x3 − 58x2 − 24x − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' This point satisfies 2T4 = T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As the prime 67 splits into four primes of residue degrees 67, 67, 672, and 678 in the ring of integers of K, the point T4 explains two of the 2-divisors of T2 modulo 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' As there are only two 2-divisors of T2 in J mod 67, we conclude that T2 doesn’t have a 2-divisor over Q, and {0, T2} is the full torsion subgroup of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' References [BSD65] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Birch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Swinnerton-Dyer, Notes on elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 218 (1965), 79–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 [code] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' van Bommel, genus3torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Magma code, available at: https://git hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='com/rbommel/genus3torsion 3, 10, 14 [Can87] David G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Cantor, Computing in the Jacobian of a hyperelliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 48 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 177, 95–101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5 15 [Cos15] Edgar Costa, Effective computations of Hasse–Weil zeta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Thesis (Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=')–New York University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' ISBN: 978-1321-95392-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' ProQuest LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4 [FOR08] St´ephane Flon, Roger Oyono, Christophe Ritzenthaler, Fast addition on non-hyperelliptic genus 3 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Algebraic geometry and its applications, 1–28, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Number Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=', 5, Word Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=', Hackensack, NJ, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5, 7 [Hal10] Safia Haloui, The characteristic polynomials of abelian varieties of dimen- sions 3 over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Number Theory 130 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 12, 2745–2752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 12 [Har77] Robin Hartshorne, Algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Graduate Texts in Mathematics, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Springer-Verlag, New York-Heidelberg, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 6, 7 [Khu04] Kamal Khuri-Makdisi, Linear algebra algorithms for divisors on an al- gebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 73 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 245, 333-357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5, 6 [Khu07] Kamal Khuri-Makdisi, Asymptotically fast group operations on Jacobi- ans of general curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 76 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 260, 2213–2239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5 [Khu18] Kamal Khuri-Makdisi, On Jacobian group arithmetic for typical divisors on curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Number Theory 4 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 3, 29 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 5 [LLL82] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Lenstra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Lenstra Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Lov´asz, Factoring polynomials with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 261 (1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 4, 515–534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 11 [Katz81] Nicholas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Katz, Galois properties of torsion points on abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 62 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 3, 481–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2 [Mas20] Nicolas Mascot, Hensel-lifting torsion points on Jacobians and Galois representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 89 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 323, 1417–1455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 10 [Maz78] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Mazur, Modular curves and the Eisenstein ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' With an appendix by Mazur and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 47 (1977), 33–186 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 [Mer96] Lo¨ıc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 124 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1–3, 437–449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 [Mor22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Cambr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 21 (1922), 179–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 [M¨uRe22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Steffen M¨uller, Berno Reitsma, Computing torsion subgroups of Jac- obians of hyperelliptic curves of genus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='03372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2 [Neu18] Christian Neurohr, Efficient integration on Riemann surfaces & applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' PhD thesis (2018), https://oops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='uni-oldenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='de/3607/1/ne ueff18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 8 [Sto02] Michael Stoll, On the height constant of genus two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Acta Arith.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Sutherland, A database of nonhyperelliptic genus 3 curves over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), Open Book Series 2 (2019), 443–459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 3, 6, 14 16 [Wang81] Paul S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Wang, A p-adic algorithm for univariate partial fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Sym- bolic and algebraic computation, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' AMC Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=', Snowbird/Utah 1981, 212–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 10 [Weil29] Andr´e Weil, L’arithm´etique sur les courbes alg´ebriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1, 281–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 1 [WGD92] Paul S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Guy, James H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' Davenport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' P-adic reconstruction of rational numbers, SIGSAM Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 16 (1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 2, 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} +page_content=' 10 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E_T4oBgHgl3EQfxhwl/content/2301.08312v1.pdf'} diff --git a/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf b/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..51e4bd1789ad5f23f5301fb93a57e36479c67c71 --- /dev/null +++ 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+Thorben Petersen1,∗ Ulrich K. R¨oßler1, and Liviu Hozoi1† +1Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, +Helmholtzstraße 20, D-01069, Dresden, Germany +(Dated: January 10, 2023) +Electron correlation effects are ubiquitous in transition-metal compounds. +Here we identify a +material platform that allows to map out correlations across a whole group of the d block, most +importantly, for the same kind of leading ground-state configuration and on the same type of crys- +talline lattice. The wave-function analysis that we provide, in terms of either localized, atomic-like +or multisite orbitals, makes these materials — the group-V lacunar spinels — a distinct correlated- +electron model system. It also defines the basic theoretical frame when addressing the different +types of skyrmionic phases and magneto-electric couplings in lacunar spinels. +Introduction. A toy model to illustrate electronic cor- +relations is the H2 molecule: +by increasing the bond +length, correlations become more and more important +until the Heitler-London limit of strong correlations. +Here, ionic configurations are suppressed in the ground- +state wave-function and each electron is localized on one +atom. In this study, we establish a platform to illustrate +correlations in d-electron setting — the group-V lacu- +nar spinels AM4X8, where A can be either Ga or Al, M +stands for V, Nb, or Ta, and X is either S or Se. +A +unique feature of these materials is realizing the same +kind of leading ground-state configuration, (i) across a +whole group of the d block, i. e., 3d to 4d and 5d, and +(ii) in the same crystallographic frame. The respective +leading electron configuration is (...)t1 +2 but the underlying +orbital basis is multisite in nature, since transition-metal +ions are clustered as M4 tetrahedra in lacunar spinels +(see Fig. 1). +Using ab initio wave-function theory, we uncover what +exactly “leading” means in group-V lacunar spinels. We +clarify this way the basic multiorbital correlations on M4 +clusters and the inner structure of the effective pseu- +dospins in these materials. Our quantum chemical com- +putational results reveal a very colorful landscape, much +richer than the generic (...)t1 +2 single-configuration de- +scription applied so far to all materials in this family. +The analysis clearly shows that the strongest correlations +show up in the vanadates: in terms of localized, atomic- +like orbitals, the weight of t1 +2gt2 +2gt2 +2gt2 +2g (V4+V3+V3+V3+) +configurations is 85-90%. This is reduced in the Nb and +Ta variants since intersite M 3+M 3+→M 4+M 2+ charge +fluctuations are enhanced for more delocalized 4d and 5d +electrons. When recasting the wave-functions in terms +of molecular-like orbitals extended over the M4 tetra- +hedron, a contribution of only ≈20% is obtained for +the t1 +2 configuration in the vanadates. While larger in +the 4d and 5d compounds, ≈70%, it is still significantly +deviating from the 100% t1 +2 ground-state wave-function +∗ t.petersen@ifw-dresden.de +† l.hozoi@ifw-dresden.de +presently assumed in the community. +The reported +data defines the proper, realistic frame for more involved +many-body analysis to understand inter-tetrahedral su- +perexchange and the differences as concerns the mag- +netism of these materials. +Basic structure and properties. +The transition-metal +ions form M4 tetrahedra in the lacunar spinels, as part of +cubane-like [M4X4] units of edge-sharing MX6 octahedra +(see Fig. 1). The electronic structure of these transition- +metal clusters implies in a simplistic picture one unpaired +electron distributed over the four M sites, or localized +on one of them. This unit is believed to shape the pe- +culiar properties of the lacunar spinels, which experience +distinct structural instabilities driving transitions from +the non-centrosymmetric cubic to polar and ferroelectric +or anti-polar crystal structures [1] with magneto-electric +couplings (for a recent review, see [2]). +The collective +properties of the coupled M4 tetrahedra display magnetic +order for the V-based systems, specifically cycloidal he- +limagnetism with a field-induced skyrmion lattice phase +[3]. The Nb- and Ta-based systems have nonmagnetic +Ga +V/Nb/Ta +S/Se +Xo +e +Xi +cECP +embedded +cluster +a1 +t2 +electronic +structure +FIG. 1. +[M4X28Ga6]25− embedded cluster used in this study +(M = V, Nb, Ta and X = S, Se). Small atoms indicate the +capped effective core potentials (cECPs). Xi and Xo labels +indicate X atoms inside and outside the M4 tetrahedron, re- +spectively. A basic level diagram is also provided. +arXiv:2301.03392v1 [cond-mat.str-el] 9 Jan 2023 + +2 +ground states [4, 5], which turn metallic and also super- +conducting under pressure [6, 7]. So far, the mechanism +for the superconducting state is poorly understood [8]. +Theoretically understanding these phenomena obvi- +ously should start from the electronic structure of the +M4 cluster. Its leading electronic configuration is (...)t1 +2, +with a multisite orbital basis [7, 9–13]. But, calculating +the properties of the lacunar spinels based on conven- +tional or extended density functional theory (DFT) evi- +dences severe deficiencies, DFT being unable to predict +this electronic configuration correctly as the ground state +[14]. Also, a high sensitivity of the calculated properties +on structural parameters is observed [7, 9, 10, 15]. As +a result, DFT calculations are not precise enough to re- +produce the correct low-temperature crystal structures. +While introducing in the DFT+U approach a sufficiently +large Coulomb repulsion parameter U for 3d vanadate la- +cunar spinels seems crucial to stabilize the desired mag- +netic ground state, it results in wrong electro-structural +and magnetic properties in the case of GaV4Se8 [16]. +Indications of genuine many-body physics are available +from both ab initio quantum chemistry [13, 17] and dy- +namical mean field theory (DMFT) calculations [12] but +an in-depth profile of correlation effects across the 3d-4d- +5d lacunar-spinel series is missing, which is the scope of +our present study. +Quantum chemical calculations. +To describe the ba- +sic building block in the lacunar-spinel structure, a +[M 4X 28Ga6]25– embedded cluster model (M = V, Nb, +Ta; X = S, Se) was used (see Fig. 1). Experimentally +determined high-temperature lattice parameters were +adopted, as reported by Stefancic et al. [18] for GaV4S8 +and by Pocha et al. for GaNb4Se8 and GaTa4Se8 [7]. The +influence of the surrounding bulk atoms was modelled by +a finite point charge field (PCF) generated through the +Ewald program [19, 20]. A buffer region of 60 capped +effective core potentials (cECPs) was set up between +the quantum cluster and PCF (indicated by the smaller +atoms in Fig. 1) (for details, see Supplemental Material +[21]). +As initial step in our study, +quasi-restricted or- +bitals (QROs [22]) were generated from an unrestricted +Kohn-Sham B3LYP calculation for a single-configuration +S = 5/2 state with initial-guess orbitals from Hueckel +theory. The Hueckel guess ensures that the QROs ful- +fill Td point group symmetry. Subsequently, 12 [M4]13+ +molecular orbitals around the HOMO-LUMO gap were +identified from the QROs and used as a starting point for +complete active space self-consistent field (CASSCF) cal- +culations. Major convergence problems as encountered in +earlier quantum chemical studies [17] are circumvented in +this way. The valence-space multiplet structure was de- +rived from state averaged (SA) CASSCF optimizations +with those 12 orbitals and seven valence electrons defin- +ing the active space (denoted in quantum chemistry as +(7e,12o) CASSCF), consequently corrected for dynamical +correlation by N -electron valence 2nd order perturbation +theory (NEVPT2). Both methods were accelerated by +the resolution of identity (RI [23]) and chain-of-spheres +(COS [24]) approximations for Coulomb and exchange in- +tegrals with automatically generated auxiliary basis sets +[25]. All calculations were done using the program pack- +age Orca, v5.0.3 [26]. +High-temperature, +tetrahedral-symmetry +multiplet +structure. +The M4-tetrahedron multiplet structure, as +computed for the high-temperature F¯43m cubic lattice +arrangement of group-V lacunar spinels, is displayed in +Fig. 2. For GaV4S8 (Fig. 2.(a) ), the (7e,12o) CASSCF +yields a high-spin (S = 3/2) ground state but this +is corrected in the subsequent NEVPT2 treatment. +The near degeneracy of low- and high-spin states is +an effect often seen in 3d systems due to the similar +magnitude of Coulomb interactions and crystal-field +splittings [27]; +in solid-state context, +a well-known +example is LaCoO3 (see [28] for a quantum chemical +investigation). Spin-crossover effects were also observed +in DMFT calculations on GaV4S8 [12]. In contrast, for +the Nb- (Fig. 2(b)) and Ta-based materials (Fig. 2(c)), +the CASSCF(7e,12o) methodology already ensures a +reasonably good description — the NEVPT2 scheme +provides now only minor corrections to the relative +energies (see Supplemental Material [21]). +In all three compounds, the 2T2 state constitutes the +ground state, with a dominant a2 +1e4t1 +2 electronic configu- +ration, where a1, e, and t1 are symmetry-adapted (Td +point group), molecular-like orbitals. +SOC splits the +degenerate 2T2 components into a Jeff = 3/2 ground +and a Jeff = 1/2 excited state in all instances, with +the magnitude of this splitting increasing from 12 meV +(3d) to 97 meV (4d) and 345 meV (5d ions). The sepa- +ration between the ground 2T and the first excited 4T +state also increases for heavier transition-metal species: +41 meV in GaV4S8, 475 meV in GaNb4Se8, and 598 meV +in GaTa4Se8, which reflects the stronger ligand fields ex- +perienced by electrons in more extended (4d and 5d) or- +bitals. +Ground-state correlations in cubic group-V +spinels. +A major difference between how the single-tetrahedron +electronic structure is presently depicted in the litera- +ture and our quantum chemical results is the composi- +tion of the ground-state 2T2 term. +Different from the +100% a2 +1e4t1 +2 ground state assumed so far on the basis of +DFT computations for these materials, we find weights +of 72% in GaTa4Se8, 65% in GaNb4Se8, and as little as +21% in GaV4S8 for the a2 +1e4t1 +2 configuration. +Double- +excitations into higher-lying t1 and t2 levels represent +further configurations contributing to the ground-state +wave-functions, in each case with an overall weight of +≲10%. +The weight of the dominant 4T excited-state +configuration a2 +1e3t2 +2 follows the same trend, with ra- +tios of 24%, 69%, and 77% for GaV4S8, GaNb4Se8, and +GaTa4Se8, respectively (see also Supplemental Material +[21]). +The much more pronounced multiconfigurational char- +acter of the vanadate ground-state wave-function in the +symmetry-adapted orbital basis is also reflected in the + +3 +0 +200 +400 +600 +800 +1000 +1200 +CASSCF +SOC +(b) [Nb4Se28Ga6]25- +2T +4T +4T +2A +2E,2A +2T +2E +6A +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +CASSCF +SOC +(c) [Ta4Se28Ga6]25- +2T +4T +4T +2T +2E +2A +2T 6A +2E +2A +0 +50 +100 +150 +200 +250 +CASSCF +NEVPT2 +SOC +Energy (meV) +(a) [V4S28Ga6]25- +S = 5/2 +S = 3/2 +S = 1/2 +2T +2T +2A +2E +2A +4T +4T +6E +6A +6E +6A +4T +4T +2E +2A +2A +Jeff 1/2 +3/2 +Jeff +3/2 +1/2 +Jeff +3/2 +1/2 +2T +FIG. 2. +Calculated excitation energies for (a) [V4S28Ga6]25–, (b) [Nb4Se28Ga6]25–, and (c) [Ta4Se28Ga6]25– embedded clusters +(CAS(7e,12o)). States corresponding to different spin multiplicities (S = 1/2, 3/2, 5/2) are shown in green, red, and blue, +respectively. The corrections brought by NEVPT2 are minor for the Nb- and Ta-based compounds and therefore not depicted. +correlation index proposed by Ramos-Cordoba et al. [29] +as measure for the extent of near-degeneracy effects +(those are also referred to as nondynamical correlation +in quantum chemistry). +Using the (7e,12o) CASSCF +natural-orbital occupation numbers, we derived ‘nondy- +namical’ correlation indices IND [29] ranging from ≈2 in +vanadates (1.95 for GaV4S8 and 1.97 for GaV4Se8) to +≈1.05 in the 4d variants (1.05 for GaNb4S8 and 1.07 for +GaNb4Se8) and 0.82 in GaTa4Se8. To put this in per- +spective, along the H2 dissociation curve, IND evolves +from less than 0.1 at equilibrium distance to 0.5 towards +dissociation [29], although a quantitative comparison of +IND values between chemically different systems is not +straightforward. +Looking for further insight, we re-expressed the many- +body ground-state wave-functions in terms of atomic- +like, single-site orbitals. +The orbital localization mod- +ule available in Orca was employed for this purpose; +from the 12 symmetry-adapted orbitals (1×a1, 1×e, +1×t1, 2×t2) obtained in the (7e,12o) CASSCF we arrive +then to 12 t2g-like functions (three per transition-metal +ion, see Supplemental Material [21]). For both orbital +bases, symmetry-adapted or site-centered, the weights of +the leading configurations are illustrated in Fig. 3, for +GaV4S8, GaV4Se8, GaNb4S8, GaNb4Se8, GaTa4Se8, and +an A-site-substituted member of the family (see also Sup- +plemental Material [21]). Interestingly, a low weight of +the a2 +1e4t1 +2 configuration in the symmetry-adapted orbital +basis is associated with large weight of the t1 +2gt2 +2gt2 +2gt2 +2g +(i. e., V4+V3+V3+V3+) configurations in the localized- +orbital representation; for the AlV4S8, GaV4S8, and +GaV4Se8 vanadates, in particular, 15-20% a2 +1e4t1 +2 trans- +lates into 85-90% configurations of t1 +2gt2 +2gt2 +2gt2 +2g type. +The remaining part stems mainly from triply-occupied +transition-metal centers, i. e., excited-state configura- +tions of t1 +2gt2 +2gt1 +2gt3 +2g (V4+V3+V4+V2+) type. +Weights +of only ≈9% for the latter in the case of the V-based +lacunar spinels indicate much stronger correlations: in- +tersite fluctuations are strongly suppressed, compared to +the 4d and 5d compounds. In a Mott-Hubbard picture, +stronger localization is the result of larger U/t ratio. In +other words, correlations are moderate in the 4d and 5d +GaV4S8 +GaNb4Se8 +GaTa4Se8 +72% +5d +GaV4Se8 +GaNb4S8 +3d +4d +65% +67% +15% +21% + +weight of a1 e4 t2 +weight of t2g t2g t2g t2g +88% +89% +54% +55% +48% +AlV4S8 +89% +18% +2 +2 +2 +2 +1 +1 +FIG. 3. +Polar plot with weights for the leading a2 +1e4t1 +2 +(symmetry-adapted orbital basis, in red) and t1 +2gt2 +2gt2 +2gt2 +2g +(localized-orbital basis, in blue) electronic configurations in +the ground-state CASSCF wave-function for group-V lacunar- +spinel compounds (CAS(7e,12o)). + +4 +compounds and strong in the vanadates — for the latter, +an expansion in terms of four V4+V3+V3+V3+ resonat- +ing valence structures [30] already provides a reasonably +good description. +Discussion. +To analyze in detail how correlations +evolve from 3d to 4d and 5d ions for the same type +of leading ground-state configuration and in the same +crystallographic setting is difficult. +3d5 (Mn2+, Fe3+) +species, for example, tend to adopt a t3 +2ge2 +g ground-state +electron configuration, while 4d5 (Ru3+, Rh4+) and 5d5 +(Ir4+) varieties display a t5 +2g valence-orbital occupation. +Thinking of lower d-shell filling, Mo5+ 4d1 and Os7+ 5d1 +ions, for instance, can be found in double-perovskite fcc +settings [31], but that is not the case for Ti3+ or V4+ +3d1. Here we individualize the group-V lacunar spinels +as a unique platform that makes it possible to illustrate +how correlations shape many-body wave-functions across +a given group of the d block — 3d to 4d and 5d, for +the same kind of leading electron configuration and in +the same crystallographic setting. In particular, we pro- +vide new, important insights by expressing the many- +body M4-tetrahedron wave-function in terms of local- +ized, single-ion t2g orbitals. We show that in the vana- +dates strong correlations yield a weight of 85–90% for the +t1 +2gt2 +2gt2 +2gt2 +2g (i. e., V4+V3+V3+V3+) configurations; inter- +estingly, with such a reference wave-function, ferromag- +netic and antiferromagnetic intra-tetrahedron exchange +mechanisms are in rather good balance, which leads to +near degeneracy of the low-lying low- and high-spin states +— the low-spin (S = 1/2) state is obtained as single- +tetrahedron ground-state term only through a rather ad- +vanced many-body treatment. In contrast, with 4d (Nb) +and 5d (Ta) ions, charge fluctuations are much stronger +and the weight of t1 +2gt2 +2gt2 +2gt2 +2g configurations is reduced +to ∼50%; this reduces ferromagnetic double exchange, es- +pecially through reducing the role of M 4+M 3+M 3+M 3+ +resonances on the M4 tetrahedron, and firmly stabilizes +the low-spin ground-state term. +This result seems to +agree with the persistence of the Jeff = 3/2 ground state +under pressure even within the metallic state for the 4d +and 5d systems [8, 32]. The more extended cluster states, +involving significant contributions from ionic configura- +tions, also suggest a physical picture for the nonmagnetic +ground state in the Nb- and Ta-based lacunar spinels: +as these states are prone to stronger magnetic fluctua- +tions with larger spatial spread, inter-cluster couplings +are able to create spin singlets or valence bond states, +as concluded from experiment [4, 5]. The peculiar pseu- +dospin structure must play a role in the superconductiv- +ity mechanism under pressure, which is speculated to be +unconventional owing to closeness of magnetic states and +spin fluctuations. +Our quantum chemical data provide unparalleled +specifics as concerns the correlated electronic structure +of the group-V lacunar spinels, well beyond the feature- +less a2 +1e4t1 +2 picture circulated so far in the literature. +Stronger correlations in the vanadates imply substan- +tially less weight for the a2 +1e4t1 +2 configuration, compared +to the Nb and Ta compounds. Remarkably, spin-orbit +coupling is still effective, even for the vanadates — the +predicted fine structure with a splitting of ≈10 meV be- +tween the J ≈ 3/2 and J ≈ 1/2 terms should be de- +tectable experimentally. The stronger intersite fluctua- +tions (M 3+M 3+→M 1+M 4+) and the larger weight (65- +75%) of the a2 +1e4t1 +2 ‘molecular-orbital’ configuration in +the Nb and Ta spinels indicate that the 4d and 5d systems +are closer to the Hartree-Fock limit. Again, thinking of +the H2 molecule, the simplest model to illustrate elec- +tronic correlations, the regime of large inter-atomic sepa- +ration parallels the case of the vanadate lacunar spinels. +Assessing cooperative effects in group-V lacunar spinels +through calculation of inter-cluster couplings will require +approaches able to incorporate the correlated nature of +the M4-cluster ground states. +Acknowledgments. +We thank I. K´ezsm´arki, P. Fulde +and R. C. Morrow for discussions and U. Nitzsche for +technical assistance. +This work was supported by the +German Research Foundation (Deutsche Forschungsge- +meinschaft, DFG), Project No. 437124857. +T.P. carried out the quantum chemistry calculations +with assistance from U.K.R. and L.H. 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Olbrich, +Advanced aspects of ab initio theoretical optical spec- +troscopy of transition metal complexes: Multiplets, spin- +orbit coupling and resonance Raman intensities, Coord. +Chem. Rev. 251, 288 (2007). +[28] L. Hozoi, U. Birkenheuer, H. Stoll, and P. Fulde, Spin- +state transition and spin-polaron physics in cobalt oxide +perovskites: ab initio approach based on quantum che- +mical methods, New J. Phys. 11, 023023 (2009). +[29] E. Ramos-Cordoba, P. Salvador, and E. Matito, Sep- +aration of dynamic and nondynamic correlation, Phys. +Chem. Chem. Phys. 18, 24015 (2016). +[30] For perspectives on valence bond theory, see e. g. [33, 34]. +[31] G. Chen, R. Pereira, and L. Balents, Exotic phases in- +duced by strong spin-orbit coupling in ordered double +perovskites, Phys. Rev. B 82, 174440 (2010). +[32] M. Y. Jeong, S. H. Chang, B. H. Kim, J.-H. Sim, A. Said, +D. Casa, T. Gog, E. Janod, L. Cario, S. Yunoki, M. J. +Han, and J. Kim, Direct experimental observation of the +molecular Jeff = 3/2 ground state in the lacunar spinel +GaTa4Se8, Nat. Commun. 8, 782 (2017). +[33] R. Hoffmann, S. Shaik, and P. C. Hiberty, A conversa- +tion on VB vs MO theory: A never-ending rivalry ?, Acc. +Chem. Res. 36, 750 (2003). +[34] D. G. Truhlar, Valence bond theory for chemical dynam- +ics, J. Comput. Chem. 28, 73 (2007). + +Supplemental Material: +Luxuriant correlation landscape in lacunar spinels: +multiconfiguration expansions in molecular-orbital basis vs resonant valence structures +Thorben Petersen1,∗ Ulrich K. R¨oßler1, and Liviu Hozoi1† +1Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, +Helmholtzstraße 20, D-01069, Dresden, Germany +I. +EMBEDDED CLUSTER PROTOCOL +The quantum mechanical cluster model [M 4X 28A6]25– (with M = V, Nb, Ta X = S, Se and A = Al, Ga) was +embedded in a field of point charges (PCs), which was created using the Ewald program [1, 2] and experimentally +determined geometries for the high-temperature cubic phase [3–6]. To initialize the Ewald optimization, the initial +charges given in Supplementary Tab. I were employed. Those were determined under two constraints: +(1) the difference between the modified charges of the reciprocal unit cell and the formal charges necessary for +extracting the quantum cluster is zero and +(2) the difference between these initial charge values and to atomic charges of the quantum cluster fitted to the +molecular electrostatic potential (CHELPG, charges from electrostatic potentials using a grid-based method +[7–10]) is minimal. +Between optimized PCs and quantum cluster, a total of 60 atoms (48 M, 12 Xi) were equipped with pseudopotentials +for V from Dolg et al. [11], for Nb and Ta from from Andrae et al. [12], and for S/Se from Bergner et al. [13]. For +GaNb4Se8 and GaTa4Se8 in particular, also smaller [M 4Se16]19– cluster models were used to make computationally +demanding calculations (NEVPT2 in particular) feasible. Here, the outermost Ga (6 atoms) and Seo (12 atoms) were +assigned cECPs from Leininger et al. [14] and Bergner et al. [13] with the appropriate charges given in Supplementary +Table I, respectively. +Supplementary Table I. Initial point charge values for the employed quantum cluster models. Xi and Xo refer to X ligand +atoms inside and outside of the V4/Nb4/Ta4 unit. Additionally, the reference used for the HT-phase crystal structure is given. +Compound +Quantum cluster +Ga/Al +V/Nb/Ta +Si/Sei +So/Seo +Ref. HT-cryst. +GaV4S8 +[V4S28Ga6]25– +1.48 +2.07 +−0.82 +−1.62 +[3] +GaV4Se8 +[V4Se28Ga6]25– +1.48 +2.07 +−0.82 +−1.62 +[3] +AlV4S8 +[V4S28Al6]25– +1.48 +2.07 +−0.82 +−1.62 +[4] +GaNb4S8 +[Nb4S28Ga6]25– +1.40 +2.05 +−0.80 +−1.60 +[5] +GaNb4Se8 +[Nb4Se28Ga6]25– +1.40 +2.05 +−0.80 +−1.60 +[6] +[Nb4Se16]19– +3.00 +2.00 +−0.75 +−2.00 +[6] +GaTa4Se8 +[Ta4Se28Ga6]25– +1.40 +2.05 +−0.80 +−1.60 +[6] +[Ta4Se16]19– +3.00 +2.00 +−0.75 +−2.00 +[6] +II. +COMPUTATIONAL METHODS +The complete active space self-consistent field (CASSCF) approach is fully ab initio – therefore, the basis sets of +the atoms in the quantum cluster are crucial to chose appropriately. While large basis sets are computationally too +demanding, small basis sets do not yield a satisfactorily accuracy. A sufficient accuracy-speed trade-off was found +∗ t.petersen@ifw-dresden.de +† l.hozoi@ifw-dresden.de +arXiv:2301.03392v1 [cond-mat.str-el] 9 Jan 2023 + +2 +for mainly triple-ζ valence polarized (TZVP) basis sets with an additional polarization function. For each atomic +species, the employed basis sets are given in the following Supplementary Table II. Since a strong effect of spin-orbit +coupling (SOC) on the electronic states was found in both GaNb4Se8 and GaTa4Se8 [15–17], we additionally enable +the Douglas-Kroll-Hess (DKH) approximation [18, 19] and used appropriately decontracted basis set variants. +Supplementary Table II. Basis set per atom of each quantum cluster. X1 (with X = S, Se) refer to atoms directly bonding +to the [M4]13+ cluster (16 atoms), while X2 atoms are the outermost atoms bonding to Ga/Al (12 atoms). “�” denotes the +overall number of contracted basis functions. The reference citation is also given. +Quantum cluster +Atom +Basis set +Ref. +Quantum cluster +Atom +Basis set +Ref. +[V4S28Ga6]25– +V +cc-pVTZ-DK +[20] +[V4Se28Ga6]25– +V +cc-pVTZ-DK +[20] +S1 +cc-pVTZ-DK +[21] +Se1 +cc-pVTZ-DK +[21] +Ga +cc-pVDZ-DK +[21] +Ga +cc-pVDZ-DK +[21] +S2 +cc-pVDZ-DK +[21] +Se2 +cc-pVTZ-DK +[21] +� +1194 +� +1446 +[V4S28Al6]25– +V +cc-pVTZ-DK +[20] +[Nb4S28Ga6]25– +Nb +SARC-DKH-TZVPP +[22] +S1 +cc-pVTZ-DK +[21] +S1 +DKH-DEF2-TZVPP +[23] +Al +cc-pVDZ-DK +[21] +Ga +DKH-DEF2-SVP +[23] +S2 +cc-pVTZ-DK +[21] +S2 +DKH-DEF2-SVP +[23] +� +1140 +� +1556 +[Nb4Se28Ga6]25– +Nb +SARC-DKH-TZVPP +[22] +[Ta4S28Ga6]25– +Ta +SARC-DKH-TZVPP +[24] +Se1 +DKH-DEF2-TZVPP +[23] +Se1 +DKH-DEF2-TZVPP +[23] +Ga +DKH-DEF2-SVP +[23] +Ga +DKH-DEF2-SVP +[23] +Se2 +DKH-DEF2-SVP +[23] +Se2 +DKH-DEF2-SVP +[23] +� +2124 +� +2212 +[Nb4Se16]19– +Nb +SARC-DKH-TZVPP +[22] +[Ta4Se16]19– +Ta +SARC-DKH-TZVPP +[24] +Se1 +DKH-DEF2-TZVPP +[23] +Se1 +DKH-DEF2-TZVPP +[23] +Se2 +DKH-DEF2-TZVPP +[23] +Se2 +DKH-DEF2-TZVPP +[23] +� +1368 +� +1456 +In Supplementary Fig. 1, the active space orbitals of the employed CAS(7e,12o) are depicted in natural (1(a)) and +localized orbital (1(b)) representations. For the former, the corresponding irreducible representations according to Td +point group symmetry are also given. + +3 +a1 +e +e +t2 +t2 +t2 +t2 +t2 +t2 +t1 +t1 +t1 +(a) +(b) +dyz (V1) +dxz (V1) +dxy (V1) +dyz (V2) +dxz (V2) +dxy (V2) +dyz (V3) +dxz (V3) +dxy (V3) +dyz (V4) +dxz (V4) +dxy (V4) +Supplementary Figure 1. +Active space orbitals used in CAS(7e,12o). (a) Natural orbital representation. (b) Localized orbital +representation. +III. +LOW-ENERGY EXCITATION ENERGIES OF GROUP V LACUNAR SPINELS +In the following Supplementary Tables III, IV and V the calculated low-energy excitation energies for an active space +CASSCF(7e,12o) and subsequent NEVPT2 correction are given for the three lacunar spinel compounds investigated +in detail in the manuscript. Note, that for 4d-GaNb4Se8 and 5d-GaTa4Se8, a smaller quantum cluster size was chosen +in order to make the NEVPT2 correction computationally feasible. +Supplementary Table III. Excitation energies (meV) of GaV4S8 using a [V4S28Ga6]25– cluster model (CAS(7e,12o)). Three +sextets, six quartets and seven doublets were included in the state-average procedure. Notation according to Td point group +symmetry. For each leading configuration its respective weight in the overall wavefunction is given. +State +Leading config. +ECASSCF +ENEVPT2 +ESOC +NEVPT2 +2T2 +a2 +1 e4 t1 +2 t0 +1 t0 +2 (21%) +134 +0 +0 (J = 3/2), 12 (J = 1/2) +4T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (25%) +0 +41 +38, 42, 43, 48, 48, 50 +4T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (24%) +44 +82 +80, 84, 85, 88, 89, 89 +6E +a1 +1 e3 t3 +2 t0 +1 t0 +2 (47%) +31 +125 +129 (3 KD’s), 130 (3 KD’s) +6A2 +a2 +1 e2 t3 +2 t0 +1 t0 +2 (36%) +93 +143 +147 (3 KD’s) +2A1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (25%) +90 +164 +168 +2E +a2 +1 e3 t2 +2 t0 +1 t0 +2 (23%) +98 +173 +177 (2 KD’s) +2A1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (25%) +140 +207 +212 + +4 +Supplementary Table IV. +Excitation energies (meV) of GaNb4Se8 using a large [Nb4Se28Ga6]25– and a small [Nb4Se16]19– +cluster model (CAS(7e,12o)). The later was used to make highly-demanding NEVPT2 calculations feasible. Three sextets, six +quartets and seven doublets were included in the state-average procedure. Notation according to Td point group symmetry. +For each leading configuration its respective weight in the overall wavefunction is given. +[Nb4Se28Ga6]25– +[Nb4Se16]19– +State +Leading config. +ECASSCF +ESOC +CASSCF +ECASSCF +ENEVPT2 +ESOC +NEVPT2 +2T2 +a2 +1 e4 t1 +2 t0 +1 t0 +2 (64%) +0 +0 (J = 3/2) +0 +0 +0 (J = 3/2) +97 (J = 1/2) +98 (J = 1/2) +4T2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (70%) +478 +475−522 +473 +496 +492−537 +4T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (69%) +651 +653−702 +648 +648 +649−698 +2T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (67%) +832 +835−881 +826 +779 +783−838 +6A1 +a2 +1 e2 t3 +2 t0 +1 t0 +2 (76%) +862 +895 +851 +816 +848 +2A2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (70%) +869 +905 +869 +923 +963 +2E +a2 +1 e3 t2 +2 t0 +1 t0 +2 (67%) +882 +920 +879 +904 +943 +2A1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (62%) +884 +924 +881 +883 +923 +2T2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (67%) +985 +1008−1040 +981 +873 +894−921 +2E +a2 +1 e3 t2 +2 t0 +1 t0 +2 (63%) +1059 +1100 +1053 +1059 +1107 +Supplementary Table V. Excitation energies (meV) of GaTa4Se8 using a large [Ta4Se28Ga6]25– and a small [Ta4Se16]19– cluster +model (CAS(7e,12o)). The later was used to make highly-demanding NEVPT2 calculations feasible. Three sextets, six quartets +and seven doublets were included in the state-average procedure. Notation according to Td point group symmetry. For each +leading configuration its respective weight in the overall wavefunction is given. +[Ta4Se28Ga6]25– +[Ta4Se16]19– +State +Leading config. +ECASSCF +ESOC +CASSCF +ECASSCF +ENEVPT2 +ESOC +NEVPT2 +2T2 +a2 +1 e4 t1 +2 t0 +1 t0 +2 (71%) +0 +0 (J = 3/2) +0 +0 +0 (J = 3/2) +345 (J = 1/2) +347 (J = 1/2) +4T2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (77%) +639 +598−717 +629 +632 +588−683 +4T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (77%) +864 +880−1042 +859 +811 +834−993 +2T1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (75%) +994 +1046, 1192 +985 +912 +979, 1104 +2A2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (77%) +1114 +1261 +1111 +1128 +1289 +2E +a2 +1 e3 t2 +2 t0 +1 t0 +2 (75%) +1128 +1319 +1123 +1104 +1274 +2A1 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (69%) +1139 +1305 +1132 +1068 +1234 +2T2 +a2 +1 e3 t2 +2 t0 +1 t0 +2 (76%) +1159 +1234, 1370 +1153 +1011 +1128, 1208 +6A1 +a2 +1 e2 t3 +2 t0 +1 t0 +2 (83%) +1177 +1293 +1156 +1093 +1209 +2E +a2 +1 e3 t2 +2 t0 +1 t0 +2 (71%) +1296 +1587 +1289 +1251 +1541 +IV. +LEADING GROUND STATE CONFIGURATIONS OF GROUP-V LACUNAR SPINELS +In the following Supplementary Table VI, the composition of the ground state 2T2 term for each of the investigated +lacunar spinel compounds in terms of electronic configurations with associated weights are given. These electronic +configurations are given based on natural (molecular-like picture) and localized (atomic-like) orbitals. From both +representations, it becomes clear that the 3d-vanadate and the 4d/5d-lacunar spinels form two groups that differ +strongly in the amount of mixing of the leading a2 +1e4t1 +2t0 +1t0 +2 (in molecular orbitals) or t1 +2gt2 +2gt2 +2gt2 +2g (in localized orbitals) +configurations. + +5 +Supplementary Table VI. Leading electronic configurations and associated weights of the ground state term in the investigated +group-V compounds. Both natural (molecular-like) and localized (atomic-like) orbital basis are given. +Compound +Molecular +Localized +Compound +Molecular +Localized +GaV4S8 +a2 +1e4t1 +2t0 +1t0 +2 (21%) +t1 +2gt2 +2gt2 +2gt2 +2g (88%) +GaV4Se8 +a2 +1e4t1 +2t0 +1t0 +2 (18%) +t1 +2gt2 +2gt2 +2gt2 +2g (89%) +a1 +1e3t3 +2t0 +1t0 +2 (6%) +t1 +2gt2 +2gt1 +2gt3 +2g (11%) +a1 +1e3t3 +2t0 +1t0 +2 (6%) +t1 +2gt2 +2gt1 +2gt3 +2g (9%) +a2 +1e2t1 +2t1 +1t1 +2 (3%) +... +a2 +1e2t1 +2t1 +1t1 +2 (3%) +... +... +... +AlV4S8 +a2 +1e4t1 +2t0 +1t0 +2 (19%) +t1 +2gt2 +2gt2 +2gt2 +2g (89%) +GaNb4S8 +a2 +1e4t1 +2t0 +1t0 +2 (66%) +t1 +2gt2 +2gt2 +2gt2 +2g (54%) +a1 +1e3t3 +2t0 +1t0 +2 (6%) +t1 +2gt2 +2gt1 +2gt3 +2g (9%) +a2 +1e2t3 +2t0 +1t0 +2 (5%) +t1 +2gt2 +2gt1 +2gt3 +2g (36%) +a1 +1e4t2 +2t0 +1t0 +2 (3%) +... +a2 +1e2t2 +2t1 +1t0 +2 (3%) +t0 +2gt2 +2gt2 +2gt3 +2g (5%) +... +... +... +GaNb4Se8 +a2 +1e4t1 +2t0 +1t0 +2 (64%) +t1 +2gt2 +2gt2 +2gt2 +2g (56%) +GaTa4Se8 +a2 +1e4t1 +2t0 +1t0 +2 (71%) +t1 +2gt2 +2gt2 +2gt2 +2g (48%) +a2 +1e2t3 +2t0 +1t0 +2 (6%) +t1 +2gt2 +2gt1 +2gt3 +2g (36%) +a2 +1e2t3 +2t0 +1t0 +2 (6%) +t1 +2gt2 +2gt1 +2gt3 +2g (39%) +a2 +1e2t2 +2t1 +1t0 +2 (3%) +t1 +2gt3 +2gt0 +2gt3 +2g (7%) +a2 +1e2t2 +2t1 +1t0 +2 (3%) +t1 +2gt1 +2gt1 +2gt4 +2g (9%) +... +... +... +... +[1] M. 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Comput. 4, 908 (2008). + diff --git a/UtE1T4oBgHgl3EQfuwWT/content/tmp_files/load_file.txt b/UtE1T4oBgHgl3EQfuwWT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c899f153e29074338452af8fdab56fa97a9df822 --- /dev/null +++ b/UtE1T4oBgHgl3EQfuwWT/content/tmp_files/load_file.txt @@ -0,0 +1,1100 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf,len=1099 +page_content='Luxuriant correlation landscape in lacunar spinels: multiconfiguration expansions in molecular-orbital basis vs resonant valence structures Thorben Petersen1,∗ Ulrich K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' R¨oßler1, and Liviu Hozoi1† 1Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, Helmholtzstraße 20, D-01069, Dresden, Germany (Dated: January 10, 2023) Electron correlation effects are ubiquitous in transition-metal compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Here we identify a material platform that allows to map out correlations across a whole group of the d block, most importantly, for the same kind of leading ground-state configuration and on the same type of crys- talline lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The wave-function analysis that we provide, in terms of either localized, atomic-like or multisite orbitals, makes these materials — the group-V lacunar spinels — a distinct correlated- electron model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' It also defines the basic theoretical frame when addressing the different types of skyrmionic phases and magneto-electric couplings in lacunar spinels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A toy model to illustrate electronic cor- relations is the H2 molecule: by increasing the bond length, correlations become more and more important until the Heitler-London limit of strong correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Here, ionic configurations are suppressed in the ground- state wave-function and each electron is localized on one atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In this study, we establish a platform to illustrate correlations in d-electron setting — the group-V lacu- nar spinels AM4X8, where A can be either Ga or Al, M stands for V, Nb, or Ta, and X is either S or Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A unique feature of these materials is realizing the same kind of leading ground-state configuration, (i) across a whole group of the d block, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=', 3d to 4d and 5d, and (ii) in the same crystallographic frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The respective leading electron configuration is (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=')t1 2 but the underlying orbital basis is multisite in nature, since transition-metal ions are clustered as M4 tetrahedra in lacunar spinels (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Using ab initio wave-function theory, we uncover what exactly “leading” means in group-V lacunar spinels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' We clarify this way the basic multiorbital correlations on M4 clusters and the inner structure of the effective pseu- dospins in these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Our quantum chemical com- putational results reveal a very colorful landscape, much richer than the generic (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=')t1 2 single-configuration de- scription applied so far to all materials in this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The analysis clearly shows that the strongest correlations show up in the vanadates: in terms of localized, atomic- like orbitals, the weight of t1 2gt2 2gt2 2gt2 2g (V4+V3+V3+V3+) configurations is 85-90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' This is reduced in the Nb and Ta variants since intersite M 3+M 3+→M 4+M 2+ charge fluctuations are enhanced for more delocalized 4d and 5d electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' When recasting the wave-functions in terms of molecular-like orbitals extended over the M4 tetra- hedron, a contribution of only ≈20% is obtained for the t1 2 configuration in the vanadates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' While larger in the 4d and 5d compounds, ≈70%, it is still significantly deviating from the 100% t1 2 ground-state wave-function ∗ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='petersen@ifw-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='de † l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='hozoi@ifw-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='de presently assumed in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The reported data defines the proper, realistic frame for more involved many-body analysis to understand inter-tetrahedral su- perexchange and the differences as concerns the mag- netism of these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Basic structure and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The transition-metal ions form M4 tetrahedra in the lacunar spinels, as part of cubane-like [M4X4] units of edge-sharing MX6 octahedra (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The electronic structure of these transition- metal clusters implies in a simplistic picture one unpaired electron distributed over the four M sites, or localized on one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' This unit is believed to shape the pe- culiar properties of the lacunar spinels, which experience distinct structural instabilities driving transitions from the non-centrosymmetric cubic to polar and ferroelectric or anti-polar crystal structures [1] with magneto-electric couplings (for a recent review, see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The collective properties of the coupled M4 tetrahedra display magnetic order for the V-based systems, specifically cycloidal he- limagnetism with a field-induced skyrmion lattice phase [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The Nb- and Ta-based systems have nonmagnetic Ga V/Nb/Ta S/Se Xo e Xi cECP embedded cluster a1 t2 electronic structure FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [M4X28Ga6]25− embedded cluster used in this study (M = V, Nb, Ta and X = S, Se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Small atoms indicate the capped effective core potentials (cECPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Xi and Xo labels indicate X atoms inside and outside the M4 tetrahedron, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A basic level diagram is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='03392v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='str-el] 9 Jan 2023 2 ground states [4, 5], which turn metallic and also super- conducting under pressure [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' So far, the mechanism for the superconducting state is poorly understood [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Theoretically understanding these phenomena obvi- ously should start from the electronic structure of the M4 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Its leading electronic configuration is (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=')t1 2, with a multisite orbital basis [7, 9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' But, calculating the properties of the lacunar spinels based on conven- tional or extended density functional theory (DFT) evi- dences severe deficiencies, DFT being unable to predict this electronic configuration correctly as the ground state [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Also, a high sensitivity of the calculated properties on structural parameters is observed [7, 9, 10, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' As a result, DFT calculations are not precise enough to re- produce the correct low-temperature crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' While introducing in the DFT+U approach a sufficiently large Coulomb repulsion parameter U for 3d vanadate la- cunar spinels seems crucial to stabilize the desired mag- netic ground state, it results in wrong electro-structural and magnetic properties in the case of GaV4Se8 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Indications of genuine many-body physics are available from both ab initio quantum chemistry [13, 17] and dy- namical mean field theory (DMFT) calculations [12] but an in-depth profile of correlation effects across the 3d-4d- 5d lacunar-spinel series is missing, which is the scope of our present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Quantum chemical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' To describe the ba- sic building block in the lacunar-spinel structure, a [M 4X 28Ga6]25– embedded cluster model (M = V, Nb, Ta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' X = S, Se) was used (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Experimentally determined high-temperature lattice parameters were adopted, as reported by Stefancic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [18] for GaV4S8 and by Pocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' for GaNb4Se8 and GaTa4Se8 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The influence of the surrounding bulk atoms was modelled by a finite point charge field (PCF) generated through the Ewald program [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A buffer region of 60 capped effective core potentials (cECPs) was set up between the quantum cluster and PCF (indicated by the smaller atoms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1) (for details, see Supplemental Material [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' As initial step in our study, quasi-restricted or- bitals (QROs [22]) were generated from an unrestricted Kohn-Sham B3LYP calculation for a single-configuration S = 5/2 state with initial-guess orbitals from Hueckel theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The Hueckel guess ensures that the QROs ful- fill Td point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Subsequently, 12 [M4]13+ molecular orbitals around the HOMO-LUMO gap were identified from the QROs and used as a starting point for complete active space self-consistent field (CASSCF) cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Major convergence problems as encountered in earlier quantum chemical studies [17] are circumvented in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The valence-space multiplet structure was de- rived from state averaged (SA) CASSCF optimizations with those 12 orbitals and seven valence electrons defin- ing the active space (denoted in quantum chemistry as (7e,12o) CASSCF), consequently corrected for dynamical correlation by N -electron valence 2nd order perturbation theory (NEVPT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Both methods were accelerated by the resolution of identity (RI [23]) and chain-of-spheres (COS [24]) approximations for Coulomb and exchange in- tegrals with automatically generated auxiliary basis sets [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' All calculations were done using the program pack- age Orca, v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='3 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' High-temperature, tetrahedral-symmetry multiplet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The M4-tetrahedron multiplet structure, as computed for the high-temperature F¯43m cubic lattice arrangement of group-V lacunar spinels, is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For GaV4S8 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' (a) ), the (7e,12o) CASSCF yields a high-spin (S = 3/2) ground state but this is corrected in the subsequent NEVPT2 treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The near degeneracy of low- and high-spin states is an effect often seen in 3d systems due to the similar magnitude of Coulomb interactions and crystal-field splittings [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' in solid-state context, a well-known example is LaCoO3 (see [28] for a quantum chemical investigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Spin-crossover effects were also observed in DMFT calculations on GaV4S8 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In contrast, for the Nb- (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 2(b)) and Ta-based materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 2(c)), the CASSCF(7e,12o) methodology already ensures a reasonably good description — the NEVPT2 scheme provides now only minor corrections to the relative energies (see Supplemental Material [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In all three compounds, the 2T2 state constitutes the ground state, with a dominant a2 1e4t1 2 electronic configu- ration, where a1, e, and t1 are symmetry-adapted (Td point group), molecular-like orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' SOC splits the degenerate 2T2 components into a Jeff = 3/2 ground and a Jeff = 1/2 excited state in all instances, with the magnitude of this splitting increasing from 12 meV (3d) to 97 meV (4d) and 345 meV (5d ions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The sepa- ration between the ground 2T and the first excited 4T state also increases for heavier transition-metal species: 41 meV in GaV4S8, 475 meV in GaNb4Se8, and 598 meV in GaTa4Se8, which reflects the stronger ligand fields ex- perienced by electrons in more extended (4d and 5d) or- bitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Ground-state correlations in cubic group-V spinels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A major difference between how the single-tetrahedron electronic structure is presently depicted in the litera- ture and our quantum chemical results is the composi- tion of the ground-state 2T2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Different from the 100% a2 1e4t1 2 ground state assumed so far on the basis of DFT computations for these materials, we find weights of 72% in GaTa4Se8, 65% in GaNb4Se8, and as little as 21% in GaV4S8 for the a2 1e4t1 2 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Double- excitations into higher-lying t1 and t2 levels represent further configurations contributing to the ground-state wave-functions, in each case with an overall weight of ≲10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The weight of the dominant 4T excited-state configuration a2 1e3t2 2 follows the same trend, with ra- tios of 24%, 69%, and 77% for GaV4S8, GaNb4Se8, and GaTa4Se8, respectively (see also Supplemental Material [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The much more pronounced multiconfigurational char- acter of the vanadate ground-state wave-function in the symmetry-adapted orbital basis is also reflected in the 3 0 200 400 600 800 1000 1200 CASSCF SOC (b) [Nb4Se28Ga6]25- 2T 4T 4T 2A 2E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='2A 2T 2E 6A 0 200 400 600 800 1000 1200 1400 1600 CASSCF SOC (c) [Ta4Se28Ga6]25- 2T 4T 4T 2T 2E 2A 2T 6A 2E 2A 0 50 100 150 200 250 CASSCF NEVPT2 SOC Energy (meV) (a) [V4S28Ga6]25- S = 5/2 S = 3/2 S = 1/2 2T 2T 2A 2E 2A 4T 4T 6E 6A 6E 6A 4T 4T 2E 2A 2A Jeff 1/2 3/2 Jeff 3/2 1/2 Jeff 3/2 1/2 2T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Calculated excitation energies for (a) [V4S28Ga6]25–, (b) [Nb4Se28Ga6]25–, and (c) [Ta4Se28Ga6]25– embedded clusters (CAS(7e,12o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' States corresponding to different spin multiplicities (S = 1/2, 3/2, 5/2) are shown in green, red, and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The corrections brought by NEVPT2 are minor for the Nb- and Ta-based compounds and therefore not depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' correlation index proposed by Ramos-Cordoba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [29] as measure for the extent of near-degeneracy effects (those are also referred to as nondynamical correlation in quantum chemistry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Using the (7e,12o) CASSCF natural-orbital occupation numbers, we derived ‘nondy- namical’ correlation indices IND [29] ranging from ≈2 in vanadates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='95 for GaV4S8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='97 for GaV4Se8) to ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='05 in the 4d variants (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='05 for GaNb4S8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='07 for GaNb4Se8) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='82 in GaTa4Se8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' To put this in per- spective, along the H2 dissociation curve, IND evolves from less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1 at equilibrium distance to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='5 towards dissociation [29], although a quantitative comparison of IND values between chemically different systems is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Looking for further insight, we re-expressed the many- body ground-state wave-functions in terms of atomic- like, single-site orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The orbital localization mod- ule available in Orca was employed for this purpose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' from the 12 symmetry-adapted orbitals (1×a1, 1×e, 1×t1, 2×t2) obtained in the (7e,12o) CASSCF we arrive then to 12 t2g-like functions (three per transition-metal ion, see Supplemental Material [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For both orbital bases, symmetry-adapted or site-centered, the weights of the leading configurations are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 3, for GaV4S8, GaV4Se8, GaNb4S8, GaNb4Se8, GaTa4Se8, and an A-site-substituted member of the family (see also Sup- plemental Material [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Interestingly, a low weight of the a2 1e4t1 2 configuration in the symmetry-adapted orbital basis is associated with large weight of the t1 2gt2 2gt2 2gt2 2g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=', V4+V3+V3+V3+) configurations in the localized- orbital representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' for the AlV4S8, GaV4S8, and GaV4Se8 vanadates, in particular, 15-20% a2 1e4t1 2 trans- lates into 85-90% configurations of t1 2gt2 2gt2 2gt2 2g type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The remaining part stems mainly from triply-occupied transition-metal centers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=', excited-state configura- tions of t1 2gt2 2gt1 2gt3 2g (V4+V3+V4+V2+) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Weights of only ≈9% for the latter in the case of the V-based lacunar spinels indicate much stronger correlations: in- tersite fluctuations are strongly suppressed, compared to the 4d and 5d compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In a Mott-Hubbard picture, stronger localization is the result of larger U/t ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In other words, correlations are moderate in the 4d and 5d GaV4S8 GaNb4Se8 GaTa4Se8 72% 5d GaV4Se8 GaNb4S8 3d 4d 65% 67% 15% 21% weight of a1 e4 t2 weight of t2g t2g t2g t2g 88% 89% 54% 55% 48% AlV4S8 89% 18% 2 2 2 2 1 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Polar plot with weights for the leading a2 1e4t1 2 (symmetry-adapted orbital basis, in red) and t1 2gt2 2gt2 2gt2 2g (localized-orbital basis, in blue) electronic configurations in the ground-state CASSCF wave-function for group-V lacunar- spinel compounds (CAS(7e,12o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 4 compounds and strong in the vanadates — for the latter, an expansion in terms of four V4+V3+V3+V3+ resonat- ing valence structures [30] already provides a reasonably good description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' To analyze in detail how correlations evolve from 3d to 4d and 5d ions for the same type of leading ground-state configuration and in the same crystallographic setting is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 3d5 (Mn2+, Fe3+) species, for example, tend to adopt a t3 2ge2 g ground-state electron configuration, while 4d5 (Ru3+, Rh4+) and 5d5 (Ir4+) varieties display a t5 2g valence-orbital occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Thinking of lower d-shell filling, Mo5+ 4d1 and Os7+ 5d1 ions, for instance, can be found in double-perovskite fcc settings [31], but that is not the case for Ti3+ or V4+ 3d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Here we individualize the group-V lacunar spinels as a unique platform that makes it possible to illustrate how correlations shape many-body wave-functions across a given group of the d block — 3d to 4d and 5d, for the same kind of leading electron configuration and in the same crystallographic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In particular, we pro- vide new, important insights by expressing the many- body M4-tetrahedron wave-function in terms of local- ized, single-ion t2g orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' We show that in the vana- dates strong correlations yield a weight of 85–90% for the t1 2gt2 2gt2 2gt2 2g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=', V4+V3+V3+V3+) configurations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' inter- estingly, with such a reference wave-function, ferromag- netic and antiferromagnetic intra-tetrahedron exchange mechanisms are in rather good balance, which leads to near degeneracy of the low-lying low- and high-spin states — the low-spin (S = 1/2) state is obtained as single- tetrahedron ground-state term only through a rather ad- vanced many-body treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' In contrast, with 4d (Nb) and 5d (Ta) ions, charge fluctuations are much stronger and the weight of t1 2gt2 2gt2 2gt2 2g configurations is reduced to ∼50%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' this reduces ferromagnetic double exchange, es- pecially through reducing the role of M 4+M 3+M 3+M 3+ resonances on the M4 tetrahedron, and firmly stabilizes the low-spin ground-state term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' This result seems to agree with the persistence of the Jeff = 3/2 ground state under pressure even within the metallic state for the 4d and 5d systems [8, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The more extended cluster states, involving significant contributions from ionic configura- tions, also suggest a physical picture for the nonmagnetic ground state in the Nb- and Ta-based lacunar spinels: as these states are prone to stronger magnetic fluctua- tions with larger spatial spread, inter-cluster couplings are able to create spin singlets or valence bond states, as concluded from experiment [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The peculiar pseu- dospin structure must play a role in the superconductiv- ity mechanism under pressure, which is speculated to be unconventional owing to closeness of magnetic states and spin fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Our quantum chemical data provide unparalleled specifics as concerns the correlated electronic structure of the group-V lacunar spinels, well beyond the feature- less a2 1e4t1 2 picture circulated so far in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Stronger correlations in the vanadates imply substan- tially less weight for the a2 1e4t1 2 configuration, compared to the Nb and Ta compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Remarkably, spin-orbit coupling is still effective, even for the vanadates — the predicted fine structure with a splitting of ≈10 meV be- tween the J ≈ 3/2 and J ≈ 1/2 terms should be de- tectable experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The stronger intersite fluctua- tions (M 3+M 3+→M 1+M 4+) and the larger weight (65- 75%) of the a2 1e4t1 2 ‘molecular-orbital’ configuration in the Nb and Ta spinels indicate that the 4d and 5d systems are closer to the Hartree-Fock limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Again, thinking of the H2 molecule, the simplest model to illustrate elec- tronic correlations, the regime of large inter-atomic sepa- ration parallels the case of the vanadate lacunar spinels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Assessing cooperative effects in group-V lacunar spinels through calculation of inter-cluster couplings will require approaches able to incorporate the correlated nature of the M4-cluster ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' We thank I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' K´ezsm´arki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Fulde and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Morrow for discussions and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Nitzsche for technical assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' This work was supported by the German Research Foundation (Deutsche Forschungsge- meinschaft, DFG), Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 437124857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' carried out the quantum chemistry calculations with assistance from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' All authors were involved in writing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' planned the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Geirhos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Langmann, L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 36, 750 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Truhlar, Valence bond theory for chemical dynam- ics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 28, 73 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Supplemental Material: Luxuriant correlation landscape in lacunar spinels: multiconfiguration expansions in molecular-orbital basis vs resonant valence structures Thorben Petersen1,∗ Ulrich K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' R¨oßler1, and Liviu Hozoi1† 1Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, Helmholtzstraße 20, D-01069, Dresden, Germany I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' EMBEDDED CLUSTER PROTOCOL The quantum mechanical cluster model [M 4X 28A6]25– (with M = V, Nb, Ta X = S, Se and A = Al, Ga) was embedded in a field of point charges (PCs), which was created using the Ewald program [1, 2] and experimentally determined geometries for the high-temperature cubic phase [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' To initialize the Ewald optimization, the initial charges given in Supplementary Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' I were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Those were determined under two constraints: (1) the difference between the modified charges of the reciprocal unit cell and the formal charges necessary for extracting the quantum cluster is zero and (2) the difference between these initial charge values and to atomic charges of the quantum cluster fitted to the molecular electrostatic potential (CHELPG, charges from electrostatic potentials using a grid-based method [7–10]) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Between optimized PCs and quantum cluster, a total of 60 atoms (48 M, 12 Xi) were equipped with pseudopotentials for V from Dolg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [11], for Nb and Ta from from Andrae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [12], and for S/Se from Bergner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For GaNb4Se8 and GaTa4Se8 in particular, also smaller [M 4Se16]19– cluster models were used to make computationally demanding calculations (NEVPT2 in particular) feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Here, the outermost Ga (6 atoms) and Seo (12 atoms) were assigned cECPs from Leininger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [14] and Bergner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [13] with the appropriate charges given in Supplementary Table I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Supplementary Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Initial point charge values for the employed quantum cluster models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Xi and Xo refer to X ligand atoms inside and outside of the V4/Nb4/Ta4 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Additionally, the reference used for the HT-phase crystal structure is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Compound Quantum cluster Ga/Al V/Nb/Ta Si/Sei So/Seo Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' HT-cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' GaV4S8 [V4S28Ga6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='82 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='62 [3] GaV4Se8 [V4Se28Ga6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='82 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='62 [3] AlV4S8 [V4S28Al6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='82 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='62 [4] GaNb4S8 [Nb4S28Ga6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='80 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='60 [5] GaNb4Se8 [Nb4Se28Ga6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='80 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='60 [6] [Nb4Se16]19– 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='75 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 [6] GaTa4Se8 [Ta4Se28Ga6]25– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='80 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='60 [6] [Ta4Se16]19– 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='75 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='00 [6] II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' COMPUTATIONAL METHODS The complete active space self-consistent field (CASSCF) approach is fully ab initio – therefore, the basis sets of the atoms in the quantum cluster are crucial to chose appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' While large basis sets are computationally too demanding, small basis sets do not yield a satisfactorily accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' A sufficient accuracy-speed trade-off was found ∗ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='petersen@ifw-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='de † l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='hozoi@ifw-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='03392v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='str-el] 9 Jan 2023 2 for mainly triple-ζ valence polarized (TZVP) basis sets with an additional polarization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For each atomic species, the employed basis sets are given in the following Supplementary Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Since a strong effect of spin-orbit coupling (SOC) on the electronic states was found in both GaNb4Se8 and GaTa4Se8 [15–17], we additionally enable the Douglas-Kroll-Hess (DKH) approximation [18, 19] and used appropriately decontracted basis set variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Supplementary Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Basis set per atom of each quantum cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' X1 (with X = S, Se) refer to atoms directly bonding to the [M4]13+ cluster (16 atoms), while X2 atoms are the outermost atoms bonding to Ga/Al (12 atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' “�” denotes the overall number of contracted basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The reference citation is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Quantum cluster Atom Basis set Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Quantum cluster Atom Basis set Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[V4S28Ga6]25– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='cc-pVTZ-DK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[V4Se28Ga6]25– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='cc-pVTZ-DK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='cc-pVTZ-DK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[21] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='Se1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='cc-pVTZ-DK ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='Se2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='DKH-DEF2-TZVPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='Se2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='DKH-DEF2-TZVPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='[23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1368 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='In Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1, the active space orbitals of the employed CAS(7e,12o) are depicted in natural (1(a)) and localized orbital (1(b)) representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For the former, the corresponding irreducible representations according to Td point group symmetry are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 3 a1 e e t2 t2 t2 t2 t2 t2 t1 t1 t1 (a) (b) dyz (V1) dxz (V1) dxy (V1) dyz (V2) dxz (V2) dxy (V2) dyz (V3) dxz (V3) dxy (V3) dyz (V4) dxz (V4) dxy (V4) Supplementary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Active space orbitals used in CAS(7e,12o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' (a) Natural orbital representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' (b) Localized orbital representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' LOW-ENERGY EXCITATION ENERGIES OF GROUP V LACUNAR SPINELS In the following Supplementary Tables III, IV and V the calculated low-energy excitation energies for an active space CASSCF(7e,12o) and subsequent NEVPT2 correction are given for the three lacunar spinel compounds investigated in detail in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Note, that for 4d-GaNb4Se8 and 5d-GaTa4Se8, a smaller quantum cluster size was chosen in order to make the NEVPT2 correction computationally feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Supplementary Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Excitation energies (meV) of GaV4S8 using a [V4S28Ga6]25– cluster model (CAS(7e,12o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Three sextets, six quartets and seven doublets were included in the state-average procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Notation according to Td point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For each leading configuration its respective weight in the overall wavefunction is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' State Leading config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' ECASSCF ENEVPT2 ESOC NEVPT2 2T2 a2 1 e4 t1 2 t0 1 t0 2 (21%) 134 0 0 (J = 3/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 12 (J = 1/2) 4T1 a2 1 e3 t2 2 t0 1 t0 2 (25%) 0 41 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 50 4T1 a2 1 e3 t2 2 t0 1 t0 2 (24%) 44 82 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 85,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 88,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 89 6E a1 1 e3 t3 2 t0 1 t0 2 (47%) 31 125 129 (3 KD’s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 130 (3 KD’s) 6A2 a2 1 e2 t3 2 t0 1 t0 2 (36%) 93 143 147 (3 KD’s) 2A1 a2 1 e3 t2 2 t0 1 t0 2 (25%) 90 164 168 2E a2 1 e3 t2 2 t0 1 t0 2 (23%) 98 173 177 (2 KD’s) 2A1 a2 1 e3 t2 2 t0 1 t0 2 (25%) 140 207 212 4 Supplementary Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Excitation energies (meV) of GaNb4Se8 using a large [Nb4Se28Ga6]25– and a small [Nb4Se16]19– cluster model (CAS(7e,12o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The later was used to make highly-demanding NEVPT2 calculations feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Three sextets, six quartets and seven doublets were included in the state-average procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Notation according to Td point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For each leading configuration its respective weight in the overall wavefunction is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [Nb4Se28Ga6]25– [Nb4Se16]19– State Leading config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='ECASSCF ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='2 (63%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1059 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1053 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1059 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='1107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='Supplementary Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Excitation energies (meV) of GaTa4Se8 using a large [Ta4Se28Ga6]25– and a small [Ta4Se16]19– cluster model (CAS(7e,12o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' The later was used to make highly-demanding NEVPT2 calculations feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Three sextets, six quartets and seven doublets were included in the state-average procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Notation according to Td point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' For each leading configuration its respective weight in the overall wavefunction is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [Ta4Se28Ga6]25– [Ta4Se16]19– State Leading config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' ECASSCF ESOC CASSCF ECASSCF ENEVPT2 ESOC NEVPT2 2T2 a2 1 e4 t1 2 t0 1 t0 2 (71%) 0 0 (J = 3/2) 0 0 0 (J = 3/2) 345 (J = 1/2) 347 (J = 1/2) 4T2 a2 1 e3 t2 2 t0 1 t0 2 (77%) 639 598−717 629 632 588−683 4T1 a2 1 e3 t2 2 t0 1 t0 2 (77%) 864 880−1042 859 811 834−993 2T1 a2 1 e3 t2 2 t0 1 t0 2 (75%) 994 1046,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1192 985 912 979,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1104 2A2 a2 1 e3 t2 2 t0 1 t0 2 (77%) 1114 1261 1111 1128 1289 2E a2 1 e3 t2 2 t0 1 t0 2 (75%) 1128 1319 1123 1104 1274 2A1 a2 1 e3 t2 2 t0 1 t0 2 (69%) 1139 1305 1132 1068 1234 2T2 a2 1 e3 t2 2 t0 1 t0 2 (76%) 1159 1234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1370 1153 1011 1128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 1208 6A1 a2 1 e2 t3 2 t0 1 t0 2 (83%) 1177 1293 1156 1093 1209 2E a2 1 e3 t2 2 t0 1 t0 2 (71%) 1296 1587 1289 1251 1541 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' LEADING GROUND STATE CONFIGURATIONS OF GROUP-V LACUNAR SPINELS In the following Supplementary Table VI, the composition of the ground state 2T2 term for each of the investigated lacunar spinel compounds in terms of electronic configurations with associated weights are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' These electronic configurations are given based on natural (molecular-like picture) and localized (atomic-like) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' From both representations, it becomes clear that the 3d-vanadate and the 4d/5d-lacunar spinels form two groups that differ strongly in the amount of mixing of the leading a2 1e4t1 2t0 1t0 2 (in molecular orbitals) or t1 2gt2 2gt2 2gt2 2g (in localized orbitals) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' 5 Supplementary Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Leading electronic configurations and associated weights of the ground state term in the investigated group-V compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Both natural (molecular-like) and localized (atomic-like) orbital basis are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Compound Molecular Localized Compound Molecular Localized GaV4S8 a2 1e4t1 2t0 1t0 2 (21%) t1 2gt2 2gt2 2gt2 2g (88%) GaV4Se8 a2 1e4t1 2t0 1t0 2 (18%) t1 2gt2 2gt2 2gt2 2g (89%) a1 1e3t3 2t0 1t0 2 (6%) t1 2gt2 2gt1 2gt3 2g (11%) a1 1e3t3 2t0 1t0 2 (6%) t1 2gt2 2gt1 2gt3 2g (9%) a2 1e2t1 2t1 1t1 2 (3%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' a2 1e2t1 2t1 1t1 2 (3%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' AlV4S8 a2 1e4t1 2t0 1t0 2 (19%) t1 2gt2 2gt2 2gt2 2g (89%) GaNb4S8 a2 1e4t1 2t0 1t0 2 (66%) t1 2gt2 2gt2 2gt2 2g (54%) a1 1e3t3 2t0 1t0 2 (6%) t1 2gt2 2gt1 2gt3 2g (9%) a2 1e2t3 2t0 1t0 2 (5%) t1 2gt2 2gt1 2gt3 2g (36%) a1 1e4t2 2t0 1t0 2 (3%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' a2 1e2t2 2t1 1t0 2 (3%) t0 2gt2 2gt2 2gt3 2g (5%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' GaNb4Se8 a2 1e4t1 2t0 1t0 2 (64%) t1 2gt2 2gt2 2gt2 2g (56%) GaTa4Se8 a2 1e4t1 2t0 1t0 2 (71%) t1 2gt2 2gt2 2gt2 2g (48%) a2 1e2t3 2t0 1t0 2 (6%) t1 2gt2 2gt1 2gt3 2g (36%) a2 1e2t3 2t0 1t0 2 (6%) t1 2gt2 2gt1 2gt3 2g (39%) a2 1e2t2 2t1 1t0 2 (3%) t1 2gt3 2gt0 2gt3 2g (7%) a2 1e2t2 2t1 1t0 2 (3%) t1 2gt1 2gt1 2gt4 2g (9%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE1T4oBgHgl3EQfuwWT/content/2301.03392v1.pdf'} +page_content=' Klintenberg, S.' metadata={'source': 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STUDY +LEONID DOLINSKYI AND YAN DOLINSKY +Abstract. We consider the Bachelier model with linear price impact. Expo- +nential utility indifference prices are studied for vanilla European options in +the case where the investor is required to liquidate her position at the maturity +date. Our main result is establishing a non-trivial scaling limit for a vanishing +price impact which is inversely proportional to the risk aversion. We compute +the limit of the corresponding utility indifference prices and find explicitly a +family of portfolios which are asymptotically optimal. +Mathematical Subject Classification (2010): 91B16, 91G10, 60H30 +Keywords: exponential utility, linear price impact, optimal liquidation +1. Introduction +In financial markets, trading moves prices against the trader: buying faster in- +creases execution prices, and selling faster decreases them. This aspect of liquidity, +known as market depth Black (1986) or price-impact, has received large attention +in optimal liquidation problems, see, for instance, Almgren & Chriss (2001), Schied +et al. (2009), Gatheral & Schied (2011), Bayrakatar & Ludkovski (2014), Bank & +Voß (2019), Fruth et al. (2019), and the references therein. +In this paper we consider the problem of optimal liquidation for exponential +utility function in the Almgren–Chriss model Almgren & Chriss (2001) with linear +temporary impact for the underlying asset. We compute the asymptotic behavior +of the exponential utility indifference prices where the risk aversion goes to infinity +at a rate which is inversely proportional to the linear price impact which goes to +zero. In addition we provide a family of asymptotically optimal hedging strategies. +The main motivation for the study of the asymptotic behaviour of utility indiffer- +ence prices is that in the presence of price impact, super–replication is prohibitively +costly, see Guasoni & Rasonyi (2015). Namely, in the presence of price impact, even +in market models such as the Bachelier model or the Black–Scholes model (which +are complete in the frictionless setup) there is no practical way to construct a hedg- +ing strategy which eliminates all risk from a financial position. This brings us to +utility indifference pricing. +We divide the proof of our main result (Theorem 2.1) into two main steps: +the proof of the lower bound and the proof of the upper bound. +In the proof +of the lower bound we apply Theorem 2.2 from Dolinsky (2022) which gives a +dual representation of the certainty equivalent for the case where the investor has +to liquidate her position. +This dual representation together with the Brownian +structure allows to compute the scaling limit of the utility indifference prices. The +Date: January 5, 2023. +YD Supported in part by the GIF Grant 1489-304.6/2019 and the ISF grant 230/21. +1 + +2 +proof of the upper bound is done by an explicit construction of a family of portfolios +which are asymptotically optimal. +The rest of the paper is organized as follows. In the next section we introduce +the setup and formulate the main results. In Section 3 we prove the lower bound. +In Section 4 we prove the upper bound. In Section 5 we derive an auxiliary result +from the field of deterministic variational analysis. +2. Preliminaries and Main Results +Let T < ∞ be the time horizon and let W = (Wt)t∈[0,T ] be a standard one dimen- +sional Brownian motion defined on the filtered probability space (Ω, F, (Ft)t∈[0,T ], P) +where the filtration (Ft)t∈[0,T ] satisfies the usual assumptions (right continuity and +completeness). We consider a simple financial market with a riskless savings ac- +count bearing zero interest (for simplicity) and with a risky asset S = (St)t∈[0,T ] +with Bachelier price dynamics +(2.1) +St = S0 + µt + σWt +where S0 ∈ R is the initial position of the risky asset, µ ∈ R is the constant drift +and σ > 0 is the constant volatility. +Following Almgren & Chriss (2001), we model the investor’s market impact, in a +temporary linear form and, thus, when at time t the investor turns over her position +Φt at the rate ˙Φt := dΦt +dt the execution price is St + Λ +2 ˙Φt for some constant Λ > 0. +The portfolio value at the maturity date is given by +(2.2) +V Φ +T := +� T +0 +ΦtdSt − Λ +2 +� T +0 +˙Φ2 +tdt. +In our setup the investor has to liquidate her position. Thus, the natural class of +admissible strategies which we denote by A is the set all progressively measurable +processes Φ = (Φt)t∈[0,T ] with differentiable trajectories such that +� T +0 +˙Φ2 +tdt < ∞ +and ΦT = 0 almost surely. We assume that the initial number of shares Φ0 is fixed. +Consider a vanilla European option with the payoff X = f(ST ) where f is of the +form +(2.3) +f(x) = max (0, Θ (x − K)) , +x ∈ R +for some constants Θ, K ∈ R. Observe that this form includes call/put options. +The investor will assess the quality of a hedge by the resulting expected utility. +Assuming exponential utility with constant absolute risk aversion α > 0, the util- +ity indifference price and the certainty equivalent price of the claim X (see, e.g., +Carmona (2009) for details on indifference prices) do not depend on the investor’s +initial wealth and, respectively, take the well-known forms +(2.4) +π(Λ, α, Φ0, X) := 1 +α log +� +infφ∈A EP +� +exp +� +α +� +X − V Φ +T +��� +infφ∈A EP +� +exp +� +−αV Φ +T +�� +� +and +c(Λ, α, Φ0, X) := 1 +α log +� +inf +φ∈A EP +� +exp +� +α +� +X − V Φ +T +���� +. +If the risk aversion α > 0 is fixed, then by applying standard density arguments +we obtain that for Λ ↓ 0, the above indifference price converges to the unique +price of the continuous time complete (frictionless) market given by (2.1). A more +interesting limit emerges, however, if we re-scale the investor’s risk-aversion in the + +3 +form α := A/Λ. Hence, we fix A > 0 and consider the case where the risk aversion +is α(Λ) := A +Λ . +Before we formulate the main result we need some preparations. Introduce the +functions +(2.5) g(x) := sup +y∈R +� +f(x + y) − +y2 +4σ +√ +A +� += max +� +0, Θ (x − K) + σ +√ +AΘ2� +, +x ∈ R +and +u(t, x) := EP [g(x + σWT −t)] , +(t, x) ∈ [0, T ] × R. +The term u(t, St) represents the price at time t of a European option with the +payoff g(ST ) in the complete market given by (2.1). It is well known that u ∈ +C1,2([0, T ) × R) solves the PDE +(2.6) +∂u +∂t + σ2 +2 +∂2u +∂x2 = 0 +in [0, T ) × R. +Next, let Λ > 0 and let +ρ = ρ(Λ) := σ2α(Λ) +Λ += σ2A +Λ2 +be the risk-liquidity ratio. Consider the (random) ODE on the interval [0, T ] +˙Ft = √ρ +� +cosh(√ρ(T −t)) +2 cosh2� √ρ(T −t) +2 +� ∂u +∂x(t, St − σ +√ +AFt) − tanh(√ρ(T − t))Ft +� +, +(2.7) +with the initial condition F0 = Φ0 coth(√ρT ). +From the linear growth of g it follows that for any ǫ > 0 the functions +∂u +∂x, ∂2u +∂x2 +are uniformly bounded in the domain [0, T − ǫ] × R. In particular ∂u +∂x is Lipschitz +continuous with respect to x in the domain [0, T −ǫ]×R. Observe that the functions +cosh(√ρ(T −t)) +2 cosh2� √ρ(T −t) +2 +�, tanh(√ρ(T − t)) are bounded. Hence, from the standard theory +of ODE (see Walter (1998), Chapter II, Section 6) we obtain that there exists a +unique solution to (2.7) which we denote by F Λ = (F Λ +t )t∈[0,T ) and the solution is +Lipschitz continuous, and so limt→T − F Λ +t exists. Set F Λ +T := limt→T − F Λ +t and define +(2.8) +ΦΛ +t := tanh +�� +ρ(Λ)(T − t) +� +F Λ +t , +t ∈ [0, T ]. +Theorem 2.1. For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- +aversion A/Λ with A > 0 fixed, the certainty equivalent of X = max (0, Θ (ST − K)) +has the scaling limit +(2.9) +lim +Λ↓0 c(Λ, A/Λ, Φ0, X) = u +� +0, S0 − σ +√ +AΦ0 +� ++ σ +√ +AΦ2 +0 +2 +. +Moreover, the trading strategies given by (2.8) are asymptotically optimal, i.e. +(2.10) lim +Λ↓0 +Λ +A log +� +EP +� +exp +�A +Λ +� +X − V ΦΛ +T +���� += u +� +0, S − σ +√ +AΦ0 +� ++ σ +√ +AΦ2 +0 +2 +. +From Theorem 2.1 we obtain immediately the following corollary which says that +the asymptotic value of the utility indifference prices is equal to the price of the +vanilla European option with the payoff g(ST ) and the shifted initial stock price +S0 − σ +√ +AΦ0. + +4 +Corollary 2.2. For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- +aversion A/Λ with A > 0 fixed, the utility indifference price of X has the scaling +limit +lim +Λ↓0 π(Λ, A/Λ, Φ0, X) = u +� +0, S0 − σ +√ +AΦ0 +� +. +Proof. Apply (2.8) and take X ≡ 0 for the denominator of (2.4). +□ +Remark 2.3. In the proof of the lower bound (given in the next section) we only +assume that the payoff function f is Lipschitz continuous. By a more careful anal- +ysis we can prove that in fact there is an equality, namely (2.9) holds true for any +payoff function X = f(ST ) with a Lipschitz continuous f. Unfortunately, the proof +of (2.10) (given in Section 4) uses the specific structure of the payoff given by (2.3). +This together with the fact that the most common vanilla options in real markets +are of the form (2.3) led us to assume from the beginning that the payoff is of this +form. +Let us emphasize that our results can be extended to the multi–asset case with +a similar proof. +In the multi asset case the volatility σ is replaced with a posi- +tive definite matrix and the functions coth and tanh are viewed as matrix valued +functions. +Remark 2.4. Theorem 2.1 can be viewed as an extension of the main result in +Dolinsky & Moshe (2022), for the case where the investor has to liquidate her +portfolio at the maturity date. In both cases (with or without liquidation) the scaling +limit of the utility indifference prices is equal to E [h (x + σWT )] for a modified +function h. In the present paper +h(x) = sup +y∈R +� +f(x + y) − +y2 +4σ +√ +A +� +while in Dolinsky& Moshe (2022) the modified payoff is smaller and given by +h(x) = sup +y∈R +� +f(x + y) − +y2 +2σ +√ +A +� +. +Next, we discuss the constructed asymptotically optimal portfolios. From (2.8) +we have +(2.11) +˙ΦΛ +t = +� +ρ(Λ) +� +tanh +�� +ρ(Λ) (T − t) +2 +� +ΥΛ +t − coth +�� +ρ(Λ) (T − t) +� +ΦΛ +t +� +where ρ(Λ) := σ2A +Λ2 +and ΥΛ +t := ∂u +∂x(t, St − σ +√ +AF Λ +t ), t ∈ [0, T ). +Thus, we have a mean reverting structure which combines trucking the ∆–hedging +strategy of a modified claim g and liquidating the position at the maturity date. As +time t approaches maturity the weight of the ∆–hedging trading strategy becomes +smaller and the investor trading is mainly towards liquidation. This is in contrast +to the asymptotically optimal portfolios in Dolinsky and & Moshe (2022) which are +just based on trucking the appropriate ∆–hedging strategy. +3. Proof of the Lower Bound +In this section we prove the following statement. + +5 +Proposition 3.1. For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- +aversion A/Λ with A > 0 fixed, we have the following lower bound +lim inf +Λ↓0 c(Λ, A/Λ, Φ0, X) ≥ u +� +0, S0 − σ +√ +AΦ0 +� ++ σ +√ +AΦ2 +0 +2 +. +We start with the following Lemma. +Lemma 3.2. Denote by Γ the set of all progressively measurable processes θ = +(θt)t∈[0,T ] such that θ ∈ L2(dt ⊗ P) and let M be the set of all P–martingales +M = (Mt)t∈[0,T ) which are defined on the half-open interval [0, T ) and satisfy +||M||L2(dt⊗P) := EP +�� T +0 M 2 +t dt +� +< ∞. Then, for any Λ, α > 0 we have +c(Λ, α, Φ0, X) +≥ sup(θ,M)∈Γ×M EP +� +f +� +ST + σ +� T +0 θtdt +� +− +1 +2α +� T +0 θ2 +t dt +−Φ0(M0 − S0) − +1 +2Λ +� T +0 +���S0 + µt + σ � t +0 θsds − Mt +��� +2 +dt +� +. +Proof. Denote by Q the set of all equivalent probability measures Q ∼ P with finite +entropy EQ +� +log +� dQ +dP +�� +< ∞ relative to P. For any Q ∈ Q let MQ be the set of all +Q–martingales M Q = (M Q +t )t∈[0,T ) which are defined on the half-open interval [0, T ) +and satisfy ||M Q||L2(dt⊗Q) := EQ +�� T +0 |M Q +t |2dt +� +< ∞. +From the linear growth of f it follows that EP +� +eαX� +< ∞. Thus, define the +probability measure ˜P by d˜P +dP := +eαX +EP[eαX]. The Cauchy–Schwarz inequality yields that +there exists a > 0 such that E˜P +� +exp +� +a sup0≤t≤T S2 +t +�� +< ∞. Hence, Assumption 2.1 +in Dolinsky (2022) holds true. Thus, by applying Theorem 2.2 in Dolinsky (2022) +for the probability measure ˜P and the simple equality +EQ +� +log +�dQ +d˜P +�� += EQ +� +log +�dQ +dP +� +− αX +� ++ α log +� +EP +� +eαX�� +∀Q ∈ Q +we obtain +c(Λ, α, Φ0, X) +(3.1) += supQ∈Q supMQ∈MQ EQ +� +X − 1 +α log +� dQ +dP +� +− Φ0(M Q +0 − S0) − +1 +2Λ +� T +0 |M Q +t − St|2dt +� +. +Next, Let C[0, T ] be the space of continuous functions z : [0, T ] → R equipped +with the uniform norm ||z|| := sup0≤t≤T |zt|. +Denote by ˆΓ ⊂ Γ the set of all +continuous and bounded processes θ = (θt)t∈[0,T ] of the form θ = τ(W) where +τ : C[0, T ] → C[0, T ] is Lipschitz continuous and predictable (i.e. τt(x) = τt(y) if +x[0,t] = y[0,t]). From standard density arguments and the Lipschitz continuity of f +it follows that in order to complete the proof of the Lemma it is sufficient to show +that for any (θ, M) ∈ ˆΓ × M we have +c(Λ, α, Φ0, X) +≥ EP +� +f +� +ST + σ +� T +0 θtdt +� +− +1 +2α +� T +0 θ2 +t dt +(3.2) +−Φ0(M0 − S0) − +1 +2Λ +� T +0 +���S0 + µt + σ +� t +0 θsds − Mt +��� +2 +dt +� +. + +6 +To this end let (θ, M) ∈ ˆΓ × M such that θ = τ(W) where τ as above. Consider +the stochastic differential equation (SDE) +(3.3) +dYt = dWt − τt(Y )dt, +t ∈ [0, T ] +with the initial condition Y0 = 0. Theorem 2.1 from Chapter IX in Revuz & Yor +(1999) yields that there exists a unique strong solution to the above SDE. From +the Girsanov theorem it follows that there exists a probability measure Q ∈ Q such +that W Q := Y is a Brownian motion with respect to Q. +From (2.1) and (3.3) we obtain that the distribution of (St)t∈[0,T ] under Q is +equal to the distribution of +� +St + σ � t +0 θsds +� +t∈[0,T ] under P. Moreover, +EQ +� 1 +α log +�dQ +dP +�� += EQ +� +1 +2α +� T +0 +τ 2 +t (Y )dt +� += EP +� +1 +2α +� T +0 +θ2 +t dt +� +. +Finally, choose M Q ∈ MQ such that the law of (W Q, M Q) under Q is equal to the +law of (W, M + σW) under P. We conclude, +EQ +� +X − 1 +α +� T +0 log +� dQ +dP +� +− Φ0(M Q +0 − S0) − +1 +2Λ +� T +0 |M Q +t − St|2dt +� += EP +� +f +� +ST + σ +� T +0 θtdt +� +− +1 +2α +� T +0 θ2 +t dt +−Φ0(M0 − S0) − +1 +2Λ +� T +0 +���S0 + µt + σ +� t +0 θsds − Mt +��� +2 +dt +� +. +This together with (3.1) gives (3.2) as required. +□ +Next, we prove the following. +Lemma 3.3. Denote by L2 +0(FT , P) the set of all random variables of the form +(3.4) +Z = ι + +� T +0 +κtdWt +for some ι ∈ R and a predictable and bounded process κ = (κt)t∈[0,T ] such that +κ[T −ǫ,T ] ≡ 0 for some (deterministic) ǫ > 0. Let Z ∈ L2 +0(FT , P). There exists a +constant ˆC > 0 (may depend on Z) such that for any Λ ∈ (0, 1) +sup(θ,M)∈Γ×M EP +� +f +� +ST + σ +� T +0 θtdt +� +− +1 +2α(Λ) +� T +0 θ2 +t dt +−Φ0(M0 − S0) − +1 +2Λ +� T +0 +���S0 + µt + σ � t +0 θsds − Mt +��� +2 +dt +� +≥ EP +� +f (S0 + σWT + Z) − (Z+σ +√ +AΦ0) +2 +4σ +√ +A +� ++ σ +√ +AΦ2 +0 +2 +− ˆCΛ +where, as before α(Λ) = A +Λ. +Proof. Let Z given by (3.4) and let Ξ be the map from Proposition 5.1. Define the +deterministic function ν : [0, T ] → R by ν := ΞT (Λ, ι, Φ0) and for any s < T define +the stochastic process (l·,s)·∈[s,T ] by (l·,s)·∈[s,T ] = ΞT −s(Λ, κs, 0). +Next, introduce (θ, M) ∈ Γ × M +θt := ˙νt−µ +σ ++ 1 +σ +� t +0 +∂lt,s +∂t dWs, +t ∈ [0, T ], +Mt := S0 + +� T +0 νtdt−Φ0Λ +T ++ σ +� t +0 +� +1 + +1 +T −s +� T +s lt,sdt +� +dWs, +t ∈ [0, T ]. + +7 +Observe that from the definition of Ξ we have +ν0 = 0, +νT = ι and ls,s = 0, +lT,s = κs ∀s. +This together with the Fubini theorem, the Itˆo Isometry, (2.1) and (3.4) gives +EP +� +f +� +ST + σ +� T +0 θtdt +� +− +1 +2α(Λ) +� T +0 θ2 +t dt − Φ0(M0 − S0)− +1 +2Λ +� T +0 +���S0 + µt + σ +� t +0 θsds − Mt +��� +2 +dt +� += EP [f (S0 + σWT + Z)] +(3.5) ++ µΛι +σ2A − +µ2Λ +2σ2A − I(Λ, ν) − +� T −ǫ +0 +EP [Js(Λ, l)] ds +where +I(Λ, ν) := +Λ +2σ2A +� T +0 +˙ν2 +t dt + 1 +2Λ + + +� T +0 +ν2 +t dt − 1 +T +� +Φ0Λ − +� T +0 +νtdt +�2 + +and +Js(Λ, l) := +Λ +2σ2A +� T +s +�∂lt,s +∂t +�2 +dt + 1 +2Λ + + +� T +s +l2 +t,sdt − +1 +T − s +�� T +s +lt,sdt +�2 + . +From Proposition 5.1 there exists a constant C > 0 (may depend on ι and κ) such +that +(3.6) +������� +I(Λ, ν) − +� +ι + σ +√ +AΦ0 +�2 +4σ +√ +A ++ σ +√ +AΦ2 +0 +2 +������� +≤ CΛ +and for any s ∈ [0, T − ǫ] +(3.7) +����Js(Λ, l) − +κ2 +s +4σ +√ +A +���� ≤ CΛ. +By combining the Itˆo Isometry and (3.5)–(3.7) we complete the proof. +□ +We now have all the pieces in place that we need for the completion of the +proof of Proposition 3.1. +Proof. Recall the definition of g given in (2.5). From the Lipschitz continuity of f +it follows that there exists a bounded (measurable) function ζ : R → R such that +(3.8) +g(x) = f (x + ζ(x)) − ζ2(x) +4σ +√ +A +, +∀x ∈ R. +Choose a sequence Zn ∈ L2 +0(FT , P), n ∈ N such that +lim +n→∞ Zn = ζ(S0 − σ +√ +AΦ0 + σWT ) − σ +√ +AΦ0 + +8 +where the limit is in L2(P). From Lemmas 3.2–3.3 and (3.8) we obtain +lim infΛ↓0 c (Λ, A/Λ, Φ0, X) +≥ supn∈N EP +� +f (S0 + σWT + Zn) − (Zn+σ +√ +AΦ0) +2 +4σ +√ +A +� ++ σ +√ +AΦ2 +0 +2 +≥ EP +� +g +� +S0 − σ +√ +AΦ0 + σWT +�� ++ σ +√ +AΦ2 +0 +2 += u +� +0, S0 − σ +√ +AΦ0 +� ++ σ +√ +AΦ2 +0 +2 +. +□ +4. Proof of the Upper Bound +In order to complete the proof of Theorem 2.1 it remains to establish the following +result. +Proposition 4.1. Recall the trading strategies ΦΛ, Λ > 0 given by (2.8). Then, +lim sup +Λ↓0 +Λ +A log +� +EP +� +exp +�A +Λ +� +X − V ΦΛ +T +���� +≤ u +� +0, S − σ +√ +AΦ0 +� ++ σ +√ +AΦ2 +0 +2 +. +Proof. The proof will be done in three steps. +Step I: +In this step we use the specific structure of the payoff f given by (2.3). +Let us show that for any Λ > 0 +(4.1) +g +� +ST − σ +√ +AF Λ +T +� +≥ f(ST ) − +σ +√ +A|Φ0Θ| +sinh +�� +ρ(Λ)T +� +where, as before ρ(Λ) := σ2A +Λ2 . +Fix Λ > 0. From (2.7) +d +dt + + +F Λ +t +cosh +�� +ρ(Λ)(T − t) +� + + = +� +ρ(Λ) +2 cosh2 +�√ +ρ(Λ)(T −t) +2 +�ΥΛ +t , +t ∈ [0, T ] +where, recall that ΥΛ +t := ∂u +∂x +� +t, St − σ +√ +AF Λ +t +� +, t ∈ [0, T ). Clearly, |ΥΛ +t | ≤ Θ, and +so, +|F Λ +T | ≤ +���� +F Λ +0 +cosh +�√ +ρ(Λ)T +� +���� + Θ � T +0 +√ +ρ(Λ) +2 cosh2 +� √ +ρ(Λ)(T −t) +2 +�dt +≤ +���� +Φ0 +sinh +�√ +ρ(Λ)T +� +���� + |Θ|. +This together with (2.3) and (2.5) gives (4.1). +Step II: In this step we prove that there exists a constant ˜C > 0 such tht +(4.2) +����� +� T +0 +ΥΛ +t dt − +� T +0 +ΦΛ +t dt +����� ≤ ˜CΛ, +∀Λ > 0. + +9 +Fix Λ > 0. From (2.11) +d +dt +� +ΦΛ +t +cosh +�√ +ρ(Λ)(T −t) +� +� += +√ +ρ(Λ) +2 cosh2 +� √ +ρ(Λ)(T −t) +2 +� tanh +�√ +ρ(Λ)(T −t) +2 +� +ΥΛ +t , +t ∈ [0, T ]. +We get +ΦΛ +t = Φ0 +cosh +�√ +ρ(Λ)(T −t) +� +cosh(√ +ρ(Λ)T ) ++ +� t +0 +√ +ρ(Λ) cosh +�√ +ρ(Λ)(T −t) +� +2 cosh2 +� √ +ρ(Λ)(T −s) +2 +� +tanh +�√ +ρ(Λ)(T −s) +2 +� +ΥΛ +s ds +and so, from the Fubini theorem +� T +0 +ΦΛ +t dt − +� T +0 +ΥΛ +t dt = Φ0 +tanh +�� +ρ(Λ)T +� +� +ρ(Λ) +− +� T +0 +ΥΛ +s +cosh2 +�√ +ρ(Λ)(T −s) +2 +�ds. +This together with the simple integral +� T +0 +ds +cosh2 +�√ +ρ(Λ)(T −s) +2 +� = +2 tanh +�√ +ρ(Λ)T +2 +� +� +ρ(Λ) +and the inequality |ΥΛ +t | ≤ Θ gives (4.2). +Step III: In this step we complete the proof. Fix Λ > 0 and introduce the process +M Λ +t := exp +� +A +Λ +� +u +� +t, St − σ +√ +AF Λ +t +� ++ σ +√ +AF Λ +t ΦΛ +t +2 +− V ΦΛ +t +�� +, +t ∈ [0, T ]. +From the Itˆo formula, (2.2), (2.6)–(2.8) and (2.11) we obtain +dMΛ +t +MΛ +t += A +Λ +� +ΥΛ +t − ΦΛ +t +� +dSt + σ2A2 +2Λ2 +� +ΥΛ +t − ΦΛ +t +�2 dt +− σ2A2 +Λ2 ΥΛ +t + + cosh +�√ +ρ(Λ)(T −t) +� +2 cosh2 +� √ +ρ(Λ)(T −t) +2 +�ΥΛ +t − ΦΛ +t + + dt ++ σ2A2 +2Λ2 +� +tanh +�√ +ρ(Λ)(T −t) +2 +� +ΥΛ +t − coth +�� +ρ(Λ)(T − t) +� +ΦΛ +t +�2 +dt ++ σ2A2 +2Λ2 ΦΛ +t + + cosh +�√ +ρ(Λ)(T −t) +� +2 cosh2 +� √ +ρ(Λ)(T −t) +2 +�ΥΛ +t − ΦΛ +t + + dt ++ σ2A2 +2Λ2 coth +�� +ρ(Λ)(T − t) +� +ΦΛ +t +× +� +tanh +�√ +ρ(Λ)(T −t) +2 +� +ΥΛ +t − coth +�� +ρ(Λ)(T − t) +� +ΦΛ +t +� += A +Λ +� +ΥΛ +t − ΦΛ +t +� +dSt + +10 +where the last equality follows from simple calculations. +Hence, from (2.1) it follows that the process +N Λ +t := exp +� +−µA +� t +0 +� +ΥΛ +t − ΦΛ +s +� +ds +Λ +� +M Λ +t , +t ∈ [0, T ] +is a local–martingale, and so from the obvious inequality N Λ > 0 we conclude that +this process is a super–martingale. +Finally, +Λ +A log +� +EP +� +exp +� +A +Λ +� +X − V ΦΛ +T +���� +≤ Λ +A log +� +EP[M Λ +T ] +� ++ +σ +√ +A|Φ0Θ| +sinh +�√ +ρ(Λ)T +� +≤ Λ +A log +� +EP[N Λ +T ] +� ++ ˜C|µ|Λ + +σ +√ +A|Φ0Θ| +sinh +�√ +ρ(Λ)T +� +≤ Λ +A log +� +N Λ +0 +� ++ ˜C|µ|Λ + +σ +√ +A|Φ0Θ| +sinh +�√ +ρ(Λ)T +� += u +� +0, S0 − σ +√ +AΦ0 coth +�� +ρ(Λ)T +�� ++ +σ +√ +AΦ2 +0 coth +�√ +ρ(Λ)T +� +2 ++ ˜C|µ|Λ + +σ +√ +A|Φ0Θ| +sinh +�√ +ρ(Λ)T +�. +The first inequality follows from (4.1) and the relations u(T, ·) = g(·), ΦΛ +T = 0. The +second inequality is due to (4.2). The super–martingale property of N Λ gives the +third inequality. The equality is due to (2.8). +By taking Λ ↓ 0 we complete the proof. +□ +5. Auxiliary Result +For any T ∈ (0, T ] and x ∈ R let C0,x[0, T] be the space of all continuous functions +z : [0, T] → R which satisfy z0 = 0 and zT = x. +Proposition 5.1. For any T ∈ (0, T ] there exists a measurable map ΞT : (0, 1) × +R2 → C[0, T) such that for any Λ ∈ (0, 1) and x, φ ∈ R the continuous function +ΞT(Λ, x, φ) ∈ C0,x[0, T] is the unique minimizer for the optimization problem +(5.1) +min +δ∈C0,x[0,T] + + +Λ +2σ2A +� T +0 +˙δ2 +t dt + 1 +2Λ + + +� T +0 +δ2 +t dt − 1 +T +� +φΛ − +� T +0 +δtdt +�2 + + + . +Moreover, denote the corresponding value by VT(Λ, x, φ). Then, for any ǫ > 0 and +a compact set K ⊂ R2 there exists a constant ˆC (may depend on ǫ and K) such +that +(5.2) +������� +VT(Λ, x, φ) − +� +x + σ +√ +Aφ +�2 +4σ +√ +A ++ σ +√ +Aφ2 +2 +������� +≤ ˆCΛ, +∀(T, Λ, x, φ) ∈ [ǫ, T ] × (0, 1) × K. +Proof. Fix (T, Λ, x, φ) ∈ [ǫ, T ] × (0, 1) × R2. First we minimize the pattern given +by (5.1) under the additional constraint that +� T +0 δtdt is given. Then, we will find +the optimal +� T +0 δtdt. + +11 +For any y ∈ R let Cy +0,x[0, T] ⊂ C0,x[0, T] be the subset of all functions δ ∈ +C0,x[0, T] which satisfy +� T +0 δtdt = y. Consider the minimization problem +min +δ∈Cy +0,x[0,T] +� T +0 +H( ˙δt, δt)dt +where H(v1, v2) := +Λ +2σ2Av2 +1 + +1 +2Λv2 +2 for v1, v2 ∈ R. This optimization problem is +convex and so it has a unique solution which has to satisfy the Euler–Lagrange +equation (for details see Gelfand & Fomin (1963)) +d +dt +∂H +∂ ˙δt = λ + +d +dt +∂H +∂δt for some +constant λ > 0 (lagrange multiplier due to the constraint +� T +0 δtdt = y). +Thus, +the optimizer which we denote by ˆδ solves the ODE ¨ˆδt − ρˆδ ≡ const (recall the +risk-liquidity ration ρ = ρ(Λ) := σ2A +Λ2 ). From the standard theory it follows that +(5.3) +ˆδt = c1 sinh(√ρt) + c2 sinh(√ρ(T − t)) + c3, +t ∈ [0, T] +for some constants c1, c2, c3. +From the three constraints ˆδ0 = 0, ˆδT = x and +� T +0 ˆδtdt = y we obtain +(5.4) +c1 = +x − c3 +sinh(√ρT), +c2 = − +c3 +sinh(√ρT) and c3 = +√ρy − x tanh(√ρT/2) +√ρT − 2 tanh(√ρT/2). +We argue that +ρ +� T +0 ˆδ2 +t dt + +� T +0 +˙ˆδ2 +t dt = ρ +� T +0 +� +(ˆδt − c3) + c3 +�2 +dt + +� T +0 +˙ˆδ2 +t dt += +√ρ +2 +� +c2 +1 + c2 +2 +� +sinh +� +2√ρT +� +− 2c1c2√ρ sinh(√ρT) − ρc2 +3T + 2ρc3y += √ρx2 coth(√ρT) + 2√ρc1c2 sinh(√ρT) +� +cosh(√ρT) − 1 +� +− ρc2 +3T + 2ρc3y += √ρx2 coth(√ρT) + +� +2√ρ tanh(√ρT/2) − ρT +� +c2 +3 ++2 +� +ρy − √ρ tanh(√ρT/2)x +� +c3 += √ρ +� +x2 coth(√ρT) + (x tanh(√ρT/2)−√ρy) +2 +√ρT−2 tanh(√ρT/2) +� +. +(5.5) +Indeed, the first equality is obvious. The second equality follows from (5.3) and +simple computations. The third equality is due to c1 − c2 = +x +sinh(√ρT). The fourth +equality is due to c1c2 = +c2 +3−xc3 +sinh2(√ρT). The last equality follows from substituting c3. +From (5.5) we conclude that in order to minimize (5.1) we need to find y which +minimizes the quadratic pattern +1 +2√ρΛ +� +x tanh(√ρT/2) − √ρy +�2 +√ρT − 2 tanh(√ρT/2) +− +1 +2ΛT (φΛ − y)2 . +Observe that this quadratic pattern is convex in y and so has a unique minimum +(5.6) +y = xT +2 − φΛ +�√ρT − 2 tanh(√ρT/2) +� +2 tanh(√ρT/2) +. +Thus, define ΞT(Λ, x, φ) := ˆδ where ˆδ is given by (5.3)–(5.4) and (5.6). Clearly, +ΞT(Λ, x, φ) is the unique minimizer for (5.1). + +12 +Let +VT(Λ, x, φ) := +Λ +2σ2A +� T +0 +˙ˆδ2 +t dt + 1 +2Λ + + +� T +0 +ˆδ2 +t dt − 1 +T +� +φΛ − +� T +0 +ˆδtdt +�2 + . +Finally, we prove (5.2). Choose ǫ > 0 and a compact set K ⊂ R2. Assume that +(T, x, φ) ∈ [ǫ, T ] × K. From (5.5) and the equality ρ = σ2A +Λ2 we get that there exists +a constant C1 (may depend on ǫ and K) such that +(5.7) +����VT(Λ, x, φ) − +� +x2 +2σ +√ +A ++ +y2 +σ +√ +AT2 + φy +T − +xy +σ +√ +AT +����� ≤ C1Λ +where y given by (5.6). From (5.6) we have +���y − T +2 +� +x − σ +√ +Aφ +���� ≤ C2Λ for some +constant C2 (may depend on ǫ and K). This together with (5.7) gives (5.2) and +completes the proof. +□ +References +[1] R. Almgren and N. Chriss, Optimal execution of portfolio transactions, Journal of Risk, 3, +5–39, (2001). +[2] F. Black, Noise, Journal of Finance, 41, 529–543, (1986). +[3] E. Bayrakatar and M. Ludkovski, Liquidation in Limit Order Books with Controlled Inten- +sity, Mathematical Finance, 24, 627–650, (2014). +[4] P. Bank and M. Voß, Optimal Investment with Transient Price Impact, SIAM Journal on +Financial Mathematics, 10, 723–768, (2019). +[5] R. Carmona, Indifference pricing: theory and applications, Princeton University Press, series +in Financial Engineering, (2009). +[6] Y. Dolinsky, Duality Theory for Exponential Utility Based Hedging in the Almgren-Chriss +model, submitted, arxiv: 2210.03917, (2022). +[7] Y. Dolinsky and S. Moshe, Utility Indifference Pricing with High Risk Aversion and Small +Linear Price Impact, SIAM Journal on Financial Mathematics, 13, SC-12–SC-25, (2022). +[8] A. Fruth, T. Sch¨oneborn and M. Urusov, Optimal trade execution in order books with sto- +chastic liquidity, Mathematical Finance, 29, 507–541, (2019). +[9] I.M. Gelfand and S.V. Fomin, Calculus of variations, Prentice Hall, International, (1963). +[10] P. Guasoni and M. R´asonyi, Hedging, arbitrage and optimality under superlinear friction, +Annals of Applied Probability, 25, 2066–2095, (2015). +[11] J. Gatheral and A. Schied, Optimal trade execution under geometric Brownian motion in the +Almgren and Chriss framework, International Journal of Theoretical and Applied Finance, +14, 353–368, (2011). +[12] D. Revuz and M. Yor, Continuous martingales and Brownian motion, volume 293 of +Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemati- +cal Sciences]. Springer-Verlag, Berlin, third edition, 1999. +[13] A. Schied, T. Sch¨oneborn and M. Tehranchi, Optimal Basket Liquidation for CARA In- +vestors is Deterministic, Applied Mathematical Finance, 17, 471–489, (2009). +[14] W. Walter, Ordinary Differential Equations, Springer-Verlag New-York, (1998). +DEPARTMENT OF FINANCE, NATIONAL UNIVERSITY OF KYIV-MOHYLA ACAD- +EMY. +E.MAIL: LDOLINSKYI@UKMA.EDU.UA +DEPARTMENT +OF +STATISTICS, +HEBREW +UNIVERSITY +OF +JERUSALEM. +E.MAIL: YAN.DOLINSKY@MAIL.HUJI.AC.IL + diff --git a/VNAzT4oBgHgl3EQfl_0n/content/tmp_files/load_file.txt b/VNAzT4oBgHgl3EQfl_0n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..07da72acebb51b3149a739750828f82eb70ff4fb --- /dev/null +++ b/VNAzT4oBgHgl3EQfl_0n/content/tmp_files/load_file.txt @@ -0,0 +1,453 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf,len=452 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='01555v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='MF] 4 Jan 2023 OPTIMAL LIQUIDATION WITH HIGH RISK AVERSION IN THE ALMGREN–CHRISS MODEL: A CASE STUDY LEONID DOLINSKYI AND YAN DOLINSKY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We consider the Bachelier model with linear price impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Expo- nential utility indifference prices are studied for vanilla European options in the case where the investor is required to liquidate her position at the maturity date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Our main result is establishing a non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We compute the limit of the corresponding utility indifference prices and find explicitly a family of portfolios which are asymptotically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Mathematical Subject Classification (2010): 91B16, 91G10, 60H30 Keywords: exponential utility, linear price impact, optimal liquidation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Introduction In financial markets, trading moves prices against the trader: buying faster in- creases execution prices, and selling faster decreases them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This aspect of liquidity, known as market depth Black (1986) or price-impact, has received large attention in optimal liquidation problems, see, for instance, Almgren & Chriss (2001), Schied et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' (2009), Gatheral & Schied (2011), Bayrakatar & Ludkovski (2014), Bank & Voß (2019), Fruth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' (2019), and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In this paper we consider the problem of optimal liquidation for exponential utility function in the Almgren–Chriss model Almgren & Chriss (2001) with linear temporary impact for the underlying asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We compute the asymptotic behavior of the exponential utility indifference prices where the risk aversion goes to infinity at a rate which is inversely proportional to the linear price impact which goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In addition we provide a family of asymptotically optimal hedging strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The main motivation for the study of the asymptotic behaviour of utility indiffer- ence prices is that in the presence of price impact, super–replication is prohibitively costly, see Guasoni & Rasonyi (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Namely, in the presence of price impact, even in market models such as the Bachelier model or the Black–Scholes model (which are complete in the frictionless setup) there is no practical way to construct a hedg- ing strategy which eliminates all risk from a financial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This brings us to utility indifference pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We divide the proof of our main result (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) into two main steps: the proof of the lower bound and the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In the proof of the lower bound we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 from Dolinsky (2022) which gives a dual representation of the certainty equivalent for the case where the investor has to liquidate her position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This dual representation together with the Brownian structure allows to compute the scaling limit of the utility indifference prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The Date: January 5, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' YD Supported in part by the GIF Grant 1489-304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6/2019 and the ISF grant 230/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 1 2 proof of the upper bound is done by an explicit construction of a family of portfolios which are asymptotically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In the next section we introduce the setup and formulate the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In Section 3 we prove the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In Section 4 we prove the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In Section 5 we derive an auxiliary result from the field of deterministic variational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Preliminaries and Main Results Let T < ∞ be the time horizon and let W = (Wt)t∈[0,T ] be a standard one dimen- sional Brownian motion defined on the filtered probability space (Ω, F, (Ft)t∈[0,T ], P) where the filtration (Ft)t∈[0,T ] satisfies the usual assumptions (right continuity and completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We consider a simple financial market with a riskless savings ac- count bearing zero interest (for simplicity) and with a risky asset S = (St)t∈[0,T ] with Bachelier price dynamics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) St = S0 + µt + σWt where S0 ∈ R is the initial position of the risky asset, µ ∈ R is the constant drift and σ > 0 is the constant volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Following Almgren & Chriss (2001), we model the investor’s market impact, in a temporary linear form and, thus, when at time t the investor turns over her position Φt at the rate ˙Φt := dΦt dt the execution price is St + Λ 2 ˙Φt for some constant Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The portfolio value at the maturity date is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) V Φ T := � T 0 ΦtdSt − Λ 2 � T 0 ˙Φ2 tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In our setup the investor has to liquidate her position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, the natural class of admissible strategies which we denote by A is the set all progressively measurable processes Φ = (Φt)t∈[0,T ] with differentiable trajectories such that � T 0 ˙Φ2 tdt < ∞ and ΦT = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We assume that the initial number of shares Φ0 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Consider a vanilla European option with the payoff X = f(ST ) where f is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) f(x) = max (0, Θ (x − K)) , x ∈ R for some constants Θ, K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Observe that this form includes call/put options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The investor will assess the quality of a hedge by the resulting expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Assuming exponential utility with constant absolute risk aversion α > 0, the util- ity indifference price and the certainty equivalent price of the claim X (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=', Carmona (2009) for details on indifference prices) do not depend on the investor’s initial wealth and, respectively, take the well-known forms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) π(Λ, α, Φ0, X) := 1 α log � infφ∈A EP � exp � α � X − V Φ T ��� infφ∈A EP � exp � −αV Φ T �� � and c(Λ, α, Φ0, X) := 1 α log � inf φ∈A EP � exp � α � X − V Φ T ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' If the risk aversion α > 0 is fixed, then by applying standard density arguments we obtain that for Λ ↓ 0, the above indifference price converges to the unique price of the continuous time complete (frictionless) market given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' A more interesting limit emerges, however, if we re-scale the investor’s risk-aversion in the 3 form α := A/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Hence, we fix A > 0 and consider the case where the risk aversion is α(Λ) := A Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Before we formulate the main result we need some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Introduce the functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) g(x) := sup y∈R � f(x + y) − y2 4σ √ A � = max � 0, Θ (x − K) + σ √ AΘ2� , x ∈ R and u(t, x) := EP [g(x + σWT −t)] , (t, x) ∈ [0, T ] × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The term u(t, St) represents the price at time t of a European option with the payoff g(ST ) in the complete market given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' It is well known that u ∈ C1,2([0, T ) × R) solves the PDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6) ∂u ∂t + σ2 2 ∂2u ∂x2 = 0 in [0, T ) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Next, let Λ > 0 and let ρ = ρ(Λ) := σ2α(Λ) Λ = σ2A Λ2 be the risk-liquidity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Consider the (random) ODE on the interval [0, T ] ˙Ft = √ρ � cosh(√ρ(T −t)) 2 cosh2� √ρ(T −t) 2 � ∂u ∂x(t, St − σ √ AFt) − tanh(√ρ(T − t))Ft � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) with the initial condition F0 = Φ0 coth(√ρT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the linear growth of g it follows that for any ǫ > 0 the functions ∂u ∂x, ∂2u ∂x2 are uniformly bounded in the domain [0, T − ǫ] × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In particular ∂u ∂x is Lipschitz continuous with respect to x in the domain [0, T −ǫ]×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Observe that the functions cosh(√ρ(T −t)) 2 cosh2� √ρ(T −t) 2 �, tanh(√ρ(T − t)) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Hence, from the standard theory of ODE (see Walter (1998), Chapter II, Section 6) we obtain that there exists a unique solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) which we denote by F Λ = (F Λ t )t∈[0,T ) and the solution is Lipschitz continuous, and so limt→T − F Λ t exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Set F Λ T := limt→T − F Λ t and define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) ΦΛ t := tanh �� ρ(Λ)(T − t) � F Λ t , t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- aversion A/Λ with A > 0 fixed, the certainty equivalent of X = max (0, Θ (ST − K)) has the scaling limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='9) lim Λ↓0 c(Λ, A/Λ, Φ0, X) = u � 0, S0 − σ √ AΦ0 � + σ √ AΦ2 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Moreover, the trading strategies given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) are asymptotically optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='10) lim Λ↓0 Λ A log � EP � exp �A Λ � X − V ΦΛ T ���� = u � 0, S − σ √ AΦ0 � + σ √ AΦ2 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 we obtain immediately the following corollary which says that the asymptotic value of the utility indifference prices is equal to the price of the vanilla European option with the payoff g(ST ) and the shifted initial stock price S0 − σ √ AΦ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 4 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- aversion A/Λ with A > 0 fixed, the utility indifference price of X has the scaling limit lim Λ↓0 π(Λ, A/Λ, Φ0, X) = u � 0, S0 − σ √ AΦ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) and take X ≡ 0 for the denominator of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In the proof of the lower bound (given in the next section) we only assume that the payoff function f is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' By a more careful anal- ysis we can prove that in fact there is an equality, namely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='9) holds true for any payoff function X = f(ST ) with a Lipschitz continuous f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Unfortunately, the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='10) (given in Section 4) uses the specific structure of the payoff given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with the fact that the most common vanilla options in real markets are of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) led us to assume from the beginning that the payoff is of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Let us emphasize that our results can be extended to the multi–asset case with a similar proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In the multi asset case the volatility σ is replaced with a posi- tive definite matrix and the functions coth and tanh are viewed as matrix valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 can be viewed as an extension of the main result in Dolinsky & Moshe (2022), for the case where the investor has to liquidate her portfolio at the maturity date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In both cases (with or without liquidation) the scaling limit of the utility indifference prices is equal to E [h (x + σWT )] for a modified function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' In the present paper h(x) = sup y∈R � f(x + y) − y2 4σ √ A � while in Dolinsky& Moshe (2022) the modified payoff is smaller and given by h(x) = sup y∈R � f(x + y) − y2 2σ √ A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Next, we discuss the constructed asymptotically optimal portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='11) ˙ΦΛ t = � ρ(Λ) � tanh �� ρ(Λ) (T − t) 2 � ΥΛ t − coth �� ρ(Λ) (T − t) � ΦΛ t � where ρ(Λ) := σ2A Λ2 and ΥΛ t := ∂u ∂x(t, St − σ √ AF Λ t ), t ∈ [0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, we have a mean reverting structure which combines trucking the ∆–hedging strategy of a modified claim g and liquidating the position at the maturity date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' As time t approaches maturity the weight of the ∆–hedging trading strategy becomes smaller and the investor trading is mainly towards liquidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This is in contrast to the asymptotically optimal portfolios in Dolinsky and & Moshe (2022) which are just based on trucking the appropriate ∆–hedging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof of the Lower Bound In this section we prove the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 5 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' For vanishing linear price impact Λ ↓ 0 and re-scaled high risk- aversion A/Λ with A > 0 fixed, we have the following lower bound lim inf Λ↓0 c(Λ, A/Λ, Φ0, X) ≥ u � 0, S0 − σ √ AΦ0 � + σ √ AΦ2 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We start with the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Denote by Γ the set of all progressively measurable processes θ = (θt)t∈[0,T ] such that θ ∈ L2(dt ⊗ P) and let M be the set of all P–martingales M = (Mt)t∈[0,T ) which are defined on the half-open interval [0, T ) and satisfy ||M||L2(dt⊗P) := EP �� T 0 M 2 t dt � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Then, for any Λ, α > 0 we have c(Λ, α, Φ0, X) ≥ sup(θ,M)∈Γ×M EP � f � ST + σ � T 0 θtdt � − 1 2α � T 0 θ2 t dt −Φ0(M0 − S0) − 1 2Λ � T 0 ���S0 + µt + σ � t 0 θsds − Mt ��� 2 dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Denote by Q the set of all equivalent probability measures Q ∼ P with finite entropy EQ � log � dQ dP �� < ∞ relative to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' For any Q ∈ Q let MQ be the set of all Q–martingales M Q = (M Q t )t∈[0,T ) which are defined on the half-open interval [0, T ) and satisfy ||M Q||L2(dt⊗Q) := EQ �� T 0 |M Q t |2dt � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the linear growth of f it follows that EP � eαX� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, define the probability measure ˜P by d˜P dP := eαX EP[eαX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The Cauchy–Schwarz inequality yields that there exists a > 0 such that E˜P � exp � a sup0≤t≤T S2 t �� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Hence, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 in Dolinsky (2022) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, by applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 in Dolinsky (2022) for the probability measure ˜P and the simple equality EQ � log �dQ d˜P �� = EQ � log �dQ dP � − αX � + α log � EP � eαX�� ∀Q ∈ Q we obtain c(Λ, α, Φ0, X) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) = supQ∈Q supMQ∈MQ EQ � X − 1 α log � dQ dP � − Φ0(M Q 0 − S0) − 1 2Λ � T 0 |M Q t − St|2dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Next, Let C[0, T ] be the space of continuous functions z : [0, T ] → R equipped with the uniform norm ||z|| := sup0≤t≤T |zt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Denote by ˆΓ ⊂ Γ the set of all continuous and bounded processes θ = (θt)t∈[0,T ] of the form θ = τ(W) where τ : C[0, T ] → C[0, T ] is Lipschitz continuous and predictable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' τt(x) = τt(y) if x[0,t] = y[0,t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From standard density arguments and the Lipschitz continuity of f it follows that in order to complete the proof of the Lemma it is sufficient to show that for any (θ, M) ∈ ˆΓ × M we have c(Λ, α, Φ0, X) ≥ EP � f � ST + σ � T 0 θtdt � − 1 2α � T 0 θ2 t dt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) −Φ0(M0 − S0) − 1 2Λ � T 0 ���S0 + µt + σ � t 0 θsds − Mt ��� 2 dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 6 To this end let (θ, M) ∈ ˆΓ × M such that θ = τ(W) where τ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Consider the stochastic differential equation (SDE) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) dYt = dWt − τt(Y )dt, t ∈ [0, T ] with the initial condition Y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 from Chapter IX in Revuz & Yor (1999) yields that there exists a unique strong solution to the above SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the Girsanov theorem it follows that there exists a probability measure Q ∈ Q such that W Q := Y is a Brownian motion with respect to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) we obtain that the distribution of (St)t∈[0,T ] under Q is equal to the distribution of � St + σ � t 0 θsds � t∈[0,T ] under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Moreover, EQ � 1 α log �dQ dP �� = EQ � 1 2α � T 0 τ 2 t (Y )dt � = EP � 1 2α � T 0 θ2 t dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Finally, choose M Q ∈ MQ such that the law of (W Q, M Q) under Q is equal to the law of (W, M + σW) under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We conclude, EQ � X − 1 α � T 0 log � dQ dP � − Φ0(M Q 0 − S0) − 1 2Λ � T 0 |M Q t − St|2dt � = EP � f � ST + σ � T 0 θtdt � − 1 2α � T 0 θ2 t dt −Φ0(M0 − S0) − 1 2Λ � T 0 ���S0 + µt + σ � t 0 θsds − Mt ��� 2 dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ Next, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Denote by L2 0(FT , P) the set of all random variables of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) Z = ι + � T 0 κtdWt for some ι ∈ R and a predictable and bounded process κ = (κt)t∈[0,T ] such that κ[T −ǫ,T ] ≡ 0 for some (deterministic) ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Let Z ∈ L2 0(FT , P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' There exists a constant ˆC > 0 (may depend on Z) such that for any Λ ∈ (0, 1) sup(θ,M)∈Γ×M EP � f � ST + σ � T 0 θtdt � − 1 2α(Λ) � T 0 θ2 t dt −Φ0(M0 − S0) − 1 2Λ � T 0 ���S0 + µt + σ � t 0 θsds − Mt ��� 2 dt � ≥ EP � f (S0 + σWT + Z) − (Z+σ √ AΦ0) 2 4σ √ A � + σ √ AΦ2 0 2 − ˆCΛ where, as before α(Λ) = A Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Let Z given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) and let Ξ be the map from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Define the deterministic function ν : [0, T ] → R by ν := ΞT (Λ, ι, Φ0) and for any s < T define the stochastic process (l·,s)·∈[s,T ] by (l·,s)·∈[s,T ] = ΞT −s(Λ, κs, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Next, introduce (θ, M) ∈ Γ × M θt := ˙νt−µ σ + 1 σ � t 0 ∂lt,s ∂t dWs, t ∈ [0, T ], Mt := S0 + � T 0 νtdt−Φ0Λ T + σ � t 0 � 1 + 1 T −s � T s lt,sdt � dWs, t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 7 Observe that from the definition of Ξ we have ν0 = 0, νT = ι and ls,s = 0, lT,s = κs ∀s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with the Fubini theorem, the Itˆo Isometry, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) gives EP � f � ST + σ � T 0 θtdt � − 1 2α(Λ) � T 0 θ2 t dt − Φ0(M0 − S0)− 1 2Λ � T 0 ���S0 + µt + σ � t 0 θsds − Mt ��� 2 dt � = EP [f (S0 + σWT + Z)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) + µΛι σ2A − µ2Λ 2σ2A − I(Λ, ν) − � T −ǫ 0 EP [Js(Λ, l)] ds where I(Λ, ν) := Λ 2σ2A � T 0 ˙ν2 t dt + 1 2Λ \uf8eb \uf8ed � T 0 ν2 t dt − 1 T � Φ0Λ − � T 0 νtdt �2\uf8f6 \uf8f8 and Js(Λ, l) := Λ 2σ2A � T s �∂lt,s ∂t �2 dt + 1 2Λ \uf8eb \uf8ed � T s l2 t,sdt − 1 T − s �� T s lt,sdt �2\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 there exists a constant C > 0 (may depend on ι and κ) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6) ������� I(Λ, ν) − � ι + σ √ AΦ0 �2 4σ √ A + σ √ AΦ2 0 2 ������� ≤ CΛ and for any s ∈ [0, T − ǫ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) ����Js(Λ, l) − κ2 s 4σ √ A ���� ≤ CΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' By combining the Itˆo Isometry and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ We now have all the pieces in place that we need for the completion of the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Recall the definition of g given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the Lipschitz continuity of f it follows that there exists a bounded (measurable) function ζ : R → R such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) g(x) = f (x + ζ(x)) − ζ2(x) 4σ √ A , ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Choose a sequence Zn ∈ L2 0(FT , P), n ∈ N such that lim n→∞ Zn = ζ(S0 − σ √ AΦ0 + σWT ) − σ √ AΦ0 8 where the limit is in L2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) we obtain lim infΛ↓0 c (Λ, A/Λ, Φ0, X) ≥ supn∈N EP � f (S0 + σWT + Zn) − (Zn+σ √ AΦ0) 2 4σ √ A � + σ √ AΦ2 0 2 ≥ EP � g � S0 − σ √ AΦ0 + σWT �� + σ √ AΦ2 0 2 = u � 0, S0 − σ √ AΦ0 � + σ √ AΦ2 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof of the Upper Bound In order to complete the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1 it remains to establish the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Recall the trading strategies ΦΛ, Λ > 0 given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Then, lim sup Λ↓0 Λ A log � EP � exp �A Λ � X − V ΦΛ T ���� ≤ u � 0, S − σ √ AΦ0 � + σ √ AΦ2 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The proof will be done in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Step I: In this step we use the specific structure of the payoff f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Let us show that for any Λ > 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) g � ST − σ √ AF Λ T � ≥ f(ST ) − σ √ A|Φ0Θ| sinh �� ρ(Λ)T � where, as before ρ(Λ) := σ2A Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Fix Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) d dt \uf8ee \uf8f0 F Λ t cosh �� ρ(Λ)(T − t) � \uf8f9 \uf8fb = � ρ(Λ) 2 cosh2 �√ ρ(Λ)(T −t) 2 �ΥΛ t , t ∈ [0, T ] where, recall that ΥΛ t := ∂u ∂x � t, St − σ √ AF Λ t � , t ∈ [0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Clearly, |ΥΛ t | ≤ Θ, and so, |F Λ T | ≤ ���� F Λ 0 cosh �√ ρ(Λ)T � ���� + Θ � T 0 √ ρ(Λ) 2 cosh2 � √ ρ(Λ)(T −t) 2 �dt ≤ ���� Φ0 sinh �√ ρ(Λ)T � ���� + |Θ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Step II: In this step we prove that there exists a constant ˜C > 0 such tht (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) ����� � T 0 ΥΛ t dt − � T 0 ΦΛ t dt ����� ≤ ˜CΛ, ∀Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 9 Fix Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='11) d dt � ΦΛ t cosh �√ ρ(Λ)(T −t) � � = √ ρ(Λ) 2 cosh2 � √ ρ(Λ)(T −t) 2 � tanh �√ ρ(Λ)(T −t) 2 � ΥΛ t , t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We get ΦΛ t = Φ0 cosh �√ ρ(Λ)(T −t) � cosh(√ ρ(Λ)T ) + � t 0 √ ρ(Λ) cosh �√ ρ(Λ)(T −t) � 2 cosh2 � √ ρ(Λ)(T −s) 2 � tanh �√ ρ(Λ)(T −s) 2 � ΥΛ s ds and so, from the Fubini theorem � T 0 ΦΛ t dt − � T 0 ΥΛ t dt = Φ0 tanh �� ρ(Λ)T � � ρ(Λ) − � T 0 ΥΛ s cosh2 �√ ρ(Λ)(T −s) 2 �ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with the simple integral � T 0 ds cosh2 �√ ρ(Λ)(T −s) 2 � = 2 tanh �√ ρ(Λ)T 2 � � ρ(Λ) and the inequality |ΥΛ t | ≤ Θ gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Step III: In this step we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Fix Λ > 0 and introduce the process M Λ t := exp � A Λ � u � t, St − σ √ AF Λ t � + σ √ AF Λ t ΦΛ t 2 − V ΦΛ t �� , t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the Itˆo formula, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='11) we obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='dMΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='MΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='= A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='dSt + σ2A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2Λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�2 dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='− σ2A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='Λ2 ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8ed cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8f8 dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='+ σ2A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2Λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='tanh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − coth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='+ σ2A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2Λ2 ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8ed cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='\uf8f8 dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='+ σ2A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2Λ2 coth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='tanh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T −t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − coth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ρ(Λ)(T − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='= A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='ΥΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t − ΦΛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='dSt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='where the last equality follows from simple calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Hence, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) it follows that the process N Λ t := exp � −µA � t 0 � ΥΛ t − ΦΛ s � ds Λ � M Λ t , t ∈ [0, T ] is a local–martingale, and so from the obvious inequality N Λ > 0 we conclude that this process is a super–martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Finally, Λ A log � EP � exp � A Λ � X − V ΦΛ T ���� ≤ Λ A log � EP[M Λ T ] � + σ √ A|Φ0Θ| sinh �√ ρ(Λ)T � ≤ Λ A log � EP[N Λ T ] � + ˜C|µ|Λ + σ √ A|Φ0Θ| sinh �√ ρ(Λ)T � ≤ Λ A log � N Λ 0 � + ˜C|µ|Λ + σ √ A|Φ0Θ| sinh �√ ρ(Λ)T � = u � 0, S0 − σ √ AΦ0 coth �� ρ(Λ)T �� + σ √ AΦ2 0 coth �√ ρ(Λ)T � 2 + ˜C|µ|Λ + σ √ A|Φ0Θ| sinh �√ ρ(Λ)T �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The first inequality follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) and the relations u(T, ·) = g(·), ΦΛ T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The second inequality is due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The super–martingale property of N Λ gives the third inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The equality is due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' By taking Λ ↓ 0 we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Auxiliary Result For any T ∈ (0, T ] and x ∈ R let C0,x[0, T] be the space of all continuous functions z : [0, T] → R which satisfy z0 = 0 and zT = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' For any T ∈ (0, T ] there exists a measurable map ΞT : (0, 1) × R2 → C[0, T) such that for any Λ ∈ (0, 1) and x, φ ∈ R the continuous function ΞT(Λ, x, φ) ∈ C0,x[0, T] is the unique minimizer for the optimization problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) min δ∈C0,x[0,T] \uf8ee \uf8f0 Λ 2σ2A � T 0 ˙δ2 t dt + 1 2Λ \uf8eb \uf8ed � T 0 δ2 t dt − 1 T � φΛ − � T 0 δtdt �2\uf8f6 \uf8f8 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Moreover, denote the corresponding value by VT(Λ, x, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Then, for any ǫ > 0 and a compact set K ⊂ R2 there exists a constant ˆC (may depend on ǫ and K) such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) ������� VT(Λ, x, φ) − � x + σ √ Aφ �2 4σ √ A + σ √ Aφ2 2 ������� ≤ ˆCΛ, ∀(T, Λ, x, φ) ∈ [ǫ, T ] × (0, 1) × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Fix (T, Λ, x, φ) ∈ [ǫ, T ] × (0, 1) × R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' First we minimize the pattern given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) under the additional constraint that � T 0 δtdt is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Then, we will find the optimal � T 0 δtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 11 For any y ∈ R let Cy 0,x[0, T] ⊂ C0,x[0, T] be the subset of all functions δ ∈ C0,x[0, T] which satisfy � T 0 δtdt = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Consider the minimization problem min δ∈Cy 0,x[0,T] � T 0 H( ˙δt, δt)dt where H(v1, v2) := Λ 2σ2Av2 1 + 1 2Λv2 2 for v1, v2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This optimization problem is convex and so it has a unique solution which has to satisfy the Euler–Lagrange equation (for details see Gelfand & Fomin (1963)) d dt ∂H ∂ ˙δt = λ + d dt ∂H ∂δt for some constant λ > 0 (lagrange multiplier due to the constraint � T 0 δtdt = y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, the optimizer which we denote by ˆδ solves the ODE ¨ˆδt − ρˆδ ≡ const (recall the risk-liquidity ration ρ = ρ(Λ) := σ2A Λ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the standard theory it follows that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) ˆδt = c1 sinh(√ρt) + c2 sinh(√ρ(T − t)) + c3, t ∈ [0, T] for some constants c1, c2, c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From the three constraints ˆδ0 = 0, ˆδT = x and � T 0 ˆδtdt = y we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) c1 = x − c3 sinh(√ρT), c2 = − c3 sinh(√ρT) and c3 = √ρy − x tanh(√ρT/2) √ρT − 2 tanh(√ρT/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' We argue that ρ � T 0 ˆδ2 t dt + � T 0 ˙ˆδ2 t dt = ρ � T 0 � (ˆδt − c3) + c3 �2 dt + � T 0 ˙ˆδ2 t dt = √ρ 2 � c2 1 + c2 2 � sinh � 2√ρT � − 2c1c2√ρ sinh(√ρT) − ρc2 3T + 2ρc3y = √ρx2 coth(√ρT) + 2√ρc1c2 sinh(√ρT) � cosh(√ρT) − 1 � − ρc2 3T + 2ρc3y = √ρx2 coth(√ρT) + � 2√ρ tanh(√ρT/2) − ρT � c2 3 +2 � ρy − √ρ tanh(√ρT/2)x � c3 = √ρ � x2 coth(√ρT) + (x tanh(√ρT/2)−√ρy) 2 √ρT−2 tanh(√ρT/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) Indeed, the first equality is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The second equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3) and simple computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The third equality is due to c1 − c2 = x sinh(√ρT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The fourth equality is due to c1c2 = c2 3−xc3 sinh2(√ρT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' The last equality follows from substituting c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) we conclude that in order to minimize (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1) we need to find y which minimizes the quadratic pattern 1 2√ρΛ � x tanh(√ρT/2) − √ρy �2 √ρT − 2 tanh(√ρT/2) − 1 2ΛT (φΛ − y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Observe that this quadratic pattern is convex in y and so has a unique minimum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6) y = xT 2 − φΛ �√ρT − 2 tanh(√ρT/2) � 2 tanh(√ρT/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Thus, define ΞT(Λ, x, φ) := ˆδ where ˆδ is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='3)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Clearly, ΞT(Λ, x, φ) is the unique minimizer for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' 12 Let VT(Λ, x, φ) := Λ 2σ2A � T 0 ˙ˆδ2 t dt + 1 2Λ \uf8eb \uf8ed � T 0 ˆδ2 t dt − 1 T � φΛ − � T 0 ˆδtdt �2\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Finally, we prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Choose ǫ > 0 and a compact set K ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Assume that (T, x, φ) ∈ [ǫ, T ] × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='5) and the equality ρ = σ2A Λ2 we get that there exists a constant C1 (may depend on ǫ and K) such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) ����VT(Λ, x, φ) − � x2 2σ √ A + y2 σ √ AT2 + φy T − xy σ √ AT ����� ≤ C1Λ where y given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='6) we have ���y − T 2 � x − σ √ Aφ ���� ≤ C2Λ for some constant C2 (may depend on ǫ and K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' This together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='7) gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='2) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' □ References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Almgren and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Chriss, Optimal execution of portfolio transactions, Journal of Risk, 3, 5–39, (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Black, Noise, Journal of Finance, 41, 529–543, (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' Bayrakatar and M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' DEPARTMENT OF FINANCE, NATIONAL UNIVERSITY OF KYIV-MOHYLA ACAD- EMY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='MAIL: LDOLINSKYI@UKMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='EDU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content='UA DEPARTMENT OF STATISTICS, HEBREW UNIVERSITY OF JERUSALEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfl_0n/content/2301.01555v1.pdf'} +page_content=' E.' metadata={'source': 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problems requiring +high-level predictive power. Despite this success, +the inner workings of DNNs are often not trans- +parent, making them difficult to interpret or un- +derstand. This lack of interpretability has led +to increased research on inherently interpretable +neural networks in recent years. Models such as +Neural Additive Models (NAMs) achieve visual +interpretability through the combination of clas- +sical statistical methods with DNNs. However, +these approaches only concentrate on mean re- +sponse predictions, leaving out other properties +of the response distribution of the underlying data. +We propose Neural Additive Models for Loca- +tion Scale and Shape (NAMLSS), a modelling +framework that combines the predictive power +of classical deep learning models with the inher- +ent advantages of distributional regression while +maintaining the interpretability of additive mod- +els. +1. Introduction +Deep learning models have shown impressive performances +on a variety of predictive tasks. They are state-of-the-art +models for tasks involving unstructured data, such as image +classification (Yu et al., 2022; Dosovitskiy et al., 2020), text +classification (Huang et al., 2021; Lin et al., 2021), audio +classification (Nagrani et al., 2021), time-series forecast- +ing (Zhou et al., 2022; Zeng et al., 2022) and many more. +However, the predictive performance comes not only at +*Equal contribution. +Correspondence to: +Anton Thielmann anton.thielmann@tu- +clausthal.de, +Ren´e-Marcel +Kruse +rene-marcel.kruse@uni- +goettingen.de +the price of computational demands. The black-box nature +of deep neural networks poses hard challenges for inter- +pretability. To achieve sample-level interpretability, existing +methods resort to model-agnostic methods. Locally Inter- +pretable Model Explanations (LIME) (Ribeiro et al., 2016) +or Shapley values (Shapley, 1953) and their extensions (Sun- +dararajan & Najmi, 2020) try to explain model predictions +via local approximation and feature importance. Sensitivity- +based approaches (Horel & Giesecke, 2020), exploiting +significance statistics, can only be applied to single-layer +feed-forward neural networks and can hence not be used to +model difficult non-linear effects, requiring more complex +model structures. +Subsequently, high-risk domains, such as e.g. medical ap- +plications often cannot exploit the advantages of complex +neural networks due to their lack of innate interpretability. +The creation of these innately interpretable models hence re- +mains an important challenge. Achieving the interpretability +from flexible statistical models as e.g. Generalized Linear +Models (GLMs) (Nelder & Wedderburn, 1972) or Gener- +alized Additive Models (GAMs) (Hastie, 2017), in deep +neural networks, however, is inherently difficult. Recently, +Agarwal et al. (2021) introduced Neural Additive Models +(NAMs), a framework that models all features individually +and thus creates visual interpretability of the single fea- +tures. While this is an important step towards interpretable +deep neural networks, any insightfulness of aspects beyond +the mean is lost in the model structure. To counter that, +we propose the neural counterpart to Generalized Additive +Models for Location, Scale and Shape (GAMLSS) (Rigby +& Stasinopoulos, 2005), the Neural Additive Model for +Location, Scale and Shape (NAMLSS). NAMLSS adopts +and iterates on the model class of GAMLSS, in the same +scope as NAMs (Agarwal et al., 2021) on GAMs. +The GAMLSS framework relaxes the exponential family +assumption and replaces it with a general distribution family. +The systematic part of the model is expanded to allow not +only the mean (location) but all the parameters of the condi- +1 +arXiv:2301.11862v1 [stat.ML] 27 Jan 2023 + +tional distribution of the dependent variable to be modelled +as additive nonparametric functions of the features, resulting +in the following model notation: +θ(k) = g(k)−1 +� +�β(k) + +Jk +� +j=1 +f (k) +j +(x(k) +j ) +� +� = ηθ(k), +with the superscript k = 1, . . . , K denoting the k-th param- +eter and j = 1, . . . , J denoting the features. +The model assumes that the underlying response obser- +vations yi for i = 1, 2, . . . , n are conditionally indepen- +dent given the covariates. The assumed conditional density +can depend on up to K different distributional parameters*. +Each of these distribution parameters θ(k) can be modelled +using its additive predictor ηθ(k) for k = 1, . . . , K, allowing +for complex relationships between the response and predic- +tor variables, as well as the flexibility to choose different +distributions for different parts of the response variable. An +additional important component of the GAMLSS model is +the link function g(k)(·), which allows each parameter of +the distribution vector to be conditional on different sets of +covariates. In the case that the distribution under consider- +ation features only one distribution parameter, the model +simplifies to an ordinary GAM model. Therefore, GAMLSS +is to be seen as a conceptual extension of the GAM idea +and is suitable for the extension and generalisation of ap- +proaches such as NAMs which are themselves built upon +the GAM idea. For an overview of the current state of re- +gression models that focus on the full response distribution +approaches see Kneib et al. (2021). +While the NAM learns linear combinations of different in- +put features to learn arbitrary complex functions and at the +same time provides improved interpretability, these models, +like their statistical counterparts the GAM models, focus +exclusively on modelling mean and dispersion. This is in +contrast to the GAMLSS and subsequently, the proposed +NAMLSS models, which substantially broadens the scope +by allowing all underlying parameters of the response dis- +tribution to potentially depend on the information of the +covariates. +Contributions +The contributions of the paper hence can +be summarized as follows: +• We present a novel architecture for Neural Additive +Models for Location, Scale and Shape. +• Compared to state-of-the-art GAM, GAMLSS and +*In practice most application focus on up to four θi += +� +θ(1) +i +, θ(2) +i +, θ(3) +i +, θ(4) +i +� +. +DNNs our NAMLSS achieves similar results on bench- +mark datasets. +• We demonstrate that NAMLSS effectively captures the +information underlying the data. +• Lastly, we show that the NAMLSS approach allows to +go beyond the mean prediction of the response and to +model the entire response distribution. +The rest of the paper is structured as follows: Section 2 +gives an overview of the current state of the NAM and other +more interpretable deep learning methods on which our +model is based. Section 3 provides an introduction to the +underlying architectural and mathematical concepts of the +proposed NAMLSS approach, as well as contrasts it with +its methodological sibling the GAMLSS approach. Section +4 analyzes the properties of the implemented methods by +contrasting their performance in comparison to popular and +common modelling techniques from statistics, machine and +deep learning. The last Section 5 offers a deeper discussion +on the improved interpretability of the presented method, +contrasting the results with those of other recent methods in +the field of interpretable deep learning and thereby offering +an outlook on upcoming applications and possible further +research questions. +2. Literature Review +The idea of generating feature-level interpretability in deep +neural networks by translating GAMs into a neural frame- +work was already introduced by Potts (1999) and expanded +by de Waal & du Toit (2007). While the framework was +remarkably parameter-sparse, it did not use backpropaga- +tion and hence did not achieve as good predictive results as +GAMs, while remaining less interpretable. More recently, +Agarwal et al. (2021) introduced NAMs, a more flexible +approach than the Generalized Additive Neural Networks +(GANNs) introduced by de Waal & du Toit (2007) that +leverages the recent advances in the field of Deep Learning. +NAMs are a class of flexible and powerful machine learn- +ing models that combine the strengths of neural networks +and GAMs. These models can be used to model complex, +non-linear relationships between response and predictor +variables, and can be applied to a wide range of tasks includ- +ing regression, classification, and time series forecasting. +The basic structure of a NAM consists of a sum of mul- +tiple components, each representing a different aspect of +the relationship between the response and predictor vari- +ables. These components can be linear, non-linear, or a +combination of both, and can be learned using a variety +of optimization algorithms. One of the key advantages of +NAMs is their inherent ability to learn the interactions be- +tween different predictor variables and the response without + +the need for manual feature engineering. This allows NAMs +to capture complex relationships in the data that may not be +easily apparent to the human eye. +The general form of a NAM can be written as: +E(y) = h +� +�β + +J +� +j=1 +fj(xj) +� +� , +(1) +where h(·) is the activation function used in the output layer, +x ∈ Rj are the input features, β is the global intercept term, +and fj : R → R represents the Multi-Layer Perceptrons +(MLPs) corresponding to the j-th feature. The similarity +to GAMs is apparent, as the two frameworks mostly distin- +guish in the form the individual features are modelled. h(·) +is comparable to the link function g(·). +Several extensions to the NAM framework have already +been introduced. Yang et al. (2021) extends NAMs to ac- +count for pairwise interaction effects. Chang et al. (2021) +introduced NODE-GAM, a differentiable model based on +forgetful decision trees developed for high-risk domains. All +these models follow the additive framework from GAMs +and learn the nonlinear additive features with separate net- +works, one for each feature or feature interaction, either +leveraging MLPs (Potts, 1999; de Waal & du Toit, 2007; +Agarwal et al., 2021; Yang et al., 2021) or using decision +trees (Chang et al., 2021). +The applications of such models range from nowcasting (Jo +& Kim, 2022), financial applications (Chen & Ye, 2022), +to survival analysis (Peroni et al., 2022). While the linear +combination of neural subnetworks provides a visual in- +terpretation of the results, any interpretability beyond the +feature-level representation of the model predictions is lost +in their black-box subnetworks. +The idea of focusing on more than the underlying mean +prediction is certainly relevant and has been an important +part, especially of the statistical literature in recent years. +There has been a strong focus on the GAMLSS (Rigby & +Stasinopoulos, 2005) framework, conditional transforma- +tion models (Hothorn et al., 2014), density regression (Wang +et al., 1996) or quantile and expectile regression frameworks. +However, these methods are inferior to machine and deep +learning techniques in terms of pure predictive power; the +disadvantage of not being able to deal with unstructured +data forms such as images, text or audio files; or the inher- +ent problems of statistical models in dealing with extremely +large and complex data sets. One resulting development to +deal with these drawbacks is frameworks that utilize statis- +tical modelling methods and combine them with machine +learning techniques such as boosting to create new types of +distributional regression models such as boosted generalized +additive model for location, scale and shape as presented +by Hofner et al. (2014b). However, the models leveraging +boosting techniques, while successfully modelling all distri- +butional parameters, lack the inherent interpretability from +GAMLSS or even the visual interpretability from NAMs. +3. Methodology +While NAMs incorporate some feature-level interpretabil- +ity and hence entail easy interpretability of the estimated +regression effects, they are unable to capture skewness, het- +eroskedasticity or kurtosis in the underlying data distribu- +tion due to their focus on mean prediction. Therefore, the +presented method is the neural counterpart to GAMLSS, +offering the flexibility and predictive performance of neu- +ral networks while maintaining feature-level interpretability +and which allows estimation of the underlying total response +distribution. +Let D = {(x(i), y(i))}n +i=1 be the training dataset of size +n. Each input x = (x1, x2, . . . , xJ) contains J features. +y denotes the target variable and can be arbitrarily dis- +tributed. NAMLSS are trained by minimizing the negative +log-likelihood as the loss function, − log (L(θ|y)) by op- +timally approximating the distributional parameters, θ(k). +Each parameter, θ(k), is defined as: +θ(k) = h(k) +� +�β(k) + +J +� +j=1 +f (k) +j +(xj) +� +� , +where h(k)(·) denotes the output layer activation functions +dependent on the underlying distributional parameter, β(k) +denotes the parameter-specific intercept and f (k) +j +: R → R +represents the feature network for parameter k for the j-th +feature, subsequently called the parameter-feature network. +Just as in GAMLSS, θ(k) can be derived from a subset +of the J features, however, due to the inherent flexibility +of the neural networks, defining each θ(k) over all J is +sufficient, as the individual feature importance for each +parameter, θ(k), is learned automatically. Each parameter- +feature network, f (k) +j +, can be regularized employing regular +dropout coefficients in conjunction with feature dropout +coefficients, λ(k) +1j and λ(k) +2j respectively, as also implemented +by Agarwal et al. (2021). +For e.g. a normal distribution, NAMLSS would hence mini- +mize +− log +� +L(ˆµ, ˆσ2|y) +� += − +� +−n +2 log(2πˆσ2) − +1 +2ˆσ2 +n +� +i=1 +(yi − ˆµ)2 +� +, +(2) + +where +ˆµ = β(1) + +J +� +j=1 +f (1) +j +(xj) +(3) +and +ˆσ2 = log +� +�1 + exp +� +�β(2) + +J +� +j=1 +f (2) +j +(xj) +� +� +� +� , +(4) +utilizing a Softplus activation function for the scale parame- +ter and a linear activation for the location parameter. +Figure 1. The network structure of a simple NAMLSS model. Each +input variable as well as each distributional parameter is handled +by a different neural network. h(k) are different activation func- +tions depending on the distributional parameter that is modelled. +E.g. a quadratic transformation for modelling the variance in a nor- +mally distributed variable to ensure the non-negativity constraint. +The presented structure demonstrates a NAMLSS modelling a +distribution with two parameters, e.g. a normal distribution. +We propose two different network architectures that can +both flexibly model all distributional parameters. The first +is depicted in Figure 1 and creates J subnetworks for each +of the K distributional parameters. Each distributional sub- +network is comprised of the sum of the parameter-feature +networks f (k) +j +. Hence we create K × J parameter-feature +networks. To account for distributional restrictions, each +distributional subnetwork is specified with possibly differ- +ing activation functions in the output layer. The second +model architecture, possible due to the flexibility of neural +networks, leverages the architecture of NAMs (see formula +(1)) and is depicted in figure 5 in the Appendix. Here, only +J subnetworks are created, with each subnetwork having a +K-dimensional output layer. This architecture thus creates +the same number of subnetworks as a common NAM. Each +distributional parameter, θ(k), is subsequently obtained by +summing over the k-th output of the J subnetworks. Every +dimension in the output layer can be activated using differ- +ent activation functions, according to parameter restrictions. +This allows the capture of interaction effects between the +given model parameters in each of the subnetworks*. +Integrating possible feature interactions can easily be +achieved in both architectures by training a fully connected +MLP on the residuals after the NAMLSS has converged. +Hence, again leveraging the example of a normally dis- +tributed y, equation (3) for that interaction part becomes: +− log +� +L(˜µ, ˜σ2|e) +� += − +� +−n +2 log(2π˜σ2) − +1 +2˜σ2 +n +� +i=1 +(ei − ˜µ)2 +� +, +(5) +where ei denote the residuals yi − ˆyi. +As neural networks can achieve optimal approximation rates, +NAMLSS can learn any non-linearity or dependence be- +tween distribution parameters and features. And unlike +GAMLSS, NAMLSS can model jagged shape functions and +easily incorporate huge amounts of data. +While predicting all parameters from a distribution may not +always improve predictive power, understanding the under- +lying data distribution is crucial in high-risk domains and +can provide valuable insights about feature effects. As an +example, Figure 2 illustrates the fit of our approach on data +following a Johnson’s SU distribution, including 3 features, +compared to the fit of a MLP that minimizes the Mean +Squared Error (MSE). The MLP has a better predictive per- +formance with an MSE of 0.0002, however, NAMLSS is +able to reflect the underlying data distribution much more +accurately (as shown in Figure 2), even though it has an +MSE of 0.0005. +*Note, that for distributions where only one parameter is mod- +elled, the two proposed NAMLSS structures are identical. + +r1 +(1) +C.J +C1 +h(2) +(2) +C.JTable 1. Results for Synthetic data: The benchmark results for the simulated data. We compare the models based on log-likelihoods, ℓ. +A larger log-likelihood represents a better fit. Means and standard deviations of 5-fold cross validation are reported. See the Appendix +A.1, for a comprehensive list of the used log-likelihoods. Note, that for distributions where only one parameter is modelled, the two +proposed NAMLSS structures are identical. Hence, we report only one value for the Binomial as well as the Poisson distribution. +Model +Binomial +Poisson +Normal +Inverse Gaussian +Weibull +Johnson’s SU +GAMLSS +-397 ± (4.0) +-800 ± (19.5) +-600 ± (23.7) +-385 ± (30.4) +-625 ± (31.1) +-370 ± (13.8) +gamboostLSS1 +-315 ± (7.4) +-800 ± (19.8) +-575 ± (15.6) +-366 ± (29.0) +-648 ± (22.1) +-478 ± (15.5) +DNN +-260 ± (53.7) +-802 ± (20.2) +-558 ± (16.3) +-343 ± (31.1) +-624 ± (23.6) +-285 ± (14.1) +NAMLSS2 +- +- +-589 ± (15.1) +-377 ± (26.6) +-621 ± (25.3) +-326 ± (12.1) +NAMLSS3 +-274 ± (27.1) +-802 ± (18.6) +-577 ± (16.8) +-362 ± (24.9) +-620 ± (25.5) +-327 ± (12.4) +1 For boosting one-parametric families like the Binomial or Poisson distribution, a model-based boosting algorithm is used +(Hofner et al., 2014a). +2 With K ×5 subnetworks. See Table 1 for an exemplary network structure. +3 With 5 subnetworks and each subnetwork returning a parameter for the location and shape respectively. See Table 5 for +an exemplary network structure. +Figure 2. Johnson’s SU Distribution: Simulated Johnson’s SU +distribution and the fit of a simple NAMLSS (see Figure 1) and a +MLP. While the MLP achieves an impressive fit concerning the +quadratic loss, it clearly cannot capture the underlying distribution +adequately. +4. Benchmarking +To demonstrate the competitiveness of the presented method, +we perform several analyses. We compare the performance +of NAMLSS with several state-of-the-art models including +neural as well as non-neural approaches and orientate on the +benchmarks performed by Agarwal et al. (2021). Addition- +ally, we compare related methods of distribution-focused +data analysis approaches that overcome the focus on relating +the conditional mean of the response to features and instead +target the complete conditional response distribution. We +analyze multiple datasets as well as conduct experiments on +synthetic data. We choose the following baselines for the +comparisons: +• Multilayer Perceptron (MLP): Unrestricted fully +connected deep neural network trained with either a +mean squared error loss function (regression) or binary +cross entropy (logistic regression). +• Gradient Boosted Trees (XGBoost): Decision tree +based gradient boosting. We use the implementation +provided by Chen & Guestrin (2016). +• Neural Additive Models (NAMs): Linear combina- +tion of DNNs as described in equation (1) and pre- +sented by Agarwal et al. (2021). +• Explainable Boosting Machines (EBMs): State-of- +the-art Generalized Additive Models leveraging shal- +low boosted trees (Lou et al., 2013). +• Deep Neural Network (DNN): Similar to the Mul- +tilayer Perceptron a fully connected neural network. +However, not trained to minimize the previously men- +tioned loss functions but to minimize the negative log- +likelihood of the specified distribution. All distribu- +tional parameters are predicted. +• Generalized Additive Models for Location Scale +and Shape (GAMLSS): Standard GAMLSS models +using the R implementation from Rigby & Stasinopou- +los (2005). +• gamboost for Location Scale and Shape (gamboost- +LSS): Fitting GAMLSS by employing boosting tech- +niques as proposed by Hofner et al. (2014b). +We preprocess all used datasets exactly as done by Agarwal +et al. (2021). We perform 5-fold cross-validation for all +datasets and report the average performances over all folds +as well as the standard deviations. For reproducibility, we + +Johrson su Distribution +1D +quantile 0.05 +quantile 0.95 +0.B +0.6 +NAMLSS +MLP +True + +0.4 +1 +I +1 +0.2 +0.0 +0.900 +0.925 +0.95 +SL6O +LDO0 +1D25 +LD5 +1075 +1101Table 2. Benchmark results for the regression comparison datasets: For models not explicitly modelling a shape parameter, the shape +is approximated with a constant as the true standard deviation of the dependent variable. Higher log-likelihoods (ℓ) and lower MSEs are +better. We report results on two commonly used regression datasets. The California Housing dataset for predicting house prices (Pace & +Barry, 1997a) and an Insurance dataset for predicting billed medical expenses (Lantz, 2019). +CA Housing +Insurance +Model +ℓ (↑) +MSE (↓) +ℓ (↑) +MSE (↓) +MLP +-4191 ±(41.7) +0.197 ± (0.005) +-266.8 ± (10.9) +0.163 ± (0.022) +XGBoost +-4219 ±(39.7) +0.211 ±(0.005) +-266.8 ± (9.2) +0.161 ± (0.014) +NAM +-8344 ±(1764.4) +0.273 ±(0.037) +-474.7 ± (72.5) +0.249 ± (0.029) +EBM +-4202 ±(42.2) +0.203 ±(0.004) +-263.8 ± (10.1) +0.139 ± (0.017) +DNN +-2681 ±(1279) +0.197 ± (0.005) +-178.2 ± (29.6) +0.165 ± (0.026) +GAMLSS +-3512 ±(66.7) +0.390 ±(0.035) +-175.5 ± (28.3) +0.269 ± (0.050) +gamboostLSS +-3812 ±(51.7) +0.415 ± (0.026) +-141.4 ± (31.2) +0.268 ± (0.050) +NAMLSS1 +-2667 ± (91.4) +0.245 ± (0.004) +-172.7 ± (22.6) +0.268 ± (0.043) +NAMLSS2 +-2329 ± (176.2) +0.265 ± (0.005) +-172.6 ± (19.5) +0.265 ± (0.040) +1 With J × K subnetworks. See Table 1 for an exemplary network structure. +2 With J subnetworks and each subnetwork returning a parameter for the location and shape +respectively. See Table 5 for an exemplary network structure. +have only chosen publicly available datasets. The datasets, +as well as the preprocessing and the seeds set for obtaining +the folds, are described in detail in the Appendix, C. We fit +all models without an intercept and explicitly do not model +feature interaction effects. +4.1. Beyond the mean: Synthetic data comparison +study +The synthetic data used for this task is generated from the +same underlying processes. Five features are included in +each application. The data-generating functions used to +generate the true underlying distributional parameters can +be found in Appendix C.1. Each of the five input vectors +xj is sampled from a uniform distribution U(0, 1), with a +total of n = 3000 observations per data set. The remaining +parameters are generated based on the input vectors and +the chosen distribution. We selected distributions that are +widely used, popular in science, or relatively complex to +reflect a diverse range of scenarios. We compare models +that specifically model all distributional parameters in this +simulation study. The results can be found in Table 1. +We find that the presented NAMLSS perform similarly to +fully connected DNNs that specifically minimize the log- +likelihood and perform better than GAMLSS or gamboost- +LSS, all while maintaining visual intelligibility. We can +hence confirm the findings from Agarwal et al. (2021) that +additive neural networks can achieve similar results to fully +connected DNNs. +4.2. Experiments with Real World Data +Normal Distribution +For a regression benchmark, we +use the California Housing (CA Housing) dataset (Pace & +Barry, 1997a) from sklearn (Pedregosa et al., 2011b) and +the Insurance dataset (Lantz, 2019) and standard normalize +the response variables. As comparison metrics we use the +log-likelihood, ℓ (see Appendix, A.1), as well as the mean +squared error (MSE). A larger log-likelihood thus repre- +sents a better model fit, while a smaller mean squared error +represents a better predictive performance in terms of the +mean. Thus, we try to illustrate the trade-off between pure +predictive performance, interpretability and overall data fit. +Table 2 presents the results obtained from the analyses +carried out on the two popular regression data bench- +mark datasets. In both applications, a normal distribution +N +� +µ, σ2I +� +of the underlying response variable was as- +sumed. As the log-likelihood of a normal distribution (see +equation (A.1)) is dependent on two parameters, but models +as an MLP or XGBoost only predict a single parameter, we +adjust the computation accordingly and use the standard +deviation calculated from the underlying data for XGBoost, +EBM, NAM and MLP. For NAMLSS, independent of the +implementation, we use a Softplus activation for the scale +parameter σ2 to ensure non-negativity and a linear activa- +tion for the mean µ. The NAMLSS approach achieves the +highest log-likelihood values of all presented approaches +which speaks for its good approximation capabilities for +the California Housing dataset. It can also be seen that the +trade-off between MSE performance and the possibility of +modelling the response distribution is relatively moderate +in its impact as NAMLSS even outperforms the predictive +performance of a NAM. The results could have been further +improved by accounting for feature interactions, resulting +in a log-likelihood of -1654 and an MSE of 0.20. +One of the advantages of NAMLSS compared to DNNs +is the feature level interpretability. Similar to NAMs, we +can plot and visually analyze the results (see Figures 3 and + +Figure 3. California Housing: Graphs for median income and +population respectively learned by the NAMLSS model. We see +an increase in housing prices with a larger median income. Addi- +tionally, we find a larger variance in housing prices in less densely +populated areas. +Figure 4. California Housing: Graphs for longitude and latitude +respectively learned by the NAMLSS model. The house price +jumps around the location of Los Angeles are depictable. Addi- +tionally, we find a decrease in variance for areas further away from +the large cities. +4). Additionally, we are able to accurately depict shifts in +variance in the underlying data. It is, for example, clearly +distinguishable, that with a larger median income, the house +prices tend to vary much stronger than with a smaller median +income (see Figure 3). A piece of information, that is lost in +the models focusing solely on mean predictions. Addition- +ally, we are capable of accurately representing sharp price +jumps around the location of San Francisco, depicted by the +jumps in the graphs for longitude and latitude (see Figure +4) as compared to GAMLSS, NAMLSS are additionally +capable of representing jagged shape functions. +For the Insurance dataset, which is only comprised of 1338 +observations, we unsurprisingly find strong performances of +the classical statistical models. NAMLSS perform similarly +to NAMs in terms of pure predictive power and perform +similarly to DNNs in terms of log-likelihoods. +Inverse Gamma +For the AirBnB dataset, also analyzed +by R¨ugamer et al. (2020), we assume an Inverse Gamma dis- +tribution IG(α, β) as the underlying data distribution (see +equation (A.1) for the log-likelihood). For NAMLSS as well +Table 3. Benchmark results for the AirBnB dataset: For mod- +els not explicitly modelling the distributional parameters, transfor- +mations are performed. Similar for models resulting in distribu- +tional parameters but not mean predictions. A larger log-likelihood, +ℓ, and smaller average gamma deviance (see Appendix A.2) are +better. +AirBnB +Model +ℓ (↑) +Avg. Gamma Dev. (↓) +MLP +-6827 ± (177.8) +0.55 ± (0.04) +XGBoost +-5618 ± (152.4) +0.48 ± (0.09) +NAM +-5892 ± (37.0) +0.72 ±(0.10) +EBM +-5474 ± (56.2) +0.49 ±(0.09) +DNN +-5555 ± (33.6) +0.69 ± (0.04) +GAMLSS +-5419 ± (60.6) +0.71 ± (0.44) +gamboostLSS +-5421 ± (33.1) +0.54 ± (0.07) +NAMLSS1 +-5383 ± (23.7) +0.59 ± (0.09) +NAMLSS2 +-5422 ± (22.0) +0.59 ± (0.10) +1 With J × k subnetworks. See Table 1 for an exemplary +network structure. +2 With J subnetworks and each subnetwork returning a +parameter for the location and shape respectively. See Table +5 for an exemplary network structure. +as the DNN we have to adjust the activation functions, as +both models minimize the log-likelihood via the parameters +α and β. However, the mean prediction resulting from these +parameters is defined via: +µ = +β +α − 1 +and is hence only defined for α > 1. The activation func- +tions thus need to ensure an α prediction that is larger than 1 +and a β prediction that is larger than 0. Hence we again use +a Softplus activation for the β output layer. For the α predic- +tion, we use the following activation function element-wise: +h(x) = +� +log(1 + exp(x)), +if log(1 + exp(x)) > 1, +1 +log(1+exp(x)), +else. +(6) +To compute the log-likelihood for the models resulting in +a mean prediction we compute the parameters α and β as +follows: +α = +µ2 +σ2 + 2, +β = µ +µ2 +σ2 + 1, +with σ2 denoting the variance of the mean predictions. For +XGBoost and EBM we use a simple transformation of the +target variable to ensure that µ > 0. Hence we fit the model +on log(y) and re-transform the predictions accordingly with + +6 +4 +Features Contribution +2 +-2 +-4 +-1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Medinc6 +4 +Features Contribution +2 +2 +-4 +1.00 +-0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Population2 +-1.00 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +-1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Latitude +Longitudleexp(ˆy). Interestingly, the NAM did not converge using +the Softplus activation function as the MLP did. Using +the Softplus activation resulted in tremendously large mean +gamma deviances and log-likelihoods, as the model kept +predicting values that were nearly zero. Hence, we were +only able to achieve good results for the NAM using the +activation function given by formula (6). +Logistic Distribution +For a (binary) classification bench- +mark we use the FICO dataset (FICO, 2018). A logistic +distribution, LO (µ, s), of the underlying response variable +was assumed (see equation (A.1) for the log-likelihood). +Again, we use the true standard deviation of the underlying +data for the models only resulting in a mean prediction. For +evaluating the sole predictive performance we use the Area +Under the Curve (AUC). The models resulting in a mean +prediction use binary cross-entropy as the loss function and +hence a sigmoid activation function on the output layer. +NAMLSS outperforms all models in terms of log-likelihood +and maintains a reasonable predictive performance, compa- +rable to NAMs, XGBoost and EBMs (see Table 4). +Table 4. Benchmark results for the FICO dataset: For all mod- +els, modelling distributional parameters, a logistic distribution is +assumed. For all other models, the binary cross-entropy loss func- +tion is minimized during training and the distributional parameters +for computing the log-likelihood, ℓ, are approximated. The Area +under the Curve (AUC) (see Appendix A.2) is used. +FICO +Model +ℓ (↑) +AUC (↑) +MLP +-1813 ±(6.3) +0.79 ± (0.007) +XGBoost +-1976 ±(13.4) +0.73 ±(0.010) +NAM +-1809 ± (7.6) +0.73 ±(0.010) +EBM +-1944 ±(20.6) +0.73 ±(0.010) +DNN +-1230 ± (47.5) +0.72 ± (0.006) +GAMLSS +-1321 ± (30.0) +0.78 ± (0.009) +gamboostLSS +-1191 ± (29.7) +0.79 ± (0.008) +NAMLSS1 +-1201 ± (40.9) +0.73 ± (0.010) +NAMLSS2 +-1160 ± (48.8) +0.72 ± (0.008) +1 With J × k subnetworks. +See Table 1 for an +exemplary network structure. +2 With J subnetworks and each subnetwork returning a +parameter for the location and shape respectively. See +Table 5 for an exemplary network structure. +5. Conclusion & Future Work +We have presented Neural Additive Models for Location, +Scale and Shape and their theoretical foundation as the +neural counterpart to GAMLSS. NAMLSS can model an +arbitrary number of parameters of the underlying data distri- +bution while preserving the predictive quality of NAMs. The +visual intelligibility achieved by NAMs is also maintained +by NAMLSS, with the added benefit of gaining further +insights from knowledge of additional distribution charac- +teristics. Hence, NAMLSS are a further step in the direction +of fully interpretable neural networks and already offer in- +terpretability that may make them suitable for high-risk +domains. +The extensibility of NAMLSS offers many different further +applied and theoretical research directions. One important +point is the extension of the modelling of the distribution +of the response variable. Many empirical works focus on +modelling not just one, but several responses conditionally +on covariates. One way to do this is to use copula meth- +ods, which are a valuable extension of our approach, hence +including a copula-based approach for NAMLSS models +would greatly improve the overall general usefulness. An- +other possible extension would be the adaptation to mixture +density networks, as e.g. done by Seifert et al. (2022). +Another possible focus is to switch our approach to a +Bayesian-based training approach. Bayesian approaches +are particularly well suited to deal with epistemic uncer- +tainty and to incorporate it into the modelling. 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Coca: Contrastive captioners are image- +text foundation models. arXiv preprint arXiv:2205.01917, +2022. +Zeng, A., Chen, M., Zhang, L., and Xu, Q. Are transformers +effective for time series forecasting? +arXiv preprint +arXiv:2205.13504, 2022. +Zhou, T., Ma, Z., Wen, Q., Sun, L., Yao, T., Jin, R., +et al. Film: Frequency improved legendre memory model +for long-term time series forecasting. +arXiv preprint +arXiv:2205.08897, 2022. + +A. APPENDIX +A.1. Log-Likelihoods +As the presented method minimizes negative log-likelihoods, we created a comprehensive list of all the log-likelihoods of +the distributions used in the paper. When we reference the results of NAMLSS these are the log-likelihoods we used for +fitting the models as well as evaluating them. +(Bernoulli) Logistic Distribution +The log-likelihood function for a logistic distribution is given by: +log (L(µ, σ|y)) = +n +� +i=1 +� +yi log( +1 +1 + e−( yi−µ +σ +) ) + (1 − yi) log(1 − +1 +1 + e−( yi−µ +σ +) ) +� +, +with n is as the number of observations and the parameters location µ ∈ R, scale σ ∈ R+ and x ∈ R. +Binomial Distribution +The log-likelihood function for a binomial distribution is given by: +log (L(k|n, p)) = k log(p) + (n − k) log(1 − p) + log +��n +k +�� +, +where n is the number of trials, the parameters success probability is given by p ∈ [0, 1] and the number of successes is +denoted as k ∈ N0. +Inverse Gamma Distribution +The log-likelihood function of the invers gamma distribution is defined as: +log (L(α, β|y)) = −n (α + 1) log y − n log Γ(α) + nα log β − +n +� +i=1 +βy−1 +i +. +with α > 0 and β > 0 and where the upper bar operand indicates the arithmetic mean +Normal Distribution +The log-likelihood function for a normal distribution is given by: +log +� +L(µ, σ2|y) +� += −n +2 log(2πσ2) − +1 +2σ2 +n +� +i=1 +(yi − µ)2, +where n is the underlying number of observations and parameters y ∈ R, location µ ∈ R and scale σ ∈ R+. +Inverse Gaussian Distribution +The log-likelihood function of the inverse Gaussian distribution is given by: +log (L(µ, σ|x)) = n +2 ln(σ) − +n +� +i=1 +σ(xi − µ)2 +2µ2xi +, +with n is as the number of observations and the parameters location µ ∈ R+, scale σ ∈ R+ and x ∈ R+. +Johnson’s SU +The log-likelihood function of the Johnson’s SU distribution is defined as: +log (L(β, ω, µ, σ|y)) = n log +� +β +ω +√ +2π +� +− β2 +2ω2 +n +� +i=1 +�(yi − µ)2 +σ2 ++ ln +� +1 + (yi − µ)2 +ω2σ2 +�� +, +with n is as the number of observations and the parameters location µ ∈ R, scale σ ∈ R+, shape ω ∈ R+, skewness β ∈ R +and y ∈ R. + +Weibull +The log-likelihood function of the Weibull distribution is defined as: +log (L(λ, β, |y)) = n ln β − nβ ln λ − +n +� +i=1 +�yi +λ +�β ++ (β − 1) +n +� +i=1 +ln yi, +with n is the number of observations and with the location λ ∈ R+, the shape β ∈ R+ and y ∈ R+. +A.2. Deviance Measures +We use several deviance measures, to evaluate the model performances beyond the log-likelihoods. These deviance measures +are focused on the mere predictive power of the models. The depiction of both, the log-likelihoods as well as these +deviance measures, thus captures the trade-off between pure predictive power and the ability to capture the underlying data +distribution. +Mean Squared Error +The mean squared error is defined as : +MSE = 1 +n +n +� +i=1 +(yi − ˆyi)2. +Mean Gamma Deviance +The mean gamma deviance used for the AirBnB dataset is defined as: +D = 2 +n +n +� +i=1 +� +log( ˆyi) +yi +) + yi +ˆyi +− 1 +� +. +Area Under the Curve +We use the Riemannian formula for the AUC. Hence the area of rectangles is defined as: +AR = +� +i += 1n−1f(xi)∆x, +and hence with larger n, the definite integral of f from a to b is defined as: +� b +a +f(x)dx = lim +n→∞ +n−1 +� +i=0 +f(xi)∆x. +B. Network architecture +We propose two different network architectures that can both flexibly model all distributional parameters. The first one +is depicted in Figure 1 and creates J subnetworks for each distributional parameter. Each distributional subnetwork is +comprised of the sum of f (k) +j +. Hence we create K × J subnetworks. To account for distributional restrictions, each +distributional subnetwork is specified with possibly differing activation functions in the output layer. +The second model architecture is depicted in Figure 5. Here we only create J subnetworks and hence have the same amount +of subnetworks as a common NAM. Each subnetwork then has a k-dimensional output layer. Each distributional Parameter, +θ(k), is subsequently obtained by summing over the k-th output of the J subnetworks. Each dimension in the output layer +can be activated using different activation functions, adjusting to parameter restrictions. +Figure 5. The network structure of a simple NAMLSS model. Each input variable as well as each distributional parameter is handled by a +different neural network. hk are different activation functions depending on the distributional parameter that is modelled. E.g. a quadratic +transformation for modelling the variance in a normally distributed variable to ensure the non-negativity constraint. +C. Benchmarking +The benchmark study for used real-world datasets was performed under similar conditions. All datasets are publicly available +and we describe every preprocessing step as well as all model specifications in detail in the following. + +C.1. Synthetic Data Generation +For the simulation of the data, respectively their underlying distribution parameters θ = +� +θ(1), θ(2), θ(3), θ(4)� +, the following +assumptions are made: +θ(1) = 30 +13x1 +� +(3x2 + 1.5) − 2 sin +�x3 +2 +��−1 ++ 113 +115x4 + 0.1x5, +θ(2) = exp +� +−0.0035x1 + (x2 − 0.23)2 − 1.42x3 +� ++ 0.0001x4, +θ(3) = 1 +42(4x1 − 90x2), +θ(4) = exp (0.0323 ∗ x2 + 0.0123 − 0.0234 ∗ x4) , +where each of the five input vectors xj is sampled from a uniform distribution U(0, 1), with a total of n = 3000 observations +per data set. +C.2. Preprocessing +We implement the same preprocessing for all used datasets and only slightly adapt the preprocessing of the target variable +for the two regression problems, California Housing and Insurance. We closely follow Gorishniy et al. (2021) in their +preprocessing steps and use the preprocessing also implemented by Agarwal et al. (2021). Hence all numerical variables +are scaled between -1 and 1, all categorical features are one-hot encoded. In contrast to Gorishniy et al. (2021) we do not +implement quantile smoothing, as one of the biggest advantages of neural models is the capability to model jagged shape +functions. We use 5-fold cross-validation and report mean results as well as the standard deviations over all datasets. For +reproducibility, we use the sklearn (Pedregosa et al., 2011a) Kfold function with a random state of 101 and shuffle equal to +true for all datasets. For the two regression datasets, we implement a standard normal transformation of the target variable. +This results in better performances in terms of log-likelihood for all models only predicting a mean and is hence even +disadvantageous for the presented NAMLSS framework. +C.3. Datasets +Table 5. Statistics of the benchmarking datasets. +Dataset +No. Samples +No. Features +Distribution +Task +California Housing +20640 +8 +Normal N(µ, σ) +Regression +Insurance +1338 +6 +Normal N(µ, σ) +Regression +Fico +10459 +23 +Logistic LO(µ, s) +Classification +AirBnB +4568 +9 +Inverse Gamma IG(α, β) +Regression +California Housing +The California Housing (CA Housing) dataset (Pace & Barry, 1997b) is a popular publicly available +dataset and was obtained from sklearn (Pedregosa et al., 2011a). It is also used as a benchmark in (Agarwal et al., 2021) +and Gorishniy et al. (2021) and we achieve similar results concerning the MSE for the models which were used in both +publications. The dataset contains the house prices for California homes from the U.S. census in 1990. The dataset is +comprised of 20640 observations and besides the logarithmic median house price of the blockwise areas as the target variable +contains eight predictors. As described above, we additionally standard normalize the target variable. All other variables are +preprocessed as described above. +Insurance +The Insurance dataset is another regression type dataset for predicting billed medical expenses (Lantz, 2019). +The dataset is publicly available in the book Machine Learning with R by Lantz (2019). Additionally, the data is freely +available on Github and Kaggle. It is a small dataset with only 1338 observations. The target variable is charges, which +represents the Individual medical costs billed by health insurance. Similar to the California Housing regression we standard +normalize the response. Additionally, the dataset includes 6 feature variables. They are preprocessed as described above, +which, due to one-hot encoding leads to a feature matrix with 9 columns. +FICO +Similar to Agarwal et al. (2021) we also use the FICO dataset for our benchmarking study. However, we use it as +described on the website and hence use the Risk Performance as the target variable. A detailed description of the features and + +their meaning is available at the Explainable Machine Learning Challenge. The dataset is comprised of 10459 observations. +We did not implement any preprocessing steps for the target variable. +AirBnB +For the AirBnB data, we orientate on R¨ugamer et al. (2020) and used the data for the city of Munich. The dataset +is also publicly available and was taken from Inside AirBnB on January 15, 2023. After excluding the variables ID, Name, +Host ID, Host Name, Last Review and after removing rows with missing values the dataset contains 4568 observations. +Additionally, we drop the Neighbourhood variable as firstly the predictive power of that variable is limited at best and +secondly not to create too large feature matrices for GAMLSS. Hence, in addition to the target variable, the dataset contains +9 variables. All preprocessing steps are subsequently performed as described above and the target variable, Price, is not +preprocessed at all. +C.4. Model Architectures & Hyperparameters +As we do not implement extensive hyperparameter tuning for the presented NAMLSS framework, we do not perform +hyperparameter tuning for the comparison models. We fit all models without an intercept. However, we try to achieve +the highest comparability by choosing similar modelling frameworks, network architectures and hyperparameters where +possible. All neural models are hence fit with identical learning rates, batch sizes, hidden layer sizes, activation functions +and regularization techniques. Through all neural models and all datasets, we use the ADAM optimizer (Kingma & Ba, +2014) with a starting learning rate of 1e-04. For the larger datasets, California Housing and FICO, we orient on Agarwal +et al. (2021) and use larger batch sizes of 1024. For the smaller dataset, Insurance, we use a smaller batch size of 256 and +for the AirBnB dataset we use a batch size of 512. For every dataset and for every neural model, the maximum number of +epochs is set to 2000. However, we implement early stopping with a patience of 150 epochs and no model over no fold and +no dataset ever trained for the full 2000 epochs. Additionally, we reduce the learning rate with a factor of 0.95 with patience +of 10 epochs for all models for all datasets. We use the Rectified Linear Unit (ReLU) activation function for all hidden +layers for all models: +h(x) = +� +0, +x < 0 +x, +else. +We also experimented with the Exponential centred hidden Unit (ExU) activation function presented by Agarwal et al. +(2021) but found no improvement in model performance and even a slight deterioration for most models. +For the statistical models used from the GAMLSS and gamboostLSS frameworks, we do not optimize the model hyperparam- +eters, as with neural networks. We use the respective default settings unless otherwise stated in the modelling descriptions +included in the Appendix. We try to keep the model settings equal between all models, if applicable. All GAMLSS +models use the same RS solver proposed by (Rigby & Stasinopoulos, 2005), in cases where this approach does not lead to +convergence, the alternative CG solver presented by (Cole & Green, 1992) is employed. To exclude possible numerical +differences, the same distributions from the GAMLSS R package are used for modelling the response distribution and +calculating the log-likelihoods. gamboostLSS allows the use of different boosting approaches. Here we use the implemented +boosting methods based on GAMs and GLMs and choose the model that performs better in terms of log-likelihood and the +assumed loss. +California Housing +For the California Housing dataset, we orient again on Agarwal et al. (2021) and use the following +hidden layer sizes for all networks: [1000, 500, 100, 50, 25]. The second hidden layer is followed by a 0.25 dropout layer. +While subsequently the NAM and NAMLSS have much more trainable parameters than the MLP and the DNN, we find that +the MLP and DNN outperform the NAM and NAMLSS in terms of mean prediction. Additionally, we encountered severe +overfitting when using the same number of parameters in an MLP as in the NAM and NAMLSS implementation. For the +mean predicting models, we use a one-dimensional output layer with a linear activation. For the DNN and both NAMLSS +implementations, we use a linear activation over the mean prediction and a Softplus activation for the variance prediction +with: +h(x) = log(1 + exp(x)). + +Table 6. Hyperparameters for the neural models for the California Housing dataset +Hyperparameter +NAMLSS1 +NAMLSS2 +DNN +MLP +NAM +Learning Rate +1e-04 +1e-04 +1e-04 +1e-04 +1e-04 +Dropout +0.25 +0.25 +0.25 +0.25 +0.25 +Hidden Layers +[1000, 500, +[1000, 500, +[1000, 500, +[1000, 500, +[1000, 500, +100, 50, 25] +100, 50, 25] +100, 50, 25] +100, 50, 25] +100, 50, 25] +LR Decay, Patience +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +Activation +ReLU +ReLU +ReLU +ReLU +ReLU +Output Activation +Linear, Softplus +Linear, Softplus +Linear, Softplus +Linear +Linear +1 With 2 ×8 subnetworks. See Table 1 for an exemplary network structure. +2 With 8 subnetworks and each subnetwork returning a parameter for the location and shape respectively. See +Table 5 for an exemplary network structure. +For the NAMLSS implementation depicted in Figure 1 we use a smaller network structure for predicting the variance with +two hidden layers of sizes 50 and 25 without any form of regularization as D¨urr et al. (2020) found that using smaller +networks for predicting the scale parameters is sufficient. For XGBoost we use the default parameters from the Python +implementation. For the Explainable Boosting machines, we increased the number of maximum epochs to the default value +of 5000 but set the early stopping patience considerably lower to 10, as otherwise, the model reached far worse results +compared to the other models. We additionally increased the learning rate to 0.005 compared to the learning rate used in +the neural approaches as a too small learning rate resulted in bad results. Otherwise, we kept all other hyperparameters +as the default values. The GAMLSS and gamboostLSS models assume a normal distribution, with a location estimator µ +employing an identity link and a scale estimator σ with a log-link function. Due to numerical instabilities, we choose to use +the GLM-based boosting method instead of the default GAM-based version. +Table 7. Hyperparameters for the neural models for the Insurance dataset +Hyperparameter +NAMLSS1 +NAMLSS2 +DNN +MLP +NAM +Learning Rate +1e-04 +1e-04 +1e-04 +1e-04 +1e-04 +Dropout +0.5 +0.5 +0.5 +0.5 +0.5 +Hidden Layers +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +LR Decay, Patience +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +Activation +ReLU +ReLU +ReLU +ReLU +ReLU +Output Activation +Linear, Softplus +Linear, Softplus +Linear, Softplus +Linear +Linear +1 With 2 ×9 subnetworks. See Table 1 for an exemplary network structure. +2 With 9 subnetworks and each subnetwork returning a parameter for the location and shape respectively. See +Table 5 for an exemplary network structure. +Insurance +As the insurance dataset is considerably smaller than the California Housing dataset we use slightly different +model structures, as the model structure used for the California Housing dataset led to worse results. Hence, for all neural +models, we use hidden layers of sizes [250, 50, 25]. The first layer is followed by a 0.5 dropout layer. Again, we use a simple +linear activation for the models only predicting the mean and a linear and a Softplus activation for the models predicting the +mean and the variance respectively. For the first NAMLSS implementation (see Figure 1) we again use a smaller network +for predicting the variance with just one hidden layer with 50 neurons. +For XGBoost and EBM we use the same hyperparameter specifications as for the California Housing dataset. +The GAMLSS and gamboostLSS models assume a normal distribution, with a location estimator µ employing an identity +link and a scale estimator σ with a log-link function. The boosting for location, scale and shape method employed uses the +GLM based, instead of the GAM, based version. +FICO +For the FICO dataset, we use the exact same model structure as for the Insurance dataset, as the model structures +implemented for the California Housing dataset resulted in worse results. However, as it is a binary classification problem +we use a Sigmoid activation for the MLP as well as the NAM. For the DNN and both NAMLSS implementations, we use a +Sigmoid activation for the location and a Softplus activation for the scale. To generate the log-likelihoods for the models +only predicting a mean, we again use the true standard deviation of the underlying data. + +Table 8. Hyperparameters for the neural models for the FICO dataset +Hyperparameter +NAMLSS1 +NAMLSS2 +DNN +MLP +NAM +Learning Rate +1e-04 +1e-04 +1e-04 +1e-04 +1e-04 +Dropout +0.5 +0.5 +0.5 +0.5 +0.5 +Hidden Layers +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +[250, 50, 25] +LR Decay, Patience +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +Activation +ReLU +ReLU +ReLU +ReLU +ReLU +Output Activation +Sigmoid, Softplus +Sigmoid, Softplus +Sigmoid, Softplus +Sigmoid +Sigmoid +1 With 2 ×23 subnetworks. See Table 1 for an exemplary network structure. +2 With 23 subnetworks and each subnetwork returning a parameter for the location and shape respectively. See Table +5 for an exemplary network structure. +For XGBoost and EBM we had to adjust the hyperparameters in order to get results comparable to the MLP, NAM or +NAMLSS. Hence, for EBM we use 10 as the maximum number of leaves, 100 early stopping rounds and again the same +learning rate of 0.005. +For XGboost we use 500 estimators with a maximum depth of 15. η is set to 0.05. +For the GAMLSS and gamboost models we use a logistic distribution to model the response distribution, where µ estimator +uses identity and the σ estimator uses a log-link function. +Table 9. Hyperparameters for the neural models for the AirBnB dataset +Hyperparameter +NAMLSS1 +NAMLSS2 +DNN +MLP +NAM +Learning Rate +1e-04 +1e-04 +1e-04 +1e-04 +1e-04 +Dropout +0.5 +0.5 +0.5 +0.5 +0.5 +Hidden Layers +[512, 256, 50] +[512, 256, 50] +[512, 256, 50] +[512, 256, 50] +[512, 256, 50] +LR Decay, Patience +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +0.95 - 10 +Activation +ReLU +ReLU +ReLU +ReLU +ReLU +Output Activation +Gamma∗, Softplus +Gamma∗, Softplus +Gamma∗, Softplus +Linear +Linear +1 With 2 ×23 subnetworks. See Table 1 for an exemplary network structure. +2 With 23 subnetworks and each subnetwork returning a parameter for the location and shape respectively. See Table 5 for +an exemplary network structure. +∗ See formula (6) for the detailed element-wise activation function. +AirBnB +We fit the AirBnB dataset, with an Inverse Gamma distribution where applicable. However, we train the models +that only predict the mean with the squared error loss function. While one might suspect worse performances due to that, we +find that using the squared error actually leads to much smaller gamma deviances compared to the models leveraging the +Inverse Gamma distribution. Additionally, we use slightly smaller model structures than for the California Housing dataset. +For all neural models, we use hidden layers of sizes [512, 256, 50]. The first hidden layer is followed by a 0.5 dropout +layer. Throughout the hidden layers, we use ReLU activation functions. However, we deviate from that for the output layer +activation functions. For the MLP we use a Softplus activation function for the output layer, ensuring that strictly positive +values are predicted. For NAMLSS as well as the DNN we have to adjust the activation functions, as both models minimize +the log-likelihood via the parameters α and β. However, the mean prediction resulting from these parameters is defined via: +µ = +β +α − 1 +and is hence only defined for α > 1. The activation functions thus need to ensure a α prediction that is larger than 1 and a β +prediction that is larger than 0. Hence we again use a Softplus activation for the β output layer. For the α prediction, we use +the following activation function element-wise: +h(x) = +� +log(1 + exp(x)), +if log(1 + exp(x)) > 1 +1 +log(1+exp(x)), +else. + +To compute the log-likelihood for the models resulting in a mean prediction we compute the parameters α and β as follows: +α = +µ2 +σ2 + 2, +β = µ +µ2 +σ2 + 1, +with σ2 denoting the variance of the mean predictions. +For XGBoost and EBM we use a simple transformation of the target variable in order to ensure that µ > 0. Hence we fit the +model on log(y) and re-transform the predictions accordingly with exp(ˆy). Otherwise, we use the same hyperparameters as +for the California Housing dataset. +Interestingly, the NAM did not converge using the Softplus activation function as the MLP did. Using the Softplus activation +resulted in tremendously large mean gamma deviances and log-likelihoods, as the model kept predicting values that were +nearly zero. Hence, we were only able to achieve good results for the NAM using the activation function given by formula +(6). +The presented GAMLSS and gamboostLSS models assume an Inverse Gamma distribution with both µ and σ utilizing the +log-link function. It should be noted that the RS algorithm does not converge with GAMLSS, which is why CG is used. + diff --git a/WdFKT4oBgHgl3EQfni5N/content/tmp_files/load_file.txt b/WdFKT4oBgHgl3EQfni5N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..74b5dd51ab0e8eb25da244d370912f8ba415cb13 --- /dev/null +++ b/WdFKT4oBgHgl3EQfni5N/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf,len=1084 +page_content='Neural Additive Models for Location Scale and Shape: A Framework for Interpretable Neural Regression Beyond the Mean Anton Thielmann*1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Ren´e-Marcel Kruse*2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Thomas Kneib2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' and Benjamin S¨afken1 1Chair of Data Science and Applied Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' TU Clausthal 2Chair of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' University of G¨ottingen January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2023 Abstract Deep neural networks (DNNs) have proven to be highly effective in a variety of tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' making them the go-to method for problems requiring high-level predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Despite this success, the inner workings of DNNs are often not trans- parent, making them difficult to interpret or un- derstand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This lack of interpretability has led to increased research on inherently interpretable neural networks in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Models such as Neural Additive Models (NAMs) achieve visual interpretability through the combination of clas- sical statistical methods with DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, these approaches only concentrate on mean re- sponse predictions, leaving out other properties of the response distribution of the underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We propose Neural Additive Models for Loca- tion Scale and Shape (NAMLSS), a modelling framework that combines the predictive power of classical deep learning models with the inher- ent advantages of distributional regression while maintaining the interpretability of additive mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Introduction Deep learning models have shown impressive performances on a variety of predictive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' They are state-of-the-art models for tasks involving unstructured data, such as image classification (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2020), text classification (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021), audio classification (Nagrani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021), time-series forecast- ing (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2022) and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, the predictive performance comes not only at Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Correspondence to: Anton Thielmann anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='thielmann@tu- clausthal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='de, Ren´e-Marcel Kruse rene-marcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='kruse@uni- goettingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='de the price of computational demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The black-box nature of deep neural networks poses hard challenges for inter- pretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To achieve sample-level interpretability, existing methods resort to model-agnostic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Locally Inter- pretable Model Explanations (LIME) (Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2016) or Shapley values (Shapley, 1953) and their extensions (Sun- dararajan & Najmi, 2020) try to explain model predictions via local approximation and feature importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Sensitivity- based approaches (Horel & Giesecke, 2020), exploiting significance statistics, can only be applied to single-layer feed-forward neural networks and can hence not be used to model difficult non-linear effects, requiring more complex model structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Subsequently, high-risk domains, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' medical ap- plications often cannot exploit the advantages of complex neural networks due to their lack of innate interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The creation of these innately interpretable models hence re- mains an important challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Achieving the interpretability from flexible statistical models as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Generalized Linear Models (GLMs) (Nelder & Wedderburn, 1972) or Gener- alized Additive Models (GAMs) (Hastie, 2017), in deep neural networks, however, is inherently difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Recently, Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) introduced Neural Additive Models (NAMs), a framework that models all features individually and thus creates visual interpretability of the single fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While this is an important step towards interpretable deep neural networks, any insightfulness of aspects beyond the mean is lost in the model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To counter that, we propose the neural counterpart to Generalized Additive Models for Location, Scale and Shape (GAMLSS) (Rigby & Stasinopoulos, 2005), the Neural Additive Model for Location, Scale and Shape (NAMLSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMLSS adopts and iterates on the model class of GAMLSS, in the same scope as NAMs (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021) on GAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The GAMLSS framework relaxes the exponential family assumption and replaces it with a general distribution family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The systematic part of the model is expanded to allow not only the mean (location) but all the parameters of the condi- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='11862v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='ML] 27 Jan 2023 tional distribution of the dependent variable to be modelled as additive nonparametric functions of the features, resulting in the following model notation: θ(k) = g(k)−1 � �β(k) + Jk � j=1 f (k) j (x(k) j ) � � = ηθ(k), with the superscript k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' , K denoting the k-th param- eter and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' , J denoting the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The model assumes that the underlying response obser- vations yi for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' , n are conditionally indepen- dent given the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The assumed conditional density can depend on up to K different distributional parameters*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each of these distribution parameters θ(k) can be modelled using its additive predictor ηθ(k) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' , K, allowing for complex relationships between the response and predic- tor variables, as well as the flexibility to choose different distributions for different parts of the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' An additional important component of the GAMLSS model is the link function g(k)(·), which allows each parameter of the distribution vector to be conditional on different sets of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' In the case that the distribution under consider- ation features only one distribution parameter, the model simplifies to an ordinary GAM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Therefore, GAMLSS is to be seen as a conceptual extension of the GAM idea and is suitable for the extension and generalisation of ap- proaches such as NAMs which are themselves built upon the GAM idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For an overview of the current state of re- gression models that focus on the full response distribution approaches see Kneib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While the NAM learns linear combinations of different in- put features to learn arbitrary complex functions and at the same time provides improved interpretability, these models, like their statistical counterparts the GAM models, focus exclusively on modelling mean and dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This is in contrast to the GAMLSS and subsequently, the proposed NAMLSS models, which substantially broadens the scope by allowing all underlying parameters of the response dis- tribution to potentially depend on the information of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Contributions The contributions of the paper hence can be summarized as follows: We present a novel architecture for Neural Additive Models for Location, Scale and Shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Compared to state-of-the-art GAM, GAMLSS and In practice most application focus on up to four θi = � θ(1) i , θ(2) i , θ(3) i , θ(4) i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' DNNs our NAMLSS achieves similar results on bench- mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We demonstrate that NAMLSS effectively captures the information underlying the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Lastly, we show that the NAMLSS approach allows to go beyond the mean prediction of the response and to model the entire response distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The rest of the paper is structured as follows: Section 2 gives an overview of the current state of the NAM and other more interpretable deep learning methods on which our model is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Section 3 provides an introduction to the underlying architectural and mathematical concepts of the proposed NAMLSS approach, as well as contrasts it with its methodological sibling the GAMLSS approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Section 4 analyzes the properties of the implemented methods by contrasting their performance in comparison to popular and common modelling techniques from statistics, machine and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The last Section 5 offers a deeper discussion on the improved interpretability of the presented method, contrasting the results with those of other recent methods in the field of interpretable deep learning and thereby offering an outlook on upcoming applications and possible further research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Literature Review The idea of generating feature-level interpretability in deep neural networks by translating GAMs into a neural frame- work was already introduced by Potts (1999) and expanded by de Waal & du Toit (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While the framework was remarkably parameter-sparse, it did not use backpropaga- tion and hence did not achieve as good predictive results as GAMs, while remaining less interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' More recently, Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) introduced NAMs, a more flexible approach than the Generalized Additive Neural Networks (GANNs) introduced by de Waal & du Toit (2007) that leverages the recent advances in the field of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMs are a class of flexible and powerful machine learn- ing models that combine the strengths of neural networks and GAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' These models can be used to model complex, non-linear relationships between response and predictor variables, and can be applied to a wide range of tasks includ- ing regression, classification, and time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The basic structure of a NAM consists of a sum of mul- tiple components, each representing a different aspect of the relationship between the response and predictor vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' These components can be linear, non-linear, or a combination of both, and can be learned using a variety of optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' One of the key advantages of NAMs is their inherent ability to learn the interactions be- tween different predictor variables and the response without the need for manual feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This allows NAMs to capture complex relationships in the data that may not be easily apparent to the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The general form of a NAM can be written as: E(y) = h � �β + J � j=1 fj(xj) � � , (1) where h(·) is the activation function used in the output layer, x ∈ Rj are the input features, β is the global intercept term, and fj : R → R represents the Multi-Layer Perceptrons (MLPs) corresponding to the j-th feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The similarity to GAMs is apparent, as the two frameworks mostly distin- guish in the form the individual features are modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' h(·) is comparable to the link function g(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Several extensions to the NAM framework have already been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) extends NAMs to ac- count for pairwise interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) introduced NODE-GAM, a differentiable model based on forgetful decision trees developed for high-risk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All these models follow the additive framework from GAMs and learn the nonlinear additive features with separate net- works, one for each feature or feature interaction, either leveraging MLPs (Potts, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' de Waal & du Toit, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021) or using decision trees (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The applications of such models range from nowcasting (Jo & Kim, 2022), financial applications (Chen & Ye, 2022), to survival analysis (Peroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While the linear combination of neural subnetworks provides a visual in- terpretation of the results, any interpretability beyond the feature-level representation of the model predictions is lost in their black-box subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The idea of focusing on more than the underlying mean prediction is certainly relevant and has been an important part, especially of the statistical literature in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' There has been a strong focus on the GAMLSS (Rigby & Stasinopoulos, 2005) framework, conditional transforma- tion models (Hothorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2014), density regression (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 1996) or quantile and expectile regression frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, these methods are inferior to machine and deep learning techniques in terms of pure predictive power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' the disadvantage of not being able to deal with unstructured data forms such as images, text or audio files;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' or the inher- ent problems of statistical models in dealing with extremely large and complex data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' One resulting development to deal with these drawbacks is frameworks that utilize statis- tical modelling methods and combine them with machine learning techniques such as boosting to create new types of distributional regression models such as boosted generalized additive model for location, scale and shape as presented by Hofner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, the models leveraging boosting techniques, while successfully modelling all distri- butional parameters, lack the inherent interpretability from GAMLSS or even the visual interpretability from NAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Methodology While NAMs incorporate some feature-level interpretabil- ity and hence entail easy interpretability of the estimated regression effects, they are unable to capture skewness, het- eroskedasticity or kurtosis in the underlying data distribu- tion due to their focus on mean prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Therefore, the presented method is the neural counterpart to GAMLSS, offering the flexibility and predictive performance of neu- ral networks while maintaining feature-level interpretability and which allows estimation of the underlying total response distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Let D = {(x(i), y(i))}n i=1 be the training dataset of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each input x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' , xJ) contains J features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' y denotes the target variable and can be arbitrarily dis- tributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMLSS are trained by minimizing the negative log-likelihood as the loss function, − log (L(θ|y)) by op- timally approximating the distributional parameters, θ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each parameter, θ(k), is defined as: θ(k) = h(k) � �β(k) + J � j=1 f (k) j (xj) � � , where h(k)(·) denotes the output layer activation functions dependent on the underlying distributional parameter, β(k) denotes the parameter-specific intercept and f (k) j : R → R represents the feature network for parameter k for the j-th feature, subsequently called the parameter-feature network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Just as in GAMLSS, θ(k) can be derived from a subset of the J features, however, due to the inherent flexibility of the neural networks, defining each θ(k) over all J is sufficient, as the individual feature importance for each parameter, θ(k), is learned automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each parameter- feature network, f (k) j , can be regularized employing regular dropout coefficients in conjunction with feature dropout coefficients, λ(k) 1j and λ(k) 2j respectively, as also implemented by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' a normal distribution, NAMLSS would hence mini- mize − log � L(ˆµ, ˆσ2|y) � = − � −n 2 log(2πˆσ2) − 1 2ˆσ2 n � i=1 (yi − ˆµ)2 � , (2) where ˆµ = β(1) + J � j=1 f (1) j (xj) (3) and ˆσ2 = log � �1 + exp � �β(2) + J � j=1 f (2) j (xj) � � � � , (4) utilizing a Softplus activation function for the scale parame- ter and a linear activation for the location parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The network structure of a simple NAMLSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each input variable as well as each distributional parameter is handled by a different neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' h(k) are different activation func- tions depending on the distributional parameter that is modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' a quadratic transformation for modelling the variance in a nor- mally distributed variable to ensure the non-negativity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The presented structure demonstrates a NAMLSS modelling a distribution with two parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We propose two different network architectures that can both flexibly model all distributional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The first is depicted in Figure 1 and creates J subnetworks for each of the K distributional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each distributional sub- network is comprised of the sum of the parameter-feature networks f (k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we create K × J parameter-feature networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To account for distributional restrictions, each distributional subnetwork is specified with possibly differ- ing activation functions in the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The second model architecture, possible due to the flexibility of neural networks, leverages the architecture of NAMs (see formula (1)) and is depicted in figure 5 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Here, only J subnetworks are created, with each subnetwork having a K-dimensional output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This architecture thus creates the same number of subnetworks as a common NAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each distributional parameter, θ(k), is subsequently obtained by summing over the k-th output of the J subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Every dimension in the output layer can be activated using differ- ent activation functions, according to parameter restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This allows the capture of interaction effects between the given model parameters in each of the subnetworks*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Integrating possible feature interactions can easily be achieved in both architectures by training a fully connected MLP on the residuals after the NAMLSS has converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, again leveraging the example of a normally dis- tributed y, equation (3) for that interaction part becomes: − log � L(˜µ, ˜σ2|e) � = − � −n 2 log(2π˜σ2) − 1 2˜σ2 n � i=1 (ei − ˜µ)2 � , (5) where ei denote the residuals yi − ˆyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' As neural networks can achieve optimal approximation rates, NAMLSS can learn any non-linearity or dependence be- tween distribution parameters and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' And unlike GAMLSS, NAMLSS can model jagged shape functions and easily incorporate huge amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While predicting all parameters from a distribution may not always improve predictive power, understanding the under- lying data distribution is crucial in high-risk domains and can provide valuable insights about feature effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' As an example, Figure 2 illustrates the fit of our approach on data following a Johnson’s SU distribution, including 3 features, compared to the fit of a MLP that minimizes the Mean Squared Error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The MLP has a better predictive per- formance with an MSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0002, however, NAMLSS is able to reflect the underlying data distribution much more accurately (as shown in Figure 2), even though it has an MSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Note, that for distributions where only one parameter is mod- elled, the two proposed NAMLSS structures are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' r1 (1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='J C1 h(2) (2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='JTable 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Results for Synthetic data: The benchmark results for the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We compare the models based on log-likelihoods, ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A larger log-likelihood represents a better fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Means and standard deviations of 5-fold cross validation are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1, for a comprehensive list of the used log-likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Note, that for distributions where only one parameter is modelled, the two proposed NAMLSS structures are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, we report only one value for the Binomial as well as the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Model Binomial Poisson Normal Inverse Gaussian Weibull Johnson’s SU GAMLSS 397 ± (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0) 800 ± (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) 600 ± (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 385 ± (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 625 ± (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 370 ± (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8) gamboostLSS1 315 ± (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 800 ± (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8) 575 ± (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 366 ± (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0) 648 ± (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 478 ± (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) DNN 260 ± (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 802 ± (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 558 ± (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='3) 343 ± (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 624 ± (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 285 ± (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) NAMLSS2 589 ± (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 377 ± (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 621 ± (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='3) 326 ± (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) NAMLSS3 274 ± (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 802 ± (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 577 ± (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8) 362 ± (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='9) 620 ± (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) 327 ± (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 1 For boosting one-parametric families like the Binomial or Poisson distribution, a model-based boosting algorithm is used (Hofner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With K ×5 subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 3 With 5 subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Johnson’s SU Distribution: Simulated Johnson’s SU distribution and the fit of a simple NAMLSS (see Figure 1) and a MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While the MLP achieves an impressive fit concerning the quadratic loss, it clearly cannot capture the underlying distribution adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Benchmarking To demonstrate the competitiveness of the presented method, we perform several analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We compare the performance of NAMLSS with several state-of-the-art models including neural as well as non-neural approaches and orientate on the benchmarks performed by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Addition- ally, we compare related methods of distribution-focused data analysis approaches that overcome the focus on relating the conditional mean of the response to features and instead target the complete conditional response distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We analyze multiple datasets as well as conduct experiments on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We choose the following baselines for the comparisons: Multilayer Perceptron (MLP): Unrestricted fully connected deep neural network trained with either a mean squared error loss function (regression) or binary cross entropy (logistic regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Gradient Boosted Trees (XGBoost): Decision tree based gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We use the implementation provided by Chen & Guestrin (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Neural Additive Models (NAMs): Linear combina- tion of DNNs as described in equation (1) and pre- sented by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Explainable Boosting Machines (EBMs): State-of- the-art Generalized Additive Models leveraging shal- low boosted trees (Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Deep Neural Network (DNN): Similar to the Mul- tilayer Perceptron a fully connected neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, not trained to minimize the previously men- tioned loss functions but to minimize the negative log- likelihood of the specified distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All distribu- tional parameters are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Generalized Additive Models for Location Scale and Shape (GAMLSS): Standard GAMLSS models using the R implementation from Rigby & Stasinopou- los (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' gamboost for Location Scale and Shape (gamboost- LSS): Fitting GAMLSS by employing boosting tech- niques as proposed by Hofner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We preprocess all used datasets exactly as done by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We perform 5-fold cross-validation for all datasets and report the average performances over all folds as well as the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For reproducibility, we Johrson su Distribution 1D quantile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='05 quantile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6 NAMLSS MLP True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4 1 I 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 SL6O LDO0 1D25 LD5 1075 1101Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Benchmark results for the regression comparison datasets: For models not explicitly modelling a shape parameter, the shape is approximated with a constant as the true standard deviation of the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Higher log-likelihoods (ℓ) and lower MSEs are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We report results on two commonly used regression datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The California Housing dataset for predicting house prices (Pace & Barry, 1997a) and an Insurance dataset for predicting billed medical expenses (Lantz, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' CA Housing Insurance Model ℓ (↑) MSE (↓) ℓ (↑) MSE (↓) MLP 4191 ±(41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='197 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005) 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8 ± (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='163 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='022) XGBoost 4219 ±(39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='211 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005) 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8 ± (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='161 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='014) NAM 8344 ±(1764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='273 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='037) 474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7 ± (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='249 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='029) EBM 4202 ±(42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='203 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='004) 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8 ± (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='139 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='017) DNN 2681 ±(1279) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='197 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005) 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2 ± (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='165 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='026) GAMLSS 3512 ±(66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='390 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='035) 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 ± (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='269 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='050) gamboostLSS 3812 ±(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='415 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='026) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4 ± (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='268 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='050) NAMLSS1 2667 ± (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='245 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='004) 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7 ± (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='268 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='043) NAMLSS2 2329 ± (176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='265 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005) 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6 ± (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='265 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='040) 1 With J × K subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With J subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' have only chosen publicly available datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The datasets, as well as the preprocessing and the seeds set for obtaining the folds, are described in detail in the Appendix, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We fit all models without an intercept and explicitly do not model feature interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Beyond the mean: Synthetic data comparison study The synthetic data used for this task is generated from the same underlying processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Five features are included in each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The data-generating functions used to generate the true underlying distributional parameters can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each of the five input vectors xj is sampled from a uniform distribution U(0, 1), with a total of n = 3000 observations per data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The remaining parameters are generated based on the input vectors and the chosen distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We selected distributions that are widely used, popular in science, or relatively complex to reflect a diverse range of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We compare models that specifically model all distributional parameters in this simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The results can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We find that the presented NAMLSS perform similarly to fully connected DNNs that specifically minimize the log- likelihood and perform better than GAMLSS or gamboost- LSS, all while maintaining visual intelligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We can hence confirm the findings from Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) that additive neural networks can achieve similar results to fully connected DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Experiments with Real World Data Normal Distribution For a regression benchmark, we use the California Housing (CA Housing) dataset (Pace & Barry, 1997a) from sklearn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2011b) and the Insurance dataset (Lantz, 2019) and standard normalize the response variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' As comparison metrics we use the log-likelihood, ℓ (see Appendix, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1), as well as the mean squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A larger log-likelihood thus repre- sents a better model fit, while a smaller mean squared error represents a better predictive performance in terms of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Thus, we try to illustrate the trade-off between pure predictive performance, interpretability and overall data fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 2 presents the results obtained from the analyses carried out on the two popular regression data bench- mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' In both applications, a normal distribution N � µ, σ2I � of the underlying response variable was as- sumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' As the log-likelihood of a normal distribution (see equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1)) is dependent on two parameters, but models as an MLP or XGBoost only predict a single parameter, we adjust the computation accordingly and use the standard deviation calculated from the underlying data for XGBoost, EBM, NAM and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For NAMLSS, independent of the implementation, we use a Softplus activation for the scale parameter σ2 to ensure non-negativity and a linear activa- tion for the mean µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The NAMLSS approach achieves the highest log-likelihood values of all presented approaches which speaks for its good approximation capabilities for the California Housing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' It can also be seen that the trade-off between MSE performance and the possibility of modelling the response distribution is relatively moderate in its impact as NAMLSS even outperforms the predictive performance of a NAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The results could have been further improved by accounting for feature interactions, resulting in a log-likelihood of -1654 and an MSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' One of the advantages of NAMLSS compared to DNNs is the feature level interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Similar to NAMs, we can plot and visually analyze the results (see Figures 3 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' California Housing: Graphs for median income and population respectively learned by the NAMLSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We see an increase in housing prices with a larger median income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Addi- tionally, we find a larger variance in housing prices in less densely populated areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' California Housing: Graphs for longitude and latitude respectively learned by the NAMLSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The house price jumps around the location of Los Angeles are depictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Addi- tionally, we find a decrease in variance for areas further away from the large cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, we are able to accurately depict shifts in variance in the underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' It is, for example, clearly distinguishable, that with a larger median income, the house prices tend to vary much stronger than with a smaller median income (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A piece of information, that is lost in the models focusing solely on mean predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Addition- ally, we are capable of accurately representing sharp price jumps around the location of San Francisco, depicted by the jumps in the graphs for longitude and latitude (see Figure 4) as compared to GAMLSS, NAMLSS are additionally capable of representing jagged shape functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the Insurance dataset, which is only comprised of 1338 observations, we unsurprisingly find strong performances of the classical statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMLSS perform similarly to NAMs in terms of pure predictive power and perform similarly to DNNs in terms of log-likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Inverse Gamma For the AirBnB dataset, also analyzed by R¨ugamer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2020), we assume an Inverse Gamma dis- tribution IG(α, β) as the underlying data distribution (see equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) for the log-likelihood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For NAMLSS as well Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Benchmark results for the AirBnB dataset: For mod- els not explicitly modelling the distributional parameters, transfor- mations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Similar for models resulting in distribu- tional parameters but not mean predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A larger log-likelihood, ℓ, and smaller average gamma deviance (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' AirBnB Model ℓ (↑) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Gamma Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (↓) MLP 6827 ± (177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='55 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='04) XGBoost 5618 ± (152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='48 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='09) NAM 5892 ± (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='72 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='10) EBM 5474 ± (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='49 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='09) DNN 5555 ± (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='69 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='04) GAMLSS 5419 ± (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='71 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='44) gamboostLSS 5421 ± (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='54 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='07) NAMLSS1 5383 ± (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='59 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='09) NAMLSS2 5422 ± (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='59 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='10) 1 With J × k subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With J subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' as the DNN we have to adjust the activation functions, as both models minimize the log-likelihood via the parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, the mean prediction resulting from these parameters is defined via: µ = β α − 1 and is hence only defined for α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The activation func- tions thus need to ensure an α prediction that is larger than 1 and a β prediction that is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we again use a Softplus activation for the β output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the α predic- tion, we use the following activation function element-wise: h(x) = � log(1 + exp(x)), if log(1 + exp(x)) > 1, 1 log(1+exp(x)), else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (6) To compute the log-likelihood for the models resulting in a mean prediction we compute the parameters α and β as follows: α = µ2 σ2 + 2, β = µ µ2 σ2 + 1, with σ2 denoting the variance of the mean predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGBoost and EBM we use a simple transformation of the target variable to ensure that µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we fit the model on log(y) and re-transform the predictions accordingly with 6 4 Features Contribution 2 2 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 Medinc6 4 Features Contribution 2 2 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 Population2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='00 Latitude Longitudleexp(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Interestingly, the NAM did not converge using the Softplus activation function as the MLP did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Using the Softplus activation resulted in tremendously large mean gamma deviances and log-likelihoods, as the model kept predicting values that were nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, we were only able to achieve good results for the NAM using the activation function given by formula (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Logistic Distribution For a (binary) classification bench- mark we use the FICO dataset (FICO, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A logistic distribution, LO (µ, s), of the underlying response variable was assumed (see equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1) for the log-likelihood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Again, we use the true standard deviation of the underlying data for the models only resulting in a mean prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For evaluating the sole predictive performance we use the Area Under the Curve (AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The models resulting in a mean prediction use binary cross-entropy as the loss function and hence a sigmoid activation function on the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMLSS outperforms all models in terms of log-likelihood and maintains a reasonable predictive performance, compa- rable to NAMs, XGBoost and EBMs (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Benchmark results for the FICO dataset: For all mod- els, modelling distributional parameters, a logistic distribution is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For all other models, the binary cross-entropy loss func- tion is minimized during training and the distributional parameters for computing the log-likelihood, ℓ, are approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The Area under the Curve (AUC) (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' FICO Model ℓ (↑) AUC (↑) MLP 1813 ±(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='79 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='007) XGBoost 1976 ±(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='73 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='010) NAM 1809 ± (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='73 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='010) EBM 1944 ±(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='73 ±(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='010) DNN 1230 ± (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='72 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='006) GAMLSS 1321 ± (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='78 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='009) gamboostLSS 1191 ± (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='79 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='008) NAMLSS1 1201 ± (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='73 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='010) NAMLSS2 1160 ± (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='72 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='008) 1 With J × k subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With J subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Conclusion & Future Work We have presented Neural Additive Models for Location, Scale and Shape and their theoretical foundation as the neural counterpart to GAMLSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' NAMLSS can model an arbitrary number of parameters of the underlying data distri- bution while preserving the predictive quality of NAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The visual intelligibility achieved by NAMs is also maintained by NAMLSS, with the added benefit of gaining further insights from knowledge of additional distribution charac- teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, NAMLSS are a further step in the direction of fully interpretable neural networks and already offer in- terpretability that may make them suitable for high-risk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The extensibility of NAMLSS offers many different further applied and theoretical research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' One important point is the extension of the modelling of the distribution of the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Many empirical works focus on modelling not just one, but several responses conditionally on covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' One way to do this is to use copula meth- ods, which are a valuable extension of our approach, hence including a copula-based approach for NAMLSS models would greatly improve the overall general usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' An- other possible extension would be the adaptation to mixture density networks, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' done by Seifert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Another possible focus is to switch our approach to a Bayesian-based training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Bayesian approaches are particularly well suited to deal with epistemic uncer- tainty and to incorporate it into the modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Another advantage is that Bayesian approaches are particularly suit- able in cases where insufficiently small training datasets have to be dealt with and have been shown to have better prediction performance in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Finally, there should be a focus on incorporating unstruc- tured data to extend the previously purely tabular data with high-dimensional input structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Acknowledgements Funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within project 450330162 is gratefully acknowledged.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Seyedhosseini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', and Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Coca: Contrastive captioners are image- text foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='01917, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Zeng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', and Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Are transformers effective for time series forecasting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='13504, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Wen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Sun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Yao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Film: Frequency improved legendre memory model for long-term time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='08897, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Log-Likelihoods As the presented method minimizes negative log-likelihoods, we created a comprehensive list of all the log-likelihoods of the distributions used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' When we reference the results of NAMLSS these are the log-likelihoods we used for fitting the models as well as evaluating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (Bernoulli) Logistic Distribution The log-likelihood function for a logistic distribution is given by: log (L(µ, σ|y)) = n � i=1 � yi log( 1 1 + e−( yi−µ σ ) ) + (1 − yi) log(1 − 1 1 + e−( yi−µ σ ) ) � , with n is as the number of observations and the parameters location µ ∈ R, scale σ ∈ R+ and x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Binomial Distribution The log-likelihood function for a binomial distribution is given by: log (L(k|n, p)) = k log(p) + (n − k) log(1 − p) + log ��n k �� , where n is the number of trials, the parameters success probability is given by p ∈ [0, 1] and the number of successes is denoted as k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Inverse Gamma Distribution The log-likelihood function of the invers gamma distribution is defined as: log (L(α, β|y)) = −n (α + 1) log y − n log Γ(α) + nα log β − n � i=1 βy−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' with α > 0 and β > 0 and where the upper bar operand indicates the arithmetic mean Normal Distribution The log-likelihood function for a normal distribution is given by: log � L(µ, σ2|y) � = −n 2 log(2πσ2) − 1 2σ2 n � i=1 (yi − µ)2, where n is the underlying number of observations and parameters y ∈ R, location µ ∈ R and scale σ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Inverse Gaussian Distribution The log-likelihood function of the inverse Gaussian distribution is given by: log (L(µ, σ|x)) = n 2 ln(σ) − n � i=1 σ(xi − µ)2 2µ2xi , with n is as the number of observations and the parameters location µ ∈ R+, scale σ ∈ R+ and x ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Johnson’s SU The log-likelihood function of the Johnson’s SU distribution is defined as: log (L(β, ω, µ, σ|y)) = n log � β ω √ 2π � − β2 2ω2 n � i=1 �(yi − µ)2 σ2 + ln � 1 + (yi − µ)2 ω2σ2 �� , with n is as the number of observations and the parameters location µ ∈ R, scale σ ∈ R+, shape ω ∈ R+, skewness β ∈ R and y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Weibull The log-likelihood function of the Weibull distribution is defined as: log (L(λ, β, |y)) = n ln β − nβ ln λ − n � i=1 �yi λ �β + (β − 1) n � i=1 ln yi, with n is the number of observations and with the location λ ∈ R+, the shape β ∈ R+ and y ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Deviance Measures We use several deviance measures, to evaluate the model performances beyond the log-likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' These deviance measures are focused on the mere predictive power of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The depiction of both, the log-likelihoods as well as these deviance measures, thus captures the trade-off between pure predictive power and the ability to capture the underlying data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Mean Squared Error The mean squared error is defined as : MSE = 1 n n � i=1 (yi − ˆyi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Mean Gamma Deviance The mean gamma deviance used for the AirBnB dataset is defined as: D = 2 n n � i=1 � log( ˆyi) yi ) + yi ˆyi − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Area Under the Curve We use the Riemannian formula for the AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence the area of rectangles is defined as: AR = � i = 1n−1f(xi)∆x, and hence with larger n, the definite integral of f from a to b is defined as: � b a f(x)dx = lim n→∞ n−1 � i=0 f(xi)∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Network architecture We propose two different network architectures that can both flexibly model all distributional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The first one is depicted in Figure 1 and creates J subnetworks for each distributional parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each distributional subnetwork is comprised of the sum of f (k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we create K × J subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To account for distributional restrictions, each distributional subnetwork is specified with possibly differing activation functions in the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The second model architecture is depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Here we only create J subnetworks and hence have the same amount of subnetworks as a common NAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each subnetwork then has a k-dimensional output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each distributional Parameter, θ(k), is subsequently obtained by summing over the k-th output of the J subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each dimension in the output layer can be activated using different activation functions, adjusting to parameter restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The network structure of a simple NAMLSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Each input variable as well as each distributional parameter is handled by a different neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' hk are different activation functions depending on the distributional parameter that is modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' a quadratic transformation for modelling the variance in a normally distributed variable to ensure the non-negativity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Benchmarking The benchmark study for used real-world datasets was performed under similar conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All datasets are publicly available and we describe every preprocessing step as well as all model specifications in detail in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Synthetic Data Generation For the simulation of the data, respectively their underlying distribution parameters θ = � θ(1), θ(2), θ(3), θ(4)� , the following assumptions are made: θ(1) = 30 13x1 � (3x2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5) − 2 sin �x3 2 ��−1 + 113 115x4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='1x5, θ(2) = exp � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0035x1 + (x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='23)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='42x3 � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0001x4, θ(3) = 1 42(4x1 − 90x2), θ(4) = exp (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0323 ∗ x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0123 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='0234 ∗ x4) , where each of the five input vectors xj is sampled from a uniform distribution U(0, 1), with a total of n = 3000 observations per data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Preprocessing We implement the same preprocessing for all used datasets and only slightly adapt the preprocessing of the target variable for the two regression problems, California Housing and Insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We closely follow Gorishniy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) in their preprocessing steps and use the preprocessing also implemented by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence all numerical variables are scaled between -1 and 1, all categorical features are one-hot encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' In contrast to Gorishniy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) we do not implement quantile smoothing, as one of the biggest advantages of neural models is the capability to model jagged shape functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We use 5-fold cross-validation and report mean results as well as the standard deviations over all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For reproducibility, we use the sklearn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2011a) Kfold function with a random state of 101 and shuffle equal to true for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the two regression datasets, we implement a standard normal transformation of the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' This results in better performances in terms of log-likelihood for all models only predicting a mean and is hence even disadvantageous for the presented NAMLSS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Datasets Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Statistics of the benchmarking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Dataset No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Samples No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Features Distribution Task California Housing 20640 8 Normal N(µ, σ) Regression Insurance 1338 6 Normal N(µ, σ) Regression Fico 10459 23 Logistic LO(µ, s) Classification AirBnB 4568 9 Inverse Gamma IG(α, β) Regression California Housing The California Housing (CA Housing) dataset (Pace & Barry, 1997b) is a popular publicly available dataset and was obtained from sklearn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2011a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' It is also used as a benchmark in (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=', 2021) and Gorishniy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) and we achieve similar results concerning the MSE for the models which were used in both publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The dataset contains the house prices for California homes from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' census in 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The dataset is comprised of 20640 observations and besides the logarithmic median house price of the blockwise areas as the target variable contains eight predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' As described above, we additionally standard normalize the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All other variables are preprocessed as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Insurance The Insurance dataset is another regression type dataset for predicting billed medical expenses (Lantz, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The dataset is publicly available in the book Machine Learning with R by Lantz (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, the data is freely available on Github and Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' It is a small dataset with only 1338 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The target variable is charges, which represents the Individual medical costs billed by health insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Similar to the California Housing regression we standard normalize the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, the dataset includes 6 feature variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' They are preprocessed as described above, which, due to one-hot encoding leads to a feature matrix with 9 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' FICO Similar to Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) we also use the FICO dataset for our benchmarking study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, we use it as described on the website and hence use the Risk Performance as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' A detailed description of the features and their meaning is available at the Explainable Machine Learning Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The dataset is comprised of 10459 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We did not implement any preprocessing steps for the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' AirBnB For the AirBnB data, we orientate on R¨ugamer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2020) and used the data for the city of Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The dataset is also publicly available and was taken from Inside AirBnB on January 15, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' After excluding the variables ID, Name, Host ID, Host Name, Last Review and after removing rows with missing values the dataset contains 4568 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, we drop the Neighbourhood variable as firstly the predictive power of that variable is limited at best and secondly not to create too large feature matrices for GAMLSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, in addition to the target variable, the dataset contains 9 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All preprocessing steps are subsequently performed as described above and the target variable, Price, is not preprocessed at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Model Architectures & Hyperparameters As we do not implement extensive hyperparameter tuning for the presented NAMLSS framework, we do not perform hyperparameter tuning for the comparison models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We fit all models without an intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, we try to achieve the highest comparability by choosing similar modelling frameworks, network architectures and hyperparameters where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All neural models are hence fit with identical learning rates, batch sizes, hidden layer sizes, activation functions and regularization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Through all neural models and all datasets, we use the ADAM optimizer (Kingma & Ba, 2014) with a starting learning rate of 1e-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the larger datasets, California Housing and FICO, we orient on Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) and use larger batch sizes of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the smaller dataset, Insurance, we use a smaller batch size of 256 and for the AirBnB dataset we use a batch size of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For every dataset and for every neural model, the maximum number of epochs is set to 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, we implement early stopping with a patience of 150 epochs and no model over no fold and no dataset ever trained for the full 2000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, we reduce the learning rate with a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 with patience of 10 epochs for all models for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We use the Rectified Linear Unit (ReLU) activation function for all hidden layers for all models: h(x) = � 0, x < 0 x, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We also experimented with the Exponential centred hidden Unit (ExU) activation function presented by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) but found no improvement in model performance and even a slight deterioration for most models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the statistical models used from the GAMLSS and gamboostLSS frameworks, we do not optimize the model hyperparam- eters, as with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We use the respective default settings unless otherwise stated in the modelling descriptions included in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We try to keep the model settings equal between all models, if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' All GAMLSS models use the same RS solver proposed by (Rigby & Stasinopoulos, 2005), in cases where this approach does not lead to convergence, the alternative CG solver presented by (Cole & Green, 1992) is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To exclude possible numerical differences, the same distributions from the GAMLSS R package are used for modelling the response distribution and calculating the log-likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' gamboostLSS allows the use of different boosting approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Here we use the implemented boosting methods based on GAMs and GLMs and choose the model that performs better in terms of log-likelihood and the assumed loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' California Housing For the California Housing dataset, we orient again on Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2021) and use the following hidden layer sizes for all networks: [1000, 500, 100, 50, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The second hidden layer is followed by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While subsequently the NAM and NAMLSS have much more trainable parameters than the MLP and the DNN, we find that the MLP and DNN outperform the NAM and NAMLSS in terms of mean prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, we encountered severe overfitting when using the same number of parameters in an MLP as in the NAM and NAMLSS implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the mean predicting models, we use a one-dimensional output layer with a linear activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the DNN and both NAMLSS implementations, we use a linear activation over the mean prediction and a Softplus activation for the variance prediction with: h(x) = log(1 + exp(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hyperparameters for the neural models for the California Housing dataset Hyperparameter NAMLSS1 NAMLSS2 DNN MLP NAM Learning Rate 1e-04 1e-04 1e-04 1e-04 1e-04 Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='25 Hidden Layers [1000, 500, [1000, 500, [1000, 500, [1000, 500, [1000, 500, 100, 50, 25] 100, 50, 25] 100, 50, 25] 100, 50, 25] 100, 50, 25] LR Decay, Patience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 Activation ReLU ReLU ReLU ReLU ReLU Output Activation Linear, Softplus Linear, Softplus Linear, Softplus Linear Linear 1 With 2 ×8 subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With 8 subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the NAMLSS implementation depicted in Figure 1 we use a smaller network structure for predicting the variance with two hidden layers of sizes 50 and 25 without any form of regularization as D¨urr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' (2020) found that using smaller networks for predicting the scale parameters is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGBoost we use the default parameters from the Python implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the Explainable Boosting machines, we increased the number of maximum epochs to the default value of 5000 but set the early stopping patience considerably lower to 10, as otherwise, the model reached far worse results compared to the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' We additionally increased the learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005 compared to the learning rate used in the neural approaches as a too small learning rate resulted in bad results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Otherwise, we kept all other hyperparameters as the default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The GAMLSS and gamboostLSS models assume a normal distribution, with a location estimator µ employing an identity link and a scale estimator σ with a log-link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Due to numerical instabilities, we choose to use the GLM-based boosting method instead of the default GAM-based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hyperparameters for the neural models for the Insurance dataset Hyperparameter NAMLSS1 NAMLSS2 DNN MLP NAM Learning Rate 1e-04 1e-04 1e-04 1e-04 1e-04 Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 Hidden Layers [250, 50, 25] [250, 50, 25] [250, 50, 25] [250, 50, 25] [250, 50, 25] LR Decay, Patience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 Activation ReLU ReLU ReLU ReLU ReLU Output Activation Linear, Softplus Linear, Softplus Linear, Softplus Linear Linear 1 With 2 ×9 subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With 9 subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Insurance As the insurance dataset is considerably smaller than the California Housing dataset we use slightly different model structures, as the model structure used for the California Housing dataset led to worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, for all neural models, we use hidden layers of sizes [250, 50, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The first layer is followed by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Again, we use a simple linear activation for the models only predicting the mean and a linear and a Softplus activation for the models predicting the mean and the variance respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the first NAMLSS implementation (see Figure 1) we again use a smaller network for predicting the variance with just one hidden layer with 50 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGBoost and EBM we use the same hyperparameter specifications as for the California Housing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The GAMLSS and gamboostLSS models assume a normal distribution, with a location estimator µ employing an identity link and a scale estimator σ with a log-link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The boosting for location, scale and shape method employed uses the GLM based, instead of the GAM, based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' FICO For the FICO dataset, we use the exact same model structure as for the Insurance dataset, as the model structures implemented for the California Housing dataset resulted in worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, as it is a binary classification problem we use a Sigmoid activation for the MLP as well as the NAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the DNN and both NAMLSS implementations, we use a Sigmoid activation for the location and a Softplus activation for the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To generate the log-likelihoods for the models only predicting a mean, we again use the true standard deviation of the underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hyperparameters for the neural models for the FICO dataset Hyperparameter NAMLSS1 NAMLSS2 DNN MLP NAM Learning Rate 1e-04 1e-04 1e-04 1e-04 1e-04 Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 Hidden Layers [250, 50, 25] [250, 50, 25] [250, 50, 25] [250, 50, 25] [250, 50, 25] LR Decay, Patience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 Activation ReLU ReLU ReLU ReLU ReLU Output Activation Sigmoid, Softplus Sigmoid, Softplus Sigmoid, Softplus Sigmoid Sigmoid 1 With 2 ×23 subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With 23 subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGBoost and EBM we had to adjust the hyperparameters in order to get results comparable to the MLP, NAM or NAMLSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, for EBM we use 10 as the maximum number of leaves, 100 early stopping rounds and again the same learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGboost we use 500 estimators with a maximum depth of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' η is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the GAMLSS and gamboost models we use a logistic distribution to model the response distribution, where µ estimator uses identity and the σ estimator uses a log-link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hyperparameters for the neural models for the AirBnB dataset Hyperparameter NAMLSS1 NAMLSS2 DNN MLP NAM Learning Rate 1e-04 1e-04 1e-04 1e-04 1e-04 Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 Hidden Layers [512, 256, 50] [512, 256, 50] [512, 256, 50] [512, 256, 50] [512, 256, 50] LR Decay, Patience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='95 - 10 Activation ReLU ReLU ReLU ReLU ReLU Output Activation Gamma∗, Softplus Gamma∗, Softplus Gamma∗, Softplus Linear Linear 1 With 2 ×23 subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 1 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' 2 With 23 subnetworks and each subnetwork returning a parameter for the location and shape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' See Table 5 for an exemplary network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' ∗ See formula (6) for the detailed element-wise activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' AirBnB We fit the AirBnB dataset, with an Inverse Gamma distribution where applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, we train the models that only predict the mean with the squared error loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' While one might suspect worse performances due to that, we find that using the squared error actually leads to much smaller gamma deviances compared to the models leveraging the Inverse Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Additionally, we use slightly smaller model structures than for the California Housing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For all neural models, we use hidden layers of sizes [512, 256, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The first hidden layer is followed by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content='5 dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Throughout the hidden layers, we use ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, we deviate from that for the output layer activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the MLP we use a Softplus activation function for the output layer, ensuring that strictly positive values are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For NAMLSS as well as the DNN we have to adjust the activation functions, as both models minimize the log-likelihood via the parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' However, the mean prediction resulting from these parameters is defined via: µ = β α − 1 and is hence only defined for α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The activation functions thus need to ensure a α prediction that is larger than 1 and a β prediction that is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we again use a Softplus activation for the β output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For the α prediction, we use the following activation function element-wise: h(x) = � log(1 + exp(x)), if log(1 + exp(x)) > 1 1 log(1+exp(x)), else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' To compute the log-likelihood for the models resulting in a mean prediction we compute the parameters α and β as follows: α = µ2 σ2 + 2, β = µ µ2 σ2 + 1, with σ2 denoting the variance of the mean predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' For XGBoost and EBM we use a simple transformation of the target variable in order to ensure that µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence we fit the model on log(y) and re-transform the predictions accordingly with exp(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Otherwise, we use the same hyperparameters as for the California Housing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Interestingly, the NAM did not converge using the Softplus activation function as the MLP did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Using the Softplus activation resulted in tremendously large mean gamma deviances and log-likelihoods, as the model kept predicting values that were nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' Hence, we were only able to achieve good results for the NAM using the activation function given by formula (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' The presented GAMLSS and gamboostLSS models assume an Inverse Gamma distribution with both µ and σ utilizing the log-link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} +page_content=' It should be noted that the RS algorithm does not converge with GAMLSS, which is why CG is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFKT4oBgHgl3EQfni5N/content/2301.11862v1.pdf'} diff --git a/X9A0T4oBgHgl3EQfFf93/content/tmp_files/2301.02033v1.pdf.txt b/X9A0T4oBgHgl3EQfFf93/content/tmp_files/2301.02033v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45dae38c4f561757a84437a79d35fc7004f8f0a8 --- /dev/null +++ b/X9A0T4oBgHgl3EQfFf93/content/tmp_files/2301.02033v1.pdf.txt @@ -0,0 +1,428 @@ +arXiv:2301.02033v1 [eess.IV] 5 Jan 2023 +Physics-informed self-supervised deep learning +reconstruction for accelerated first-pass +perfusion cardiac MRI +Elena Mart´ın-Gonz´alez1[0000−0002−5922−4960], Ebraham +Alskaf2[0000−0003−4007−6854], Amedeo Chiribiri2[0000−0003−3394−4289], Pablo +Casaseca-de-la-Higuera1[0000−0003−1565−0842], Carlos +Alberola-L´opez1[0000−0003−3684−0055], Rita G. Nunes3[0000−0001−7425−5717], and +Teresa Correia2,4[0000−0002−1606−9550] +1 Laboratorio de Procesado de Imagen, ETSI Telecomunicaci´on, Universidad de +Valladolid, Valladolid, Spain +emargon@lpi.tel.uva.es +2 School of Biomedical Engineering and Imaging Sciences, King’s College London, +London, United Kingdom +3 Institute for Systems and Robotics and Department of Bioengineering, Instituto +Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal +4 Centre for Marine Sciences - CCMAR, Faro, Portugal +Abstract. First-pass perfusion cardiac magnetic resonance (FPP-CMR) +is becoming an essential non-invasive imaging method for detecting deficits +of myocardial blood flow, allowing the assessment of coronary heart dis- +ease. Nevertheless, acquisitions suffer from relatively low spatial reso- +lution and limited heart coverage. Compressed sensing (CS) methods +have been proposed to accelerate FPP-CMR and achieve higher spa- +tial resolution. However, the long reconstruction times have limited the +widespread clinical use of CS in FPP-CMR. Deep learning techniques +based on supervised learning have emerged as alternatives for speeding +up reconstructions. However, these approaches require fully sampled data +for training, which is not possible to obtain, particularly high-resolution +FPP-CMR images. Here, we propose a physics-informed self-supervised +deep learning FPP-CMR reconstruction approach for accelerating FPP- +CMR scans and hence facilitate high spatial resolution imaging. The +proposed method provides high-quality FPP-CMR images from 10x un- +dersampled data without using fully-sampled reference data. +Keywords: Deep learning reconstruction · Model-based reconstruction +· Quantitative perfusion cardiac MRI +1 +Introduction +Coronary artery disease (CAD) is the occlusion of the coronary arteries usually +caused by atherosclerosis, which causes abnormalities in blood flow to the heart. +Non-invasive imaging techniques that are widely used clinically for the evalu- +ation of CAD are single photon emission computerized tomography (SPECT) + +2 +E. Mart´ın-Gonz´alez et al. +and positron emission tomography (PET), but the reference for non-invasive +myocardial perfusion quantification is PET [8]. However, the clinical value of +first-pass perfusion cardiac magnetic resonance (FPP-CMR) has been shown in +comparison to these techniques [6,7,8,20], having emerged as an alternative way +of detecting blood flow anomalies without the use of potentially harmful ionising +radiation. In addition, FPP-CMR has other advantages, such as higher spatial +resolution, wider availability and lower scan cost compared to PET. +FPP-CMR time frames must be acquired in real-time to capture the rapid +passage of a contrast agent bolus through the heart, and hence, the spatial res- +olution and coverage of the heart is compromised. Thus, undersampled recon- +struction methods have been proposed to accelerate FPP-CMR acquisitions as a +means to improve spatial resolution and heart coverage [14,16,21]. However, these +methods can lead to long reconstruction times. In this work, we aim to speed up +reconstructions and obtain the contrast-enhanced dynamic image series from un- +dersampled FPP-CMR using deep learning (DL). Then, these images will be used +to generate quantitative perfusion maps using a tracer kinetic model [4,9,11]. DL +techniques have already been used in magnetic resonance image (MRI) recon- +struction. Work has been reported on knee [2,13,22], brain [2,5,13,22] and cardiac +[10,19] MRI, using both supervised [2,5,13] and self-supervised learning [15,22]. +Occasionally, the network is unrolled to mimic a compressed sensing (CS) it- +erative reconstruction problem, giving rise to a cascade of convolutional neural +networks (CNNs) [2,13,19]. The problem with supervised learning techniques is +the need to have fully sampled reference images to train the network, which are +not available in FPP-CMR, particularly at high spatial resolutions. +Even though the field of MRI reconstruction with DL is currently an active +area, to our knowledge, self-supervised DL techniques have not been applied +to FPP-CMR reconstruction. In this work, a SElf-Supervised aCcelerated RE- +consTruction (SECRET) DL framework for FPP-CMR is proposed to directly +reconstruct contrast-enhanced dynamic image series from undersampled (k,t)- +space data. +2 +Methods +For completeness, a conventional FPP-CMR CS reconstruction will be described. +We will also describe our proposed method, SECRET, as well as the Model Based +Deep Learning Architecture for Inverse Problems (MoDL) [2], which will be used +for comparison. +2.1 +Conventional FPP-CMR reconstruction +CS methods can be used to reconstruct dynamic images from undersampled +data. For example, FPP-CMR images s can be obtained from undersampled +data du using CS by solving the following optimisation problem: +ˆs = arg min +s +{∥du − Es∥2 +2 + λ1∥∇ss∥1 + λ2∥∇ts∥1} +(1) + +Physics-informed self-supervised DL reconstruction for FPP-CMR +3 +where E = AF, A is the (k,t)-space sampling trajectory, F is the Fourier trans- +form, λ1 and λ2 are regularization parameters and ∇s and ∇t are the finite +differences operators along the spatial and temporal dimensions, respectively. +2.2 +Supervised learning reconstruction: MoDL +MoDL combines the power of DL with model-based approaches [2]. It uses a CNN +as a denoiser and applies it as a regulariser to solve the optimisation problem +given by: +sk+1 = arg min +s +∥du − Es∥2 +2 + λ∥s − zk∥2 +2 +(2) +sk+1 = +� +EHE + λI +�−1 � +EHdu + λzk +� +(3) +where k denotes the k-th iteration and zk is the denoised version of sk, obtained +through a CNN network. MoDL requires supervised learning to optimise the +denoiser network. The data consistency layer is immediate by conjugate gradient +blocks, but as the input is zk and the output is sk+1, which, in turn, generates a +zk+1, this requires iterating until convergence. The iterative algorithm is unrolled +for a fixed number of iterations, K, in which the weights or parameters to be +optimised are shared. +The MoDL method has the zero-filled reconstruction, the coil sensitivities and +the subsampling mask as inputs, but it also needs the fully sampled images — +which are hardly available for the case of FPP-CMR at high spatial resolution— +for training. The loss is defined as the mean square error between sK and the +desired image t: C = +Nsamples +� +i=1 +∥sK(i)−t(i)∥2, where t(i) is the i-th target image. +2.3 +SECRET reconstruction +The proposed SECRET method directly reconstructs contrast-enhanced dy- +namic images from the undersampled (k,t)-space data. Considering only the +undersampled (k,t)-space data when enforcing data consistency, we can train +networks without the need for fully sampled images, simply by making use of +the physical models in the reconstruction [15]. This framework can be formulated +as follows: +ˆθ = arg min +θ +∥du − AFC(su|θ)∥2 +(4) +where C(su|θ) is the output of a CNN, with θ the parameter vector to be op- +timised. Figure 1 shows the steps necessary for training our proposed SECRET +method for FPP-CMR. First, undersampled (k,t)-space data du is transformed +to the image domain, obtaining su. Then, su enters the CNN to provide the re- +constructed contrast-enhanced dynamic images ˆs. These images are then trans- +formed back to (k,t)-space ˆd and subsampling masks are applied, thus obtaining +the undersampled version ˆdu. Finally, the loss is computed with ˆdu and the +input du, to guide the training phase. + +4 +E. Mart´ın-Gonz´alez et al. +The CNN is based on the well-known U-Net [18], widely used in medical +imaging. Skip connections are included to maintain information from previous +layers, as well as to avoid the problem of vanishing gradients during backpropa- +gation. At the end of the CNN, residual learning has been appended as in [15], +adding the average image of the input su. +CNN +Loss +60 +64 +128 +64 +60 +64 +64 +128 +256 +256 +256 +256 +512 +512 +128 +128 ++ +Fig. 1. Flow chart illustrating the proposed SECRET method for FPP-CMR. Blue +lines represent steps that only take place during training. The inputs of the framework +are the undersampled (k,t)-space data du and the (k,t)-sampling masks A, resulting in +the reconstructed contrast-enhanced dynamic images ˆs as output, and ˆdu if required. +2.4 +Dataset +Rest and stress FPP-CMR acquisitions were performed in 21 patients using a +single-bolus injection of 0.05 mmol/kg Gadobutrol (Gadovist; Bayer, Germany) +and a 1.5T CMR scanner (MAGNETOM Aera, Siemens Healthineers, Erlan- +gen, Germany) with an 18-channel chest-coil and a 32-channel spine coil. A +free-breathing FLASH perfusion dual-sequence [11] was used to acquire a low- +resolution image with low T1-sensitivity for estimating the arterial input function +and three short-axis slices (basal, mid and apical) for high resolution myocardial +perfusion imaging using the following parameters: FOV = 340 × 308 mm2, in- +plane resolution = 2.2 × 2.2 mm2, slice thickness = 10mm, TR/TE = 2.1/1ms, +flip angle = 8◦, parallel imaging acceleration factor 3, saturation recovery time += 100 ms, total scan duration = 60s, contrast agent relaxivity = 5.0L/mmol s. +Undersampled datasets were generated for 3×, 6× and 10× acceleration factors, +following a radial (k,t)-sampling trajectory. + +Average +imagePhysics-informed self-supervised DL reconstruction for FPP-CMR +5 +Preprocessing A first step to ensure that all data had the same size, both +spatially and temporally, prior to being fed to the CNN, consisted of resizing +the DICOM images to obtain a spatial resolution of 2 × 2 mm2, padding the k- +space to obtain an image size of 256×256 pixels, and interpolating each slice to a +fixed number of frames (60 frames). A final step included intensity normalisation +so that all contrast-enhanced dynamic image series present intensities between 0 +and 1, without losing the contrast variation between frames. In addition, image +pre-registration was also carried out to correct for respiratory motion. +Image quality metrics Image quality was assessed in terms of peak signal-to- +noise ratio (PSNR), structural similarity index measure (SSIM) and normalized +root mean square error (NRMSE) between the reference images and reconstruc- +tions obtained with the SECRET, MoDL and CS (10x only) methods. +2.5 +Implementation details +Patients were randomly split into training, validation and test subsets (60%, +16% and 24%, respectively). Each slice is fed into the SECRET framework so +that the time frames are stacked in depth, creating a multi-channel image. The +proposed method is implemented in Python with Tensorflow [1] and Keras [3], +and it took about half an hour of training using the Adam optimizer [12] with a +learning rate of 10−4 consuming about 3 GB of GPU memory for 100 epochs on +one Intel® CoreTM i7-4790 CPU @ 3.60GHz with 16 GB RAM and one NVIDIA +GeForce RTX 2080 Ti GPU. The MoDL training for K=1 and 100 epochs took +one hour and a half and the MoDL training for K=10 and 200 epochs took +forty-five hours using the same hardware. Note that after training the SECRET +method, it provides a reconstruction of a complete contrast-enhanced dynamic +image series in less than a second. +3 +Results and discussion +Figure 2 shows the SECRET reconstructions obtained for two representative +patients from 6× and 10× undersampled (k,t)-space data together with the +reference and MoDL (K=1) reconstructions. CS reconstruction is also shown for +10×. Three different time frames are shown, corresponding to right ventricle +(RV), left ventricle (LV) and myocardial enhancement. Although the SECRET +reconstructions are slightly blurred, due to residual learning from the average +image of the CNN input (which is blurred due to residual motion), it can be seen +that they have better quality than the images obtained with MoDL trained in +the same amount of time. Moreover, SECRET images maintain the variability +of contrast that exists between frames in addition to not losing the structure of +the heart. +Figure 3 shows results of the FPP-CMR reconstructions in terms of PSNR, +SSIM and NRMSE. While the performance of MoDL becomes noticeably worse +as the acceleration rate increases, SECRET maintains good image quality even +at high acceleration rates. For the 10x accelerated reconstructions, the median + +6 +E. Mart´ın-Gonz´alez et al. +RV enhancement +LV enhancement +Myocardial +enhancement +Patient A +Patient B +MoDL +SECRET +Reference +MoDL +SECRET +6x +10x +CS +Reference +Fig. 2. SECRET and MoDL (K=1) reconstructions obtained from 6× and 10× un- +dersampled FPP-CMR data for two representative subjects. The reference images are +displayed for comparison, in addition to CS reconstruction for 10×. The right ventricle +(RV), left ventricle (LV) and myocardial enhancement time frames are shown for one +short axis slice. + +X +0 +1Physics-informed self-supervised DL reconstruction for FPP-CMR +7 +3x +6x +10x +PSNR +NRMSE +SSIM +MoDL (K=1) +SECRET +MoDL (K=10) +CS +3x +6x +10x +3x +6x +10x +Fig. 3. PSNR, SSIM and NRMSE between the reference images and the reconstructions +obtained with SECRET and MoDL methods, for 3×, 6× and 10× acceleration factors, +for all patients in the test dataset. +(interquartile range): PSNR was 34.66 (3.47), 31.46 (3.81), 34.52 (5.43), 30.67 +(5.52); SSIM was 0.94 (0.04), 0.92 (0.07), 0.96 (0.06), 0.92 (0.06); NRMSE was +0.12 (0.06), 0.16 (0.10), 0.11 (0.09), 0.17 (0.11) for CS, MoDL (K=1), MoDL +(K=10) and SECRET methods, respectively. The image quality metrics indicate +that SECRET images maintain a more stable agreement with the reference as +the acceleration factor is increased than MoDL images, which deteriorate with +higher acceleration. CS and MoDL (K=10) show the best agreement with the +reference, but reconstructions take ∼87.08s and ∼1.99s, respectively, whereas +MoDL (K=1) takes ∼0.21s and SECRET only 0.15s. +Figure 4 shows a 1D projection of the dynamic images through time, for +a given slice. Note that although the images have been pre-registered, there is +still some residual motion. SECRET does not include any explicit regularisa- +tion term, however, due to the residual learning performed by the network all +reconstructions provided by the framework are inherently corrected. Such good +PSNR, SSIM and NRMSE values obtained when the reference images are affected +by little respiratory motion, would certainly improve if some regularisation were +added. This would enable even higher acceleration rates. Regularisation schemes +will thus be investigated in a future study. + +8 +E. Mart´ın-Gonz´alez et al. +Fully +sampled +MoDL +10x +SECRET +10x +time +Fig. 4. Representative image profile across the heart demonstrating that the SECRET +framework improves consistency across time frames. +Quantitative parameter maps were estimated from the FPP-CMR recon- +structions, showing the potential of the technique for an objective and operator- +independent analysis of myocardial perfusion. Figure 5 displays the contrast +transfer coefficient (KTrans) map estimated from fully sampled, 6× and 10× un- +dersampled patient data using the MoDL and SECRET methods, through the +Patlak model [17]. The image quality of the quantitative maps obtained from +the SECRET reconstruction at accelerations 6× and 10× is comparable to the +reference images, showing less blurring than MoDL maps. +Reference +MoDL +SECRET +6x +10x +mL/min/g +Fig. 5. Quantitative maps (KTrans) obtained from 6× and 10× undersampled data +using MoDL and the SECRET methods. The reference image is displayed for compar- +ison. + +Physics-informed self-supervised DL reconstruction for FPP-CMR +9 +4 +Conclusion +A physics-informed self-supervised deep learning reconstruction framework for +accelerating FPP-CMR scans has been described. The proposed SECRET method +provides FPP-CMR reconstructions directly from the undersampled (k,t)-space +data and does not require fully sampled reference data. Compared with state-of- +the-art approaches, the SECRET method maintains good quality reconstructions +for higher acceleration rates, with low training times and very fast reconstruc- +tion times. The proposed SECRET method shows promising results, with the +potential for improvement coupled with explicit regularization, which will be +explored in future work. +References +1. Abadi, M., et al.: TensorFlow: Large-scale machine learning on heterogeneous +systems (2015), https://www.tensorflow.org/, software available from tensor- +flow.org +2. 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Magnetic resonance in medicine 84(6), 3172–3191 +(2020) + diff --git a/X9A0T4oBgHgl3EQfFf93/content/tmp_files/load_file.txt b/X9A0T4oBgHgl3EQfFf93/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9286b6a80f3ee1522469fdcf04756f1adc9e0e52 --- /dev/null +++ b/X9A0T4oBgHgl3EQfFf93/content/tmp_files/load_file.txt @@ -0,0 +1,371 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf,len=370 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='02033v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='IV] 5 Jan 2023 Physics-informed self-supervised deep learning reconstruction for accelerated first-pass perfusion cardiac MRI Elena Mart´ın-Gonz´alez1[0000−0002−5922−4960], Ebraham Alskaf2[0000−0003−4007−6854], Amedeo Chiribiri2[0000−0003−3394−4289], Pablo Casaseca-de-la-Higuera1[0000−0003−1565−0842], Carlos Alberola-L´opez1[0000−0003−3684−0055], Rita G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Nunes3[0000−0001−7425−5717], and Teresa Correia2,4[0000−0002−1606−9550] 1 Laboratorio de Procesado de Imagen, ETSI Telecomunicaci´on, Universidad de Valladolid, Valladolid, Spain emargon@lpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='es 2 School of Biomedical Engineering and Imaging Sciences, King’s College London, London, United Kingdom 3 Institute for Systems and Robotics and Department of Bioengineering, Instituto Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal 4 Centre for Marine Sciences - CCMAR, Faro, Portugal Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' First-pass perfusion cardiac magnetic resonance (FPP-CMR) is becoming an essential non-invasive imaging method for detecting deficits of myocardial blood flow, allowing the assessment of coronary heart dis- ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Nevertheless, acquisitions suffer from relatively low spatial reso- lution and limited heart coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Compressed sensing (CS) methods have been proposed to accelerate FPP-CMR and achieve higher spa- tial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' However, the long reconstruction times have limited the widespread clinical use of CS in FPP-CMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Deep learning techniques based on supervised learning have emerged as alternatives for speeding up reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' However, these approaches require fully sampled data for training, which is not possible to obtain, particularly high-resolution FPP-CMR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Here, we propose a physics-informed self-supervised deep learning FPP-CMR reconstruction approach for accelerating FPP- CMR scans and hence facilitate high spatial resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The proposed method provides high-quality FPP-CMR images from 10x un- dersampled data without using fully-sampled reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Keywords: Deep learning reconstruction · Model-based reconstruction Quantitative perfusion cardiac MRI 1 Introduction Coronary artery disease (CAD) is the occlusion of the coronary arteries usually caused by atherosclerosis, which causes abnormalities in blood flow to the heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Non-invasive imaging techniques that are widely used clinically for the evalu- ation of CAD are single photon emission computerized tomography (SPECT) 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Mart´ın-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' and positron emission tomography (PET), but the reference for non-invasive myocardial perfusion quantification is PET [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' However, the clinical value of first-pass perfusion cardiac magnetic resonance (FPP-CMR) has been shown in comparison to these techniques [6,7,8,20], having emerged as an alternative way of detecting blood flow anomalies without the use of potentially harmful ionising radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' In addition, FPP-CMR has other advantages, such as higher spatial resolution, wider availability and lower scan cost compared to PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' FPP-CMR time frames must be acquired in real-time to capture the rapid passage of a contrast agent bolus through the heart, and hence, the spatial res- olution and coverage of the heart is compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Thus, undersampled recon- struction methods have been proposed to accelerate FPP-CMR acquisitions as a means to improve spatial resolution and heart coverage [14,16,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' However, these methods can lead to long reconstruction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' In this work, we aim to speed up reconstructions and obtain the contrast-enhanced dynamic image series from un- dersampled FPP-CMR using deep learning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Then, these images will be used to generate quantitative perfusion maps using a tracer kinetic model [4,9,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' DL techniques have already been used in magnetic resonance image (MRI) recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Work has been reported on knee [2,13,22], brain [2,5,13,22] and cardiac [10,19] MRI, using both supervised [2,5,13] and self-supervised learning [15,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Occasionally, the network is unrolled to mimic a compressed sensing (CS) it- erative reconstruction problem, giving rise to a cascade of convolutional neural networks (CNNs) [2,13,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The problem with supervised learning techniques is the need to have fully sampled reference images to train the network, which are not available in FPP-CMR, particularly at high spatial resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Even though the field of MRI reconstruction with DL is currently an active area, to our knowledge, self-supervised DL techniques have not been applied to FPP-CMR reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' In this work, a SElf-Supervised aCcelerated RE- consTruction (SECRET) DL framework for FPP-CMR is proposed to directly reconstruct contrast-enhanced dynamic image series from undersampled (k,t)- space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2 Methods For completeness, a conventional FPP-CMR CS reconstruction will be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' We will also describe our proposed method, SECRET, as well as the Model Based Deep Learning Architecture for Inverse Problems (MoDL) [2], which will be used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='1 Conventional FPP-CMR reconstruction CS methods can be used to reconstruct dynamic images from undersampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' For example, FPP-CMR images s can be obtained from undersampled data du using CS by solving the following optimisation problem: ˆs = arg min s {∥du − Es∥2 2 + λ1∥∇ss∥1 + λ2∥∇ts∥1} (1) Physics-informed self-supervised DL reconstruction for FPP-CMR 3 where E = AF, A is the (k,t)-space sampling trajectory, F is the Fourier trans- form, λ1 and λ2 are regularization parameters and ∇s and ∇t are the finite differences operators along the spatial and temporal dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='2 Supervised learning reconstruction: MoDL MoDL combines the power of DL with model-based approaches [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' It uses a CNN as a denoiser and applies it as a regulariser to solve the optimisation problem given by: sk+1 = arg min s ∥du − Es∥2 2 + λ∥s − zk∥2 2 (2) sk+1 = � EHE + λI �−1 � EHdu + λzk � (3) where k denotes the k-th iteration and zk is the denoised version of sk, obtained through a CNN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' MoDL requires supervised learning to optimise the denoiser network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The data consistency layer is immediate by conjugate gradient blocks, but as the input is zk and the output is sk+1, which, in turn, generates a zk+1, this requires iterating until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The iterative algorithm is unrolled for a fixed number of iterations, K, in which the weights or parameters to be optimised are shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The MoDL method has the zero-filled reconstruction, the coil sensitivities and the subsampling mask as inputs, but it also needs the fully sampled images — which are hardly available for the case of FPP-CMR at high spatial resolution— for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The loss is defined as the mean square error between sK and the desired image t: C = Nsamples � i=1 ∥sK(i)−t(i)∥2, where t(i) is the i-th target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='3 SECRET reconstruction The proposed SECRET method directly reconstructs contrast-enhanced dy- namic images from the undersampled (k,t)-space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Considering only the undersampled (k,t)-space data when enforcing data consistency, we can train networks without the need for fully sampled images, simply by making use of the physical models in the reconstruction [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' This framework can be formulated as follows: ˆθ = arg min θ ∥du − AFC(su|θ)∥2 (4) where C(su|θ) is the output of a CNN, with θ the parameter vector to be op- timised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Figure 1 shows the steps necessary for training our proposed SECRET method for FPP-CMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' First, undersampled (k,t)-space data du is transformed to the image domain, obtaining su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Then, su enters the CNN to provide the re- constructed contrast-enhanced dynamic images ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' These images are then trans- formed back to (k,t)-space ˆd and subsampling masks are applied, thus obtaining the undersampled version ˆdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Finally, the loss is computed with ˆdu and the input du, to guide the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Mart´ın-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The CNN is based on the well-known U-Net [18], widely used in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Skip connections are included to maintain information from previous layers, as well as to avoid the problem of vanishing gradients during backpropa- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' At the end of the CNN, residual learning has been appended as in [15], adding the average image of the input su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' CNN Loss 60 64 128 64 60 64 64 128 256 256 256 256 512 512 128 128 + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Flow chart illustrating the proposed SECRET method for FPP-CMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Blue lines represent steps that only take place during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The inputs of the framework are the undersampled (k,t)-space data du and the (k,t)-sampling masks A, resulting in the reconstructed contrast-enhanced dynamic images ˆs as output, and ˆdu if required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='4 Dataset Rest and stress FPP-CMR acquisitions were performed in 21 patients using a single-bolus injection of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='05 mmol/kg Gadobutrol (Gadovist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Bayer, Germany) and a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='5T CMR scanner (MAGNETOM Aera, Siemens Healthineers, Erlan- gen, Germany) with an 18-channel chest-coil and a 32-channel spine coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' A free-breathing FLASH perfusion dual-sequence [11] was used to acquire a low- resolution image with low T1-sensitivity for estimating the arterial input function and three short-axis slices (basal, mid and apical) for high resolution myocardial perfusion imaging using the following parameters: FOV = 340 × 308 mm2, in- plane resolution = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='2 mm2, slice thickness = 10mm, TR/TE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='1/1ms, flip angle = 8◦, parallel imaging acceleration factor 3, saturation recovery time = 100 ms, total scan duration = 60s, contrast agent relaxivity = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='0L/mmol s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Undersampled datasets were generated for 3×, 6× and 10× acceleration factors, following a radial (k,t)-sampling trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Average imagePhysics-informed self-supervised DL reconstruction for FPP-CMR 5 Preprocessing A first step to ensure that all data had the same size, both spatially and temporally, prior to being fed to the CNN, consisted of resizing the DICOM images to obtain a spatial resolution of 2 × 2 mm2, padding the k- space to obtain an image size of 256×256 pixels, and interpolating each slice to a fixed number of frames (60 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' A final step included intensity normalisation so that all contrast-enhanced dynamic image series present intensities between 0 and 1, without losing the contrast variation between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' In addition, image pre-registration was also carried out to correct for respiratory motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Image quality metrics Image quality was assessed in terms of peak signal-to- noise ratio (PSNR), structural similarity index measure (SSIM) and normalized root mean square error (NRMSE) between the reference images and reconstruc- tions obtained with the SECRET, MoDL and CS (10x only) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='5 Implementation details Patients were randomly split into training, validation and test subsets (60%, 16% and 24%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Each slice is fed into the SECRET framework so that the time frames are stacked in depth, creating a multi-channel image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The proposed method is implemented in Python with Tensorflow [1] and Keras [3], and it took about half an hour of training using the Adam optimizer [12] with a learning rate of 10−4 consuming about 3 GB of GPU memory for 100 epochs on one Intel® CoreTM i7-4790 CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='60GHz with 16 GB RAM and one NVIDIA GeForce RTX 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The MoDL training for K=1 and 100 epochs took one hour and a half and the MoDL training for K=10 and 200 epochs took forty-five hours using the same hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Note that after training the SECRET method, it provides a reconstruction of a complete contrast-enhanced dynamic image series in less than a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 3 Results and discussion Figure 2 shows the SECRET reconstructions obtained for two representative patients from 6× and 10× undersampled (k,t)-space data together with the reference and MoDL (K=1) reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' CS reconstruction is also shown for 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Three different time frames are shown, corresponding to right ventricle (RV), left ventricle (LV) and myocardial enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Although the SECRET reconstructions are slightly blurred, due to residual learning from the average image of the CNN input (which is blurred due to residual motion), it can be seen that they have better quality than the images obtained with MoDL trained in the same amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Moreover, SECRET images maintain the variability of contrast that exists between frames in addition to not losing the structure of the heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Figure 3 shows results of the FPP-CMR reconstructions in terms of PSNR, SSIM and NRMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' While the performance of MoDL becomes noticeably worse as the acceleration rate increases, SECRET maintains good image quality even at high acceleration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' For the 10x accelerated reconstructions, the median 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Mart´ın-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' RV enhancement LV enhancement Myocardial enhancement Patient A Patient B MoDL SECRET Reference MoDL SECRET 6x 10x CS Reference Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' SECRET and MoDL (K=1) reconstructions obtained from 6× and 10× un- dersampled FPP-CMR data for two representative subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The reference images are displayed for comparison, in addition to CS reconstruction for 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The right ventricle (RV), left ventricle (LV) and myocardial enhancement time frames are shown for one short axis slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' X 0 1Physics-informed self-supervised DL reconstruction for FPP-CMR 7 3x 6x 10x PSNR NRMSE SSIM MoDL (K=1) SECRET MoDL (K=10) CS 3x 6x 10x 3x 6x 10x Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' PSNR, SSIM and NRMSE between the reference images and the reconstructions obtained with SECRET and MoDL methods, for 3×, 6× and 10× acceleration factors, for all patients in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' (interquartile range): PSNR was 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='66 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='47), 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='46 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='81), 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='52 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='43), 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='67 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='52);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' SSIM was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='04), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='07), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='96 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='06), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='06);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' NRMSE was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='12 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='06), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='10), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='11 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='09), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='11) for CS, MoDL (K=1), MoDL (K=10) and SECRET methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The image quality metrics indicate that SECRET images maintain a more stable agreement with the reference as the acceleration factor is increased than MoDL images, which deteriorate with higher acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' CS and MoDL (K=10) show the best agreement with the reference, but reconstructions take ∼87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='08s and ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='99s, respectively, whereas MoDL (K=1) takes ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='21s and SECRET only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content='15s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Figure 4 shows a 1D projection of the dynamic images through time, for a given slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Note that although the images have been pre-registered, there is still some residual motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' SECRET does not include any explicit regularisa- tion term, however, due to the residual learning performed by the network all reconstructions provided by the framework are inherently corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Such good PSNR, SSIM and NRMSE values obtained when the reference images are affected by little respiratory motion, would certainly improve if some regularisation were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' This would enable even higher acceleration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Regularisation schemes will thus be investigated in a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Mart´ın-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Fully sampled MoDL 10x SECRET 10x time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Representative image profile across the heart demonstrating that the SECRET framework improves consistency across time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Quantitative parameter maps were estimated from the FPP-CMR recon- structions, showing the potential of the technique for an objective and operator- independent analysis of myocardial perfusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Figure 5 displays the contrast transfer coefficient (KTrans) map estimated from fully sampled, 6× and 10× un- dersampled patient data using the MoDL and SECRET methods, through the Patlak model [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The image quality of the quantitative maps obtained from the SECRET reconstruction at accelerations 6× and 10× is comparable to the reference images, showing less blurring than MoDL maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Reference MoDL SECRET 6x 10x mL/min/g Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Quantitative maps (KTrans) obtained from 6× and 10× undersampled data using MoDL and the SECRET methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The reference image is displayed for compar- ison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Physics-informed self-supervised DL reconstruction for FPP-CMR 9 4 Conclusion A physics-informed self-supervised deep learning reconstruction framework for accelerating FPP-CMR scans has been described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The proposed SECRET method provides FPP-CMR reconstructions directly from the undersampled (k,t)-space data and does not require fully sampled reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Compared with state-of- the-art approaches, the SECRET method maintains good quality reconstructions for higher acceleration rates, with low training times and very fast reconstruc- tion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' The proposed SECRET method shows promising results, with the potential for improvement coupled with explicit regularization, which will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9A0T4oBgHgl3EQfFf93/content/2301.02033v1.pdf'} +page_content=' Abadi, M.' 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sha256:dfb578db997fbdcf567eeb09582f51bcb5d02b1fd760f4acf8377532bafd5d65 +size 158049 diff --git a/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/2301.02506v1.pdf.txt b/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/2301.02506v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5eafc393b13949d04fc4547780484699e6bda78c --- /dev/null +++ b/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/2301.02506v1.pdf.txt @@ -0,0 +1,1383 @@ +arXiv:2301.02506v1 [math.PR] 6 Jan 2023 +Largest nearest-neighbour link and connectivity +threshold in a polytopal random sample † +Mathew D. Penrose ‡ +Xiaochuan Yang § +January 9, 2023 +Abstract +Let X1,X2,... be independent identically distributed random points in a convex +polytopal domain A ⊂ Rd. Define the largest nearest neighbour link Ln to be the +smallest r such that every point of Xn := {X1,...,Xn} has another such point within +distance r. We obtain a strong law of large numbers for Ln in the large-n limit. +A related threshold, the connectivity threshold Mn, is the smallest r such that the +random geometric graph G(Xn,r) is connected. We show that as n → ∞, almost +surely nLd +n/logn tends to a limit that depends on the geometry of A, and nMd +n/logn +tends to the same limit. +1 +Introduction +This paper is primarily concerned with the connectivity threshold and largest nearest- +neighbour link for a random sample Xn of n points specified compact region A in a d- +dimensional Euclidean space. +The connectivity threshold, here denoted Mn, is defined to be the smallest r such that +the random geometric graph G(Xn,r) is connected. For any finite X ⊂ Rd the graph +G(X ,r) is defined to have vertex set X with edges between those pairs of vertices x,y +such that ∥x −y∥ ≤ r, where ∥ · ∥ is the Euclidean norm. More generally, for k ∈ N, the +k-connectivity threshold Mn,k is the smallest r such that G(Xn,r) is k-connected (see the +definition in Section 2). +†Supported by EPSRC grant EP/T028653/1 +‡Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom. +m.d.penrose@bath.ac.uk +§Department of Mathematics, Brunel University London, Uxbridge, UB83PH, United Kingdom. +xiaochuan.yang@brunel.ac.uk ORCID:0000-0003-2435-4615 +1 + +The largest nearest neighbour link, here denoted Ln, is defined to be the the smallest +r such that every vertex in G(Xn,r) has degree at least 1. More generally, for k ∈ N with +k < n, the largest k-nearest neighbour link Ln,k is the smallest r such that every vertex +in G(Xn,r) has degree at least k. These thresholds are random variables, because the +locations of the centres are random. We investigate their probabilistic behaviour as n +becomes large. +We shall derive strong laws of large numbers showing that that nLd +n,k/logn converges +almost surely (as n → ∞) to a finite positive limit, and establishing the value of the limit. +Moreover we show that nMd +n,k/logn converges to the same limit. These strong laws carry +over to more general cases where k may vary with n, and the distribution of points may +be non-uniform. We give results of this type for A a convex polytope. +Previous results of this type (both for Ln,k and for Mn,k) were obtained for A having a +smooth boundary, and for A a d-dimensional hypercube; see [5]. It is perhaps not obvious +from the earlier results, however, how the limiting constant depends on the geometry of +∂A, the topological boundary of A, for general polytopal A, which is quite subtle. +It turns out, for example, that when d = 3 and the points are uniformly distributed +over a polyhedron, the limiting behaviour of Ln is determined by the angle of the sharpest +edge if this angle is less than π/2. We believe (but do not formally prove here) that if +this angle exceeds π/2 then the point of Xn furthest from the rest of Xn is asymptotically +uniformly distributed over ∂A, but if this angle is less than π/2 the location of this point in +is asymptotically uniformly distributed over the union of those edges which are sharpest. +Our motivation for this study is twofold. First, understanding the connectivity thresh- +old in dimension two is vital in telecommunications, for example, in 5G wireless net- +work design, with the nodes of Xn representing mobile transceivers (see for example +[1]). Second, detecting connectivity is a fundamental step for detecting all other higher +dimensional topological features in modern topological data analysis (TDA), where the +dimension of the ambient space may be very high. See [2, 3] for discussion of issues +related to the one considered here, in relation to TDA. General motivation for considering +random geometric graphs is discussed in [5]. +While our main results are presented (in Section 2) in the concrete setting of a poly- +topal sample in Rd, our proofs proceed via general lower and upper bounds (Propositions +3.2 and 3.6) that are presented in the more general setting of a random sample of points +in a metric space satisfying certain regularity conditions. This could be useful in pos- +sible future work dealing with similar problems for random samples in, for example, a +Riemannian manifold with boundary, a setting of importance in TDA. +2 + +2 +Statement of results +Throughout this paper, we work within the following mathematical framework. Let d ∈ N. +Suppose we have the following ingredients: +• A finite compact convex polytope A ⊂ Rd (i.e., one with finitely many faces). +• A Borel probability measure µ on A with probability density function f. +• On a common probability space (S,F,P), a sequence X1,X2,... of independent +identically distributed random d-vectors with common probability distribution µ, +and also a unit rate Poisson counting process (Zt,t ≥ 0), independent of (X1,X2,...) +(so Zt is Poisson distributed with mean t for each t > 0). +For n ∈ N, t > 0, let Xn := {X1,...,Xn}, and let Pt := {X1,...,XZt}. These are the +point processes that concern us here. Observe that Pt is a Poisson point process in Rd +with intensity measure tµ (see e.g. [4]). +For x ∈ Rd and r > 0 set B(x,r) := {y ∈ Rd : ∥y−x∥ ≤ r}. For r > 0, let A(r) := {x ∈ +A : B(x,r) ⊂ Ao}, the ‘r-interior’ of A. +For any point set X ⊂ Rd and any D ⊂ Rd we write X (D) for the number of points +of X in D, and we use below the convention inf(∅) := +∞. +Given n,k ∈ N, and t ∈ (0,∞), define the largest k-nearest neighbour link Ln,k by +Ln,k := inf({r > 0 : Xn(B(x,r)) ≥ k +1 +∀x ∈ Xn}). +(2.1) +Set Ln := Ln,1. Then Ln is the largest nearest-neighbour link. +We are chiefly interested in the asymptotic behaviour of Ln for large n. More generally, +we consider Ln,k where k may vary with n. +Let θd := πd/2/Γ(1 + d/2), the volume of the unit ball in Rd. Given x,y ∈ Rd, we +denote by [x,y] the line segment from x to y, that is, the convex hull of the set {x,y}. +Given m ∈ N and functions f : N∩[m,∞) → R and g : N∩[m,∞) → (0,∞), we write +f(n) = O(g(n)) as n → ∞, if limsupn→∞ | f(n)|/g(n) < ∞. We write f(n) = Ω(g(n)) +as n → ∞ if liminfn→∞( f(n)/g(n)) > 0. Given s > 0 and functions f : (0,s) → R and +g : (0,s) → (0,∞), we write f(r) = O(g(r)) as r ↓ 0 if limsupr↓0 | f(r)|/g(r) < ∞. We +write f(r) = Ω(g(r)) as r ↓ 0, if liminfr↓0( f(r)/g(r)) > 0. +Throughout this section, assume we are given a constant β ∈ [0,∞] and a sequence +k : N → N with +lim +n→∞(k(n)/logn) = β; +lim +n→∞(k(n)/n) = 0. +(2.2) +3 + +We make use of the following notation throughout: +f0 := ess infx∈A f(x); +f1 := inf +x∈∂A f(x); +(2.3) +H(t) := +� +1−t +t logt, +if t > 0 +1, +if t = 0. +(2.4) +Observe that −H(·) is unimodal with a maximum value of 0 at t = 1. Given a ∈ [0,∞), +we define the function ˆHa : [0,∞) → [a,∞) by +y = ˆHa(x) ⇐⇒ yH(a/y) = x, y ≥ a, +with ˆH0(0) := 0. Note that ˆHa(x) is increasing in x, and that ˆH0(x) = x and ˆHa(0) = a. +Throughout this paper, the phrase ‘almost surely’ or ‘a.s.’ means ‘except on a set of +P-measure zero’. For n ∈ N, we use [n] to denote {1,2,...,n}. We write f|A for the +restriction of f to A. +Let Φ(A) denote the set of all faces of the polytope A (of all dimensions up to d −1). +Also, let Φ∗(A) := Φ(A)∪{A}; it is sometimes useful for us to think of A itself as a face, +of dimension d. +Given a face ϕ ∈ Φ∗(A), denote the dimension of this face by D(ϕ). Then 0 ≤ D(ϕ) ≤ +d, and ϕ is a D(ϕ)-dimensional polytope embedded in Rd. Let ϕo denote the relative +interior of ϕ, and set ∂ϕ := ϕ \ϕo (if D(ϕ) = 0 we take ϕo := ϕ). If D(ϕ) < d then set +fϕ := infx∈ϕ f(x), and if ϕ = A then set fϕ := f0. +Then there is a cone Kϕ in Rd such that every x ∈ ϕo has a neighbourhood Ux such +that A∩Ux = (x+Kϕ)∩Ux. Define the angular volume ρϕ of ϕ to be the d-dimensional +Lebesgue measure of Kϕ ∩B(o,1). +For example, if ϕ = A then ρϕ = θd. If D(ϕ) = d −1 then ρϕ = θd/2. If D(ϕ) = 0 +then ϕ = {v} for some vertex v ∈ ∂A, and ρϕ equals the volume of B(v,r) ∩ A, divided +by rd, for all sufficiently small r. If d = 2, D(ϕ) = 0 and ωϕ denotes the angle subtended +by A at the vertex ϕ, then ρϕ = ωϕ/2. If d = 3 and D(ϕ) = 1, and αϕ denotes the angle +subtended by A at the edge ϕ (which is the angle between the two boundary planes of A +meeting at ϕ), then ρϕ = 2αϕ/3. +Theorem 2.1. Suppose A is a compact convex finite polytope in Rd. Assume that f|A is +continuous at x for all x ∈ ∂A, and that f0 > 0. Assume k(·) satisfies (2.2). Then, almost +surely, +lim +n→∞nLd +n,k(n)/k(n) = +max +ϕ∈Φ∗(A) +� +1 +fϕρϕ +� +if β = ∞; +(2.5) +lim +n→∞nLd +n,k(n)/logn = +max +ϕ∈Φ∗(A) +� ˆHβ(D(ϕ)/d) +fϕρϕ +� +if β < ∞. +(2.6) +4 + +In the next three results, we spell out some special cases of Theorem 2.1. +Corollary 2.2. Suppose that d = 2, A is a convex polygon and f|A is continuous at x for +all x ∈ ∂A. Let V denote the set of vertices of A, and for v ∈ V let ωv denote the angle +subtended by A at vertex v. Assume (2.2) holds with β < ∞. Then, almost surely, +lim +n→∞ +�nL2 +n,k(n) +logn +� += max +� ˆHβ(1) +π f0 +, 2 ˆHβ(1/2) +π f1 +,max +v∈V +� +2β +ωv f(v) +�� +. +(2.7) +In particular, for any constant k ∈ N, limn→∞ +� +nπL2 +n,k +logn +� += 1 +f0. +Corollary 2.3. Suppose d = 3 (so θd = 4π/3), A is a convex polyhedron and f|A is +continuous at x for all x ∈ ∂A. Let V denote the set of vertices of A, and E the set of edges +of A. For e ∈ E, let αe denote the angle subtended by A at edge e, and fe the infimum of f +over e. For v ∈ V let ρv denote the angular volume of vertex v. Suppose (2.2) holds with +β < ∞. Then, almost surely, +lim +n→∞ +�nL3 +n,k(n) +logn +� += max +� ˆHβ(1) +θ3 f0 +, 2 ˆHβ(2/3) +θ3 f1 +, +3 ˆHβ(1/3) +2mine∈E(αe fe),max +v∈V +� +β +ρv f(v) +�� +. +In particular, if β = 0 the above limit comes to max +� +3 +4π f0, 1 +π f1,maxe∈E +� +1 +2αe fe +�� +. +Corollary 2.4 ([5]). Suppose A = [0,1]d, and f|A is continuous at x for all x ∈ ∂A. For +1 ≤ j ≤ d let ∂j denote the union of all (d − j)-dimensional faces of A, and let f j denote +the infimum of f over ∂j. Assume (2.2) with β < ∞. Then +lim +n→∞ +�nLd +n,k(n) +logn +� += max +0≤j≤d +� +2j ˆHβ(1− j/d) +θd f j +� +, +a.s. +(2.8) +It is perhaps worth spelling out what the preceding results mean in the special case +where β = 0 (for example, if k(n) is a constant) and also µ is the uniform distribu- +tion on A (i.e. +f(x) ≡ f0 on A). In this case, the right hand side of (2.6) comes to +maxϕ∈Φ∗(A) +D(ϕ) +(d f0ρϕ). The limit in (2.7) comes to 1/(π f0), while the limit in Corollary +2.3 comes to f −1 +0 +max[1/π,maxe(1/(2αe))]. +So far we have only presented results for the largest k-nearest neighbor link. A closely +related threshold is the k-connectivity threshold defined by +Mn,k := inf{r > 0 : G(Xn,r) is k-connected}, +5 + +where a graph G of order n is said to be k-connected (k < n) if G cannot be disconnected +by the removal of at most k − 1 vertices. Set Mn,1 = Mn. Then Mn is the connectivity +threshold. +Notice that for all k,n with k < n we have +Ln,k ≤ Mn,k. +(2.9) +Indeed, if r < Ln,k, then there exists i ∈ [n] such that degXi < k in G(Xn,r). Then the +removal of all vertices adjacent to Xi disconnects G(Xn,r), implying that r < Mn,k. This +proves the claim. +Our second main result shows that (Mn,k/Ln,k) → 1 almost surely as n → ∞. For this +result we need d ≥ 2. +Theorem 2.5. Suppose d ≥ 2. Suppose A is a compact convex finite polytope in Rd. +Assume that f|A is continuous at x for all x ∈ ∂A, and that f0 > 0. Assume k(·) satisfies +(2.2) Then, almost surely, +lim +n→∞nMd +n,k(n)/k(n) = +max +ϕ∈Φ∗(A) +� +1 +fϕρϕ +� +if β = ∞; +(2.10) +lim +n→∞nMd +n,k(n)/logn = +max +ϕ∈Φ∗(A) +� ˆHβ(D(ϕ)/d) +fϕρϕ +� +if β < ∞. +(2.11) +Remark 2.6. One can spell out consequences of Theorem 2.5 in dimensions d = 2,3 and +the case of [0,1]d with exactly the same statement as in Corollaries 2.2-2.4. +Remark 2.7. Theorems 2.1 and 2.5 extend earlier work found in [5] on the case where +A is the unit cube, to more general polytopal regions. The case where A has a smooth +boundary is also considered in [5] (in this case with also k(n) = const., the result was first +given in [6] for Ln,k and in [7] for Mn,k). +Remark 2.8. In [8], similar results are given for the k-coverage threshold Rn,k, which is +given by +Rn,k := inf{r > 0 : Xn(B(x,r)) ≥ k +∀x ∈ A}; +n,k ∈ N. +(2.12) +Our results here, together with [8, Theorem 4.2], show that both Ln,k(n) and Mn,k(n) are +asymptotic to Rn,k(n) almost surely, as n → ∞. +3 +Proofs +In this section we prove the results stated in Section 2. Throughout this section we are +assuming we are given a constant β ∈ [0,∞] and a sequence (k(n))n∈N satisfying (2.2). +6 + +Recall that µ denotes the distribution of X1, and this has a density f with support A, +and that Ln,k is defined at (2.1). Recall that ˆHβ(x) is defined to be the y ≥ β such that +yH(β/y) = x, where H(·) was defined at (2.4). +For n ∈ N and p ∈ [0,1] let Bin(n, p) denote a binomial random variable with param- +eters n, p. Recall that H(·) was defined at (2.4), and Zt is a Poisson(t) variable for t > 0. +The proofs in this section rely heavily on the following lemma. +Lemma 3.1 (Chernoff bounds). Suppose n ∈ N, p ∈ (0,1), t > 0 and 0 ≤ k < n. +(a) If k ≥ np then P[Bin(n, p) ≥ k] ≤ exp(−npH(k/(np))). +(b) If k ≤ np then P[Bin(n, p) ≤ k] ≤ exp(−npH(k/(np))). +(c) If k ≥ e2np then P[Bin(n, p) ≥ k] ≤ exp(−(k/2)log(k/(np))) ≤ e−k. +(d) If k < t then P[Zt ≤ k] ≤ exp(−tH(k/t)). +(e) If k ∈ N then P[Zt = k] ≥ (2πk)−1/2e−1/(12k) exp(−tH(k/t)). +Proof. See e.g. [5, Lemmas 1.1, 1.2 and 1.3]. +3.1 +A general lower bound +In this subsection we present an asymptotic lower bound on Ln,k(n), not requiring any +extra assumptions on A. In fact, A here can be any metric space endowed with a Borel +probability measure µ which satisfies the following for some ε′ > 0 and some d > 0: +µ(B(x,r)) ≥ ε′rd, +∀ r ∈ (0,1),x ∈ A. +(3.1) +The definition of Ln,k at (2.1) carries over in an obvious way to this general setting. +Later, we shall derive the results stated in Section 2 by applying the results of this sub- +section to the different regions within A (namely interior, boundary, and lower-dimensional +faces). +Given r > 0,a > 0, define the ‘packing number’ ν(r,a) be the largest number m such +that there exists a collection of m disjoint closed balls of radius r centred on points of A, +each with µ-measure at most a. +Proposition 3.2 (General lower bound). Assume (3.1) with d,ε′ > 0. Let a > 0,b ≥ +0. Suppose ν(r,ard) = Ω(r−b) as r ↓ 0. Assume (2.2). Then almost surely, if β = ∞ +then liminfn→∞ +� +nLd +n,k(n)/k(n) +� +≥ 1/a. +If β < ∞ then liminfn→∞ +� +nLd +n,k(n)/logn +� +≥ +a−1 ˆHβ(b/d), almost surely. +Proof. First suppose β = ∞. +Let u ∈ (0,1/a). +Set rn := (uk(n)/n)1/d, n ∈ N. +By +(2.2), rn → 0 as n → ∞. Then, given n sufficiently large, we have ν(rn,ard +n) > 0 so +we can find yn ∈ A such that µ(B(yn,rn)) ≤ ard +n, and hence nµ(B(yn,rn)) ≤ auk(n). If +7 + +k(n) ≤ e2nµ(B(yn,rn)) (and hence nµ(B(yn,rn)) ≥ e−2k(n)), then since Xn(B(yn,rn)) is +binomial with parameters n and µ(B(yn,rn)), by Lemma 3.1(a) we have that +P[Xn(B(yn,rn)) ≥ k(n)] +≤ +exp +� +−nµ(B(yn,rn))H +� +k(n) +nµ(B(yn,rn)) +�� +≤ +exp +� +−e−2k(n)H +� +(au)−1�� +, +while if k(n) > e2nµ(B(yn,rn)) then by Lemma 3.1(c), P[Xn(B(yn,rn)) ≥ k(n)] ≤ e−k(n). +Therefore P[Xn(B(yn,rn)) ≥ k(n)] is summable in n because k(n)/logn → ∞ as n → ∞ +by (2.2). +Let δ0 ∈ (0,1). By (3.1) µ(B(yn,δ0rn) ≥ ε′δ d +0 uk(n)/n. Therefore by Lemma 3.1(b), +P[Xn(B(yn,δ0rn)) = 0] ≤ exp(−ε′δ d +0 uk(n)), which is summable in n. +Thus by the Borel-Cantelli lemma, almost surely event Fn := {Xn(B(yn,rn)) < k(n)}∩ +{Xn(B(yn,δ0rn)) > 0} occurs for all but finitely many n. But if Fn occurs then Ln,k(n) ≥ +(1−δ0)rn so that nLd +n,k(n)/k(n) ≥ (1−δ0)du. This gives the result for β = ∞. +Now suppose instead that β < ∞. Suppose first that b = 0, so that ˆHβ(b/d) = β. +Assume that β > 0 (otherwise the result is trivial). Choose β ′ ∈ (0,β). Let δ > 0 with +β ′ < β −2δ and with β ′H +� +β−2δ +β ′ +� +> δ. This is possible because H(β/β ′) > 0 and H(·) +is continuous. For n ∈ N, set rn := ((β ′ logn)/(an))1/d. Also set k′(n) = ⌈(β −δ)logn⌉, +and k′′(n) = ⌈(β −2δ)logn⌉. By assumption ν(rn,ard +n) = Ω(1), so for all n large enough, +we can (and do) choose xn ∈ A such that nµ(B(xn,rn)) ≤ nard +n = β ′logn. Then by a simple +coupling, and Lemma 3.1(a), +P[Xn(B(xn,rn)) ≥ k′′(n)] +≤ +P +� +Bin +� +n,(β ′logn)/n) +� +≥ k′′(n) +� +≤ +exp +� +− +� +β ′logn +� +H +�β −2δ +β ′ +�� +≤ n−δ. +Let δ ′ ∈ (0,1). By (3.1), for n large enough and all x ∈ A, +nµ(B(x,δ ′rn)) ≥ nε′(δ ′rn)d = ε′(δ ′)d(β ′/a)logn +so that by Lemma 3.1(b), P[Xn(B(x,δ ′rn)) = 0] ≤ n−ε′(δ ′)dβ ′/a. +Now choose K ∈ N such that δK > 1 and Kε′(δ ′)dβ ′/a > 1. For n ∈ N set z(n) := nK. +For all large enough n we have k′(z(n)) ≥ k′′(z(n+1)), so by the preceding estimates, +P[Xz(n+1)(B(xz(n+1),rz(n+1))) ≥ k′(z(n))] +≤ P[Xz(n+1)(B(xz(n+1),rz(n+1))) ≥ k′′(z(n+1))] ≤ (n+1)−δK, +and since xz(n+1) ∈ A, also P[Xz(n)(B(xz(n+1),δ ′rz(n))) = 0] ≤ n−ε′(δ ′)dβ ′K/a. Both of these +upper bounds are summable in n, so by the Borel-Cantelli lemma, almost surely for all +8 + +large enough n we have the event +{Xz(n+1)(B(xz(n+1),rz(n+1))) < k′(z(n))}∩{Xz(n)(B(xz(n+1),δ ′rz(n))) > 0}. +Suppose the above event occurs and suppose m ∈ N with z(n) ≤ m ≤ z(n +1). Note that +rz(n+1)/rz(n) → 1 as n → ∞. Then, provided n is large enough, +Lm,k′(z(n)) ≥ rz(n+1) −δ ′rz(n) ≥ (1−δ ′)2rm, +and moreover k′(z(n)) ≤ k(m) so that Lm,k(m) ≥ (1−δ ′)2rm. Hence it is almost surely the +case that +liminf +m→∞ (mLd +m,k(m)/logm) ≥ (1−δ ′)2d liminf +m→∞ (mrd +m/logm) = (1−δ ′)2da−1β ′, +and this yields the result for this case. +Now suppose instead that β < ∞ and b > 0. Let u ∈ (a−1β,a−1 ˆHβ(b/d)); note that +this implies uaH(β/(ua)) < b/d. Choose ε > 0 such that (1+ε)uaH(β/(ua)) < (b/d)− +9ε. Also let δ ′ ∈ (0,1). +For each n ∈ N set rn = (u(logn)/n)1/d. Let mn := ν(rn,ard +n), and choose xn,1,..., +xn,mn ∈ A such that the balls B(xn,1,rn),...,B(xn,mn,rn) are pairwise disjoint and each have +µ-measure at most ard +n. +Set λ(n) := n + n3/4 and λ −(n) := n − n3/4. For 1 ≤ i ≤ mn, if k(n) ≥ 1 then by a +simple coupling, and Lemma 3.1(e), +P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ P[Zλ(n)ardn ≤ k(n)] +≥ +� +e−1/(12k(n)) +� +2πk(n) +� +exp +� +−λ(n)ard +nH +� +k(n) +λ(n)ardn +�� +. +Now λ(n)rd +n/logn → u so by (2.2), k(n)/(λ(n)ard +n) → β/(ua) as n → ∞. Thus by the +continuity of H(·), provided n is large enough, for 1 ≤ i ≤ mn, +P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] +≥ +� +e−1/12 +� +2π(β +1)logn +� +exp +� +−(1+ε)auH +� β +au +� +logn +� +. +Hence, by our choice of ε, there is a constant c > 0 such that for all large enough n and +all i ∈ [mn] we have +P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ c(logn)−1/2n9ε−b/d ≥ n8ε−b/d. +(3.2) +Since xn,i ∈ A, by (3.1), for n large enough and 1 ≤ i ≤ mn we have µ(B(xn,i,δ ′rn)) ≥ +ε′(δ ′rn)d (as well as µ(B(xn,i,rn)) ≤ ard +n). Thus, given the value of Pλ(n)(B(xn,i,rn)), +9 + +the value of Pλ −(n)(B(xn,i,δ ′rn)) is binomially distributed with probability parameter +bounded away from zero. Also max1≤i≤mn E[Pλ(n)(B(xn,i,rn))] tends to infinity as n → +∞. Therefore there exists η > 0 such that for all large enough n, defining the event +En,i := {Pλ(n)(B(xn,i,rn)) ≤ k(n)}∩{Pλ −(n)(B(xn,i,δ ′rn) ≥ 1}, +we have for all large enough n that +inf +1≤i≤mn P[En,i|Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ η. +Hence, setting En := ∪mn +i=1En,i, for all large enough n we have +P[Ec +n] ≤ (1−ηn8ε−b/d)mn ≤ exp(−ηmnn8ε−b/d). +By assumption mn = ν(rn,ard +n) = Ω(r−b +n ) so that for large enough n we have mn ≥ +n(b/d)−ε, and therefore P[Ec +n] is is summable in n. +By Lemma 3.1(d), and Taylor expansion of H(x) about x = 1 (see the print version +of [5, Lemma 1.4] for details; there may be a typo in the electronic version), for all n +large enough P[Zλ(n) < n] ≤ exp(−1 +9n1/2). Similarly P[Zλ −(n) > n] ≤ exp(−1 +9n1/2). If En +occurs, and Zλ −(n) ≤ n, and Zλ(n) ≥ n, then for some i ≤ mn there is at least one point +of Xn in B(xn,i,δ ′rn) and at most k(n) points of Xn in B(xn,i,rn), and hence Ln,k(n) > +(1−δ ′)rn. Hence by the union bound +P[Ln,k(n) ≤ rn(1−δ ′)] ≤ P[Ec +n]+P[Zλ(n) < n]+P[Zλ −(n) > n], +which is summable in n by the preceding estimates. Therefore by the Borel-Cantelli +lemma, +P[liminf(nLd +n,k(n)/logn) ≥ u(1−δ ′)d] = 1, +u < a−1 ˆHβ(b/d),δ ′ ∈ (0,1), +so the result follows for this case too. +3.2 +Proof of Theorem 2.1 +In this subsection we assume, as in Theorem 2.1, that A is a compact convex finite poly- +tope in Rd. We also assume that the probability measure µ has density f with respect to +Lebesgue measure on Rd, and that f|A is continuous at x for all x ∈ ∂A, and that f0 > 0, +recalling from (2.3) that f0 := ess infx∈A f(x). Also we let k(n) satisfy (2.2) for some +β ∈ [0,∞]. Let Vol denote d-dimensional Lebesgue measure +Lemma 3.3. There exists ε′ > 0 depending only on f0 and A, such that (3.1) holds. +10 + +Proof. Let B0 be a (fixed) ball contained in A, and let b denote the radius of B0. For x ∈ A, +let Sx denote the convex hull of B0 ∪{x}. Then Sx ⊂ A since A is convex. If x /∈ B0, then +for r < b the set B(x,r)∩Sx is the intersection of B(x,r) with a cone having vertex x, and +since A is bounded the angular volume of this cone is bounded away from zero, uniformly +over x ∈ A \ B0. Therefore r−dVol(B(x,r) ∩ A) is bounded away from zero uniformly +over r ∈ (0,b) and x ∈ A \ B0 (and hence over x ∈ A). Since we assume f0 > 0, (3.1) +follows. +Recall that ν(r,a) was defined just before Proposition 3.2. Recall that for each face +ϕ ∈ Φ∗(A) we denote the angular volume of A at ϕ by ρϕ, and set fϕ := infϕ f(·) (if +ϕ ∈ Φ(A)) or fϕ = f0 (if ϕ = A). +Lemma 3.4. Let ϕ ∈ Φ∗(A). Assume f|A is continuous at x for all x ∈ ϕ. Then, almost +surely: +liminf +n→∞ +� +nLd +n,k(n)/k(n) +� +≥ (ρϕ fϕ)−1 +if β = ∞; +(3.3) +liminf +n→∞ +� +nLd +n,k(n)/logn +� +≥ (ρϕ fϕ)−1 ˆHβ(D(ϕ)/d) +if β < ∞. +(3.4) +Proof. Let a > fϕ. Take x0 ∈ ϕ such that f(x0) < a. If D(ϕ) > 0, assume also that +x0 ∈ ϕo. By the assumed continuity of f|A at x0, for all small enough r > 0 we have +µ(B(x0,r)) ≤ aρϕrd, so that ν(r,aρϕrd) = Ω(1) as r ↓ 0. Hence, by Proposition 3.2 +(taking b = 0), if β = ∞ then almost surely liminfn→∞ nLd +n,k(n)/k(n) ≥ 1/(aρϕ), and (3.3) +follows. +If β < ∞ and if D(ϕ) = 0, then by Proposition 3.2 (with b = 0), almost surely +liminfn→∞(nLd +n,k(n)/logn) ≥ ˆHβ(0)/(aρϕ), and hence (3.4) in this case. +Now suppose β < ∞ and D(ϕ) > 0. Take δ > 0 such that f(x) < a for all x ∈ +B(x0,2δ) ∩ A, and such that moreover B(x0,2δ) ∩ A = B(x0,2δ) ∩ (x0 + Kϕ) (the cone +Kϕ was defined in Section 2). Then for all x ∈ B(x0,δ) ∩ ϕ and all r ∈ (0,δ), we have +µ(B(x,r)) ≤ aρϕrd. +There is a constant c > 0 such that for small enough r > 0 we can find at least cr−D(ϕ) +points xi ∈ B(x0,δ)∩ϕ that are all at a distance more than 2r from each other, and there- +fore ν(r,aρϕrd) = Ω(r−D(ϕ)) as r ↓ 0. Thus by Proposition 3.2 we have +liminf +n→∞ +� +nLd +n,k(n)/k(n) +� +≥ (aρϕ)−1 ˆHβ(D(ϕ)/d), +almost surely, and (3.4) follows. +If we assumed f|A to be continuous on all of A, we would not need the next lemma +because we could instead use Lemma 3.4 for ϕ = A as well as for lower-dimensional +faces. However, in Theorem 2.1 we make the weaker assumption that f|A is continuous +at x only for x ∈ ∂A. In this situation, we also require the following lemma to deal with +ϕ = A. +11 + +Lemma 3.5. It is the case that +P[liminf(nLd +n,k(n)/k(n)) ≥ 1/(θd f0)] = 1 +if β = ∞; +(3.5) +P[liminf +n→∞ (nLd +n,k(n)/logn) ≥ ˆHβ(1)/(θd f0)] = 1 +if β < ∞. +(3.6) +Proof. Let α > f0. Then by taking B = A in [8, Lemma 6.4], +liminf +r↓0 +rdν(r,αθdrd) > 0. +(3.7) +Set rn := (k(n)/(nθdα))1/d if β = ∞, and set rn := ( ˆHβ(1)(logn)/(nθdα))1/d if β < ∞. +If β = ∞, then by (3.7) we can apply Proposition 3.2 (taking a = αθd and b = 0) to +deduce that liminfn→∞ nLd +n,k(n)/k(n) ≥ (θdα)−1, almost surely, and (3.5) follows. +Suppose instead that β < ∞. By (3.7), ν(r,αθdrd) = Ω(r−d) as r ↓ 0. Hence by +Proposition 3.2, almost surely liminfn→∞ +� +nLd +n,k(n)/logn +� +≥ (αθd)−1 ˆHβ(1). The result +follows by letting α ↓ f0. +Proof of Theorem 2.1. First suppose β < ∞. It is clear from (2.1) and (2.12) that Ln,k ≤ +Rn,k+1 for all n,k. Also by (2.2) we have (k(n)+1)/logn → β as n → ∞. Therefore using +[8, Theorem 4.2] for the second inequality below, we obtain almost surely that +limsup +n→∞ +�nLd +n,k(n) +logn +� +≤ limsup +n→∞ +�nRd +n,k(n)+1 +logn +� +≤ +max +ϕ∈Φ∗(A) +� ˆHβ(D(ϕ)/d) +fϕρϕ +� +. +(3.8) +Alternatively, this upper bound could be derived using (2.9) and the asymptotic upper +bound on Mn that we shall derive in the next section for the proof of Theorem 2.5. +By Lemmas 3.5 and 3.4, we have a.s. that +liminf +n→∞ +� +nLd +n,k(n)/logn +� +≥ +max +ϕ∈Φ∗(A) +� ˆHβ(D(ϕ)/d) +fϕρϕ +� +, +(3.9) +and combining this with (3.8) yields (2.6). +Now suppose β = ∞. In this case, again using the inequality Ln,k ≤ Rn,k+1 and [8, +Theorem 4.2], we obtain instead of (3.8) that a.s. +limsup +n→∞ +� +nLd +n,k(n)/k(n) +� +≤ +max +ϕ∈Φ∗(A) +� +1 +fϕρϕ +� +. +(3.10) +Also by Lemmas 3.5 and 3.4, instead of (3.9) we have a.s. that +liminf +n→∞ +� +nLd +n,k(n)/k(n) +� +≥ +max +ϕ∈Φ∗(A) +� +1 +fϕρϕ +� +, +and combining this with (3.10) yields (2.5). +12 + +3.3 +A general upper bound +In this subsection we present an asymptotic upper bound for Mn,k(n). As we did for the +lower bound in Section 3.1, we shall give our result (Proposition 3.6 below) in a more +general setting; we assume that A is a general metric space endowed with two Borel mea- +sures µ and µ∗ (possibly the same measure, possibly not). Assume that µ is a probability +measure and that µ∗ is a doubling measure, meaning that there is a constant c∗ (called a +doubling constant for µ∗) such that µ∗(B(x,2r)) ≤ c∗µ∗(B(x,r)) for all x ∈ A and r > 0. +We shall require further conditions on A: an ordering condition (O), a condition on balls +(B), a topological condition (T) and a geometrical condition (G) as follows: +(O) There is a total ordering of the elements of A. +(B) For all x ∈ A and r > 0, the ball B(x,r) is connected. +(T) The space A is unicoherent (see [5, Section 9.1]), and also connected. +(G) There exists δ1 > 0, and K0 ∈ (1,∞), such that for all r < δ1 and any x ∈ A, the +number of components of A\B(x,r) is at most two, and if there are two components, +at least one of these components has diameter at most K0r. +Given D ⊂ A and r > 0, we write Dr for {y ∈ A : dist(y,D) ≤ r}. Also, let κ(D,r) be the +r-covering number of D, that is, the minimal m ∈ N such that D can be covered by m balls +centred in D with radius r. +As before, given µ we assume X1,X2,... to be independent µ-distributed random +elements of A with the k-connectivity threshold Mn,k defined to be the minimal r such that +G(Xn,r) is k-connected, with Xn := {X1,...,Xn}. +Proposition 3.6 (General upper bound). Suppose that (A,µ,µ∗) are as described above +and A satisfies conditions (O), (B), (T), (G). Let ℓ ∈ N and let d > 0. For each j ∈ [ℓ] +let aj > 0,bj ≥ 0. Suppose that for each K ∈ N, there exists r0(K) > 0 such that for +all r ∈ (0,r0(K)), there is a partition {T( j,K,r), j ∈ [ℓ]} of A with the following two +properties. Firstly for each fixed K ∈ N, j ∈ [ℓ], we have +κ(T( j,K,r),r) = O(r−bj) as r ↓ 0, +(3.11) +and secondly, for all K ∈ N, j ∈ [ℓ], r ∈ (0,r0(K)) and any G ⊂ A intersecting T( j,K,r) +with diam(G) ≤ Kr, we have +µ(Gr \G) ≥ ajrd. +(3.12) +Assume (2.2). Then, almost surely, +limsup +n→∞ +� +nMd +n,k(n)/k(n) +� +≤ max +j∈[ℓ](a−1 +j ) +if β = ∞; +limsup +n→∞ +� +nMd +n,k(n)/logn +� +≤ max +j∈[ℓ](a−1 +j +ˆHβ(bj/d)) +if β < ∞. +13 + +Later we shall use Proposition 3.6 in the case where A is a convex polytope in Rd to +prove Theorem 2.5, taking µ to be the measure with density f and taking µ∗ to be the +restriction of Lebesgue measure to A (in fact, if f is bounded above then we could take +µ∗ = µ instead). The sets in the partition each represent a region near to a particular face +ϕ ∈ Φ∗(A) (if ϕ = A the corresponding set in the partition is an interior region). In this +case, coefficients aj in the measure lower bound (3.12) depend heavily on the geometry +of the determining cone near a particular face. +As a first step towards proving Proposition 3.6, we spell out some useful consequences +of the measure doubling property. In this result (and again later) we use |·| to denote the +cardinality (number of elements) of a set. +Lemma 3.7. Let µ∗ be a doubling measure on the metric space A, with doubling constant +c∗. We have the following. +(i) For any ε ∈ (0,1), there exists ρ(ε) ∈ N such that κ(B(x,r),εr) ≤ ρ(ε) for all +x ∈ A,r ∈ (0,∞). +(ii) For all r ∈ (0,1) and all D ⊂ A, we can find L ⊂ D with |L | ≤ κ(D,r/5), such +that D ⊂ ∪x∈L B(x,r), and moreover the balls B(x,r/5), x ∈ L , are disjoint. +Proof. To prove (i), let x ∈ A,r > 0. By the Vitali covering lemma, we can find a set +U ⊂ B(x,r) such that balls B(y,εr/5),y ∈ U are disjoint and that B(x,r) ⊂ ∪y∈U B(y,εr). +Set ρ(ε) := ⌈c⌈log2(15/ε)⌉ +∗ +⌉. Then by using the doubling property of µ∗ repeatedly, we have +µ∗(B(y,3r)) ≤ ρ(ε)µ∗(B(y,r/5)) for all y ∈ A. Moreover B(x,2r) ⊂ B(y,3r) for all y ∈ U . +Also ∪y∈U B(y,εr/5) ⊂ B(x,2r) and the union is disjoint. Thus +|U |µ∗(B(x,2r)) ≤ ∑ +y∈U +µ∗(B(y,3r)) ≤ ρ(ε) ∑ +y∈U +µ∗(B(y,εr/5)) ≤ ρ(ε)µ∗(B(x,2r)), +and therefore |U | ≤ ρ(ε); the claim about κ(B(x,r),εr) follows. +Now we prove (ii). Let L 0 ⊂ D with |L 0| = κ(D,r/5) and with B ⊂ ∪x∈L B(x,r/5). +By the Vitali covering lemma, we can find L ⊂ L 0 such that D ⊂ ∪x∈L B(x,r) and the +balls B(x,r/5),x ∈ L , are disjoint, and (ii) follows. +Given countable σ ⊂ A, r > 0 and k ∈ N, we say that σ is (r,k)-connected if the +geometric graph G(σ,r) is k-connected. Assuming condition (B) holds, we see that σ is +(r,1) connected if and only if σr/2 is a connected subset of A. +Lemma 3.8 (Peierls argument). Assume (O). Let ℓ ∈ N, a ∈ [1,∞). Let r ∈ (0,1/a) and +n ∈ N. Let L ⊂ A with the property that |L ∩B(x,r)| ≤ ℓ for all x ∈ A, and let x0 ∈ Lr. +Then the number of (ar,1)-connected subsets of L containing x0 with cardinality n is at +most cn, where c depends only on ℓ, a and c∗. +14 + +Proof. First we claim that |L ∩ B(x,ar)| ≤ ℓρ(1/a) for all x ∈ A, where ρ(1/a) is as +given in Lemma 3.7-(i). Indeed, we can cover B(x,ar) by ρ(1/a) balls of radius r, and +each of these balls contains at most ℓ points of L . +There is a standard algorithm (of constructing a non-decreasing sequence of lists) for +counting the connected sets of Zd; see [5, Lemma 9.3] for details of the algorithm. +The algorithm remains valid in this general setting, with the lexicographical ordering +replaced by the total ordering of A (using assumption (O)). This algorithm has to stop +at time n (cardinality of the set), and at each step the number of possibilities for the set +of the added elements is bounded by 2ℓρ(1/a) (all possible subsets of the set of points +of L within distance ar from a fixed point); hence the number of ar-connected sets of +cardinality n is at most 2ℓρ(1/a)n. +Preparing for a proof of Proposition 3.6, we recall a condition that is equivalent to +k-connectedness of a graph G. We say that non-empty sets U,W ⊂ V in a graph G with +vertex set V form a k-separating pair if (i) the subgraph of G induced by U is connected, +and likewise for W; (ii) no element of U is adjacent to any element of W; (iii) the number +of vertices of V \ (U ∪W) lying adjacent to U ∪W is at most k. We say that U is a k- +separating set for G if (i) the subgraph of G induced by U is connected, and (ii) at most k +vertices of V \U lie adjacent to U. The relevance of these definitions is presented in the +following lemma. +Lemma 3.9. [5, Lemma 13.1] Let G be a graph with more than k+1 vertices. Then G is +either (k +1)-connected, or it has k separating pair, but not both. +By Lemma 3.9, to prove Proposition 3.6 it suffices to prove, for arbitrary u > +maxj a−1 +j +ˆHβ(bj/d), the non-existence of (k(n) − 1)-separating pairs in G(Xn,rn) with +rn = (ulogn/n)1/d, as n → ∞. Notice that, for any fixed K ∈ N, if (U,W) is a (k − 1)- +separating pair, then either both U and W have diameter at least Krn, or one of them, +say U, is a (k − 1)-separating set of diameter at most Krn. Here by the diameter of a a +non-empty set U ⊂ A we mean the number diam(U) := supu,v∈U dist(u,v). +The goal is to prove that neither outcome is possible when n → ∞. Let us first eliminate +the existence of a small separating set. +Lemma 3.10. Suppose the assumptions of Proposition 3.6 hold. +If β = ∞, let u > +maxj a−1 +j +and for n ∈ N, set rn = (uk(n)/n)1/d. If β < ∞, let u > maxj∈[ℓ]a−1 +j +ˆHβ(bj/d), +and for n ∈ N set rn = (u(logn)/n)1/d. For K ∈ N, let En(K,u) be the event that there +exists a (k(n)−1)-separating set for G(Xn,rn) of diameter at most Krn. Then, given any +K ∈ N, almost surely En(K,u) occurs for only finitely many n. +Proof. First assume β < ∞. The condition on u implies that uaj > β and uajH(β/(uaj)) > +bj/d, for each j ∈ [ℓ]. Then we can and do choose β ′ > β and ε ∈ (0,1/4) such that for +15 + +each j ∈ [ℓ], (1−3ε)duaj > β ′ and +(1−3ε)duajH +� +β ′ +(1−3ε)duaj +� +> bj +d +ε. +For n ∈ N define k′(n) = ⌈β ′ logn⌉. +Let K ∈ N, and for r ∈ (0,r0(K)) let T( j,K,r) be as in the assumptions of Proposition +3.6. For j ∈ [ℓ], we claim that κ(T( j,K,rn),εrn/5) = O(r−bj +n +) as n → ∞. Indeed, +κ(T( j,K,rn),εrn/5) ≤ κ(T( j,K,rn),rn)sup +x∈A +κ(B(x,rn),εrn/5) ≤ ρκ(T( j,K,rn),rn), +where ρ = ρ(ε/5) is the constant in Lemma 3.7-(i). The claim follows from the assump- +tion (3.11). +Choose n0 ∈ N such that rn < r0(k) for all n ∈ N with n ≥ n0. By Lemma 3.7-(i), for +each j ∈ [ℓ] and n ∈ N we can find a set L j +n ⊂ T( j,K,rn), with |L j +n | ≤ κ(T( j,K,rn),εrn/5) = +O(r−bj +n +), such that T( j,K,rn) ⊂ ∪x∈L j +n B(x,εrn) and that the balls B(x,rnε/5), x ∈ L j +n , +are disjoint. Set +Ln := ∪ℓ +i=1L j +n . +(3.13) +For n ≥ n0, j ∈ [ℓ] let T j +n = {σ ⊂ Ln : diam(σ) ≤ 2Krn,σ ∩ T( j,K,rn) ̸= ∅}. We +claim that the cardinality of T j +n is O(|L j +n |) = O(r−bj +n +). Indeed, σ ∩T( j,K,rn) ̸= ∅ means +σ ∩L j +n ̸= ∅. Moreover, as explained below, +limsup +n→∞ +sup +x∈Ln +|B(x,2Krn)∩Ln| < ∞, +(3.14) +and diam(σ) ≤ 2Krn. The claim about cardinality follows from this. +Now we show (3.14). By Lemma 3.7-(i), for n large and for all x ∈ A, we can cover +B(x,2Krn) by ρ(ε/(10K)) balls of radius rnε/5, and each of these balls contains at most +ℓ points of Ln. +For n ≥ n0 and σ ⊂ Ln, set +Dσ,n := σ(1−2ε)rn \σεrn. +(3.15) +Let J ∈ N with J > 1/ε. For m ∈ N, define z(m) := mJ. For σ ⊂ Lz(m), define +Fm(σ) = {Xz(m)(Dσ,z(m)) < k′(z(m))}. +Now let n ∈ N and choose m = m(n) such that z(m) ≤ n < z(m+1). Assume z(m) ≥ +n0. Suppose that En(K,u) occurs and let U be a (k(n) − 1)-separating set of G(Xn,rn) +with diam(U) ≤ Krn. We define its ‘pixel version’ σ(U) := Lz(m(n)) ∩Uεrz(m(n)). +16 + +Since σ(U) ⊂ A, there exists j ∈ [ℓ] such that σ(U) ∩ T( j,K,rz(m(n))) ̸= ∅. By our +choice of ε, provided n is large enough we have diam(σ(U)) ≤ 2Krz(m(n)). Therefore +σ(U) ∈ ∪[ℓ] +j=1T j +z(m(n)). +Since U is (k(n) − 1)-separating for G(Xn,rn), we have Xn(Urn \U) < k(n). We +claim that Xn(Dσ(U),z(m(n))) < k(n) provided n is large enough. Indeed, by the triangle +inequality σ(U)(1−2ε)rz(m(n)) ⊂ U(1−ε)rz(m(n)) ⊂ Urn (for n large), while U ⊂ σ(U)εrz(n(m)). +Thus Dσ(U),z(m(n)) ⊂ Urn \U, and the claim follows. Also, provided n is large enough, we +have k(n) ≤ k′(z(m(n))). Thus we have the event inclusions +En(K,u) ⊂ ∪ℓ +j=1 ∪σ∈T j +z(m(n)) {Xn(Dσ,z(m(n))) < k(n)} +⊂ ∪ℓ +j=1 ∪σ∈T j +z(m(n)) Fm(n)(σ). +By (3.15), for any n ∈ N and σ ⊂ Ln we have Dσ,n ⊃ (σεrn)(1−3ε)rn \σεrn. Hence by +(3.12), for all large enough n and all σ ∈ ∪j∈[ℓ]T j +n we have µ(Dσ,n) ≥ aj(1−3ε)drd +n. A +simple coupling shows that, provided m is large, we have +P[∪j∈[ℓ] ∪σ∈T j +z(m) Fm(σ)] = +ℓ +∑ +j=1 +O(r−bj +z(m))P[Bin(z(m),(1−3ε)dajrd +z(m)) < k′(z(m))]. +By Lemma 3.1(b) and our choice of rn and ε, provided m is large, we have +P[∪j∈[ℓ] ∪σ∈T j +z(m) Fm(σ)] += O(1) +ℓ +∑ +j=1 +exp +� +(bj/d)logz(m)−(1−3ε)duajH +� +β ′ +(1−3ε)duaj +� +logz(m) +� += O(m−Jε), +which is summable in m. +It follows from the Borel-Cantelli lemma that almost surely ∪j∈[ℓ] ∪σ∈T j +z(m) Fm(σ) +occurs only for finitely many m which implies that En(K,u) occurs for only finitely many +n. This completes the proof of the case β < ∞. +Now assume β = ∞. For the rest of the proof assume also that ε ∈ (0,1) is such +that uaj(1 −ε)d > 1 for all j ∈ [ℓ]. We do not have to go through the subsequence argu- +ment as before because the growth of k(n) is super-logarithmic. Now redefine Fn(σ) := +{Xn(Dσ,n) < k(n)}. If En(K,u) happens then we now redefine the pixel version of the +separating set U as +σ(U) := Ln ∩Uεrn, +and enumerate the possible shapes σ of the pixel version. Thus we have +En(K,u) ⊂ ∪ℓ +j=1 ∪σ∈T j +n Fn(σ). +17 + +Using estimates of |T j +n |, we have +P[En(K,u)] = +ℓ +∑ +j=1 +O(r−bj +n +)P[Bin(n,(1−3ε)dajrd +n) < k(n)]. +Noticing r−1 +n += O(n1/d), and applying Lemma 3.1-(b) leads to +P[En(K,u)] = O(nbj/d) +ℓ +∑ +j=1 +exp +� +−(1−3ε)dajuk(n)H +� +k(n) +(1−3ε)dajuk(n) +�� +which is summable in n, and the claim follows by the Borel-Cantelli lemma. +The following lemma eliminates the existence of a (k(n) − 1)-separating pair with +both diameters larger than Krn. +Lemma 3.11. Let the assumptions of Proposition 3.6 hold. If β = ∞, let u > maxj a−1 +j +and for n ∈ N, set rn = (uk(n)/n)1/d. If β < ∞, let u > maxj∈[ℓ]a−1 +j +ˆHβ(bj/d), and for +n ∈ N set rn = (u(logn)/n)1/d. For K ∈ N let Hn(K,u) denote the event that there exists a +(k(n)−1)-separating pair (U,W) in G(Xn,rn) such that min(diam(U),diam(W)) ≥ Krn. +Then there exists K1 ∈ N such that almost surely Hn(K1,u) occurs for only finitely many +n. +Proof. Suppose Hn(K,u) holds. Then Urn/2 and Wrn/2 are disjoint and connected in A. +One of the components of A \Urn/2 contains W, denoted by W ′. Set U′ = A \W′. Then +U ⊂ U′, W ⊂ W ′ and A = W ′ ∪U′. Let ∂WU := W ′ ∩U′. Then ∂WU is connected by the +unicoherence of A. Moreover, any continuous path in A connecting U and W must pass +through ∂WU. +Recall δ1 and K0 in the assumption (G). We claim (and show in the next few para- +graphs) that +diam(∂WU) ≥ +1 +2K0 +2 min(δ1/3,diam(W)/3,diam(U)/3). +(3.16) +Suppose the opposite. Setting b = diam(∂WU), we can find x ∈ A such that ∂WU ⊂ B(x,b), +and we can find X ∈ U \ B(x,b),Y ∈ W \ B(x,b). Since b < δ1/3, the number of compo- +nents of A \ B(x,b) is at most two. There have to be two components because otherwise +X and Y can be connected by a path in A disjoint from ∂U, which is a contradiction. +Suppose that X lies in the component of A \ B(x,b) having diameter at most K0b, +denoted by QX, and Y lies in the other component, denoted by QY (if it is the other +way round we reverse the roles of X and Y in the rest of this argument). We claim that +there exists X′ ∈ U such that dist(X,X′) > (2K0 + 2)b. If not, then for any X1,X2 ∈ U, +18 + +we have by triangle inequality that dist(X1,X2) ≤ 2(2K0 +2)b, yielding that diam(U) ≤ +2(2K0 +2)b, contradicting diam(U) > 3(2K0 +2)b by the negation of (3.16). +We claim that dist(X,B(x,b)) ≤ K0b. To see this, using the assumed connectivity of +A, take a continuous path in A from X to Y. The first exit point of this path from QX lies +in B(x,b) (else it would not be an exit point from QX) but also in the closure of QX, and +hence in B(X,K0b). This yields the latest claim. +We show that X′ and Y have to be in the same component of A \ B(x,b). To this end, +notice first that X′ cannot be in QX, because for any z ∈ QX, +dist(X,z) ≤ K0b < (2K0 +2)b. +Secondly, X′ cannot be in B(x,b) either because for any z ∈ B(x,b), we have +dist(z,X) ≤ dist(X,B(x,b))+2b ≤ (K0 +2)b < (2K0 +2)b. +Therefore, X′ has to be in QY, and we reach again to a contradiction that X′ and Y can be +connected by a path in A disjoint from ∂U. We have thus proved (3.16). +Let ε ∈ (0,1/9) and let Ln be as defined at (3.13) (the ε does not have to be the same +as it was there). Recall that Ln has the covering property that for every x ∈ A we have +Ln ∩B(x,rnε) ̸= ∅ and the spacing property that |Ln ∩B(x,rnε/3)| ≤ ℓ for all such x. +Define DWU = {x ∈ Ln : B(x,εrn) ∩ ∂WU ̸= ∅}. Then by the covering property of +Ln, (DWU)εrn is connected and covers ∂WU. That is, DWU, as a subset of the metric +space A, is (2εrn,1)-connected. +By (3.16) and the occurrence of Hn(K,u), we have +2εrn|DWU| ≥ diam(∂WU) ≥ min(δ1/3,Krn/3)/(2K0 +2) +Therefore, provided n is large, we have |DWU| ≥ K/(6ε(2K0 +2)). +We claim that there is a constant c ∈ (0,∞), independent of n, such that for all q ∈ N, +if |DWU| = q then DWU can take at most O(r−max(bj) +n +cq) possible ’shapes’. Indeed, given +x0 ∈ Ln, set +Un,q(x0) := {σ ⊂ Ln : |σ| = q,σ is (2εrn,1)-connected,x0 ∈ σ}. +Then DWU ∈ ∪j∈[ℓ] ∪x0∈T(j,K,rn)∩Ln Un,q(x0). By Lemma 3.8, we have |Un,q(x0)| ≤ cq +for some finite constant c. Recall from the proof of Lemma 3.10 that |T( j,K,rn)∩Ln| = +O(r−bj +n +). The claim follows. +For all n ∈ N, if x ∈ ∂WU then dist(x,U) = rn/2. Therefore by the triangle inequal- +ity, (DWU)εrn/5 ⊂ Ur, while U ∩ (DWU)εrn/5 = ∅; hence Xn ∩ (DWU)εrn/5 = ∅. This, +together with the the union bound, yields that +P[Hn(K,u)] ≤ +∑ +q≥K/(6ε(2K0+2))∑ +σ +P[Xn(σεrn/5) < k(n)], +(3.17) +19 + +where the second sum is over all possible shapes σ ⊂ Ln of cardinality q that are (2εrn,1)- +connected. Since every point in A is covered at most ℓ times, by (3.12) (with G = {z}), +there exists ε1 ∈ (0,1) such that +µ(σεrn/5) ≥ (1/ℓ) ∑ +z∈σ +µ(B(z,εrn/5)) ≥ (q/ℓ)ε1(εrn/5)d. +Suppose β < ∞. Set ε2 := (ε1/ℓ)(ε/5)d. By (3.17) and Lemma 3.1(b), provided n is +large, +P[Hn(K,u)] ≤ +∑ +q≥K/(6ε(2K0+2)) +O(r−max(bj) +n +cq)P[Bin(n,ε2qrd +n) < (β +1)logn] += O(1) +∑ +q≥K/(6ε(2K0+2)) +cq exp +� +(max(bj)/d)logn−ε2quH +�β +1 +ε2qu +� +logn +� +. +By the continuity of H(·) and the fact that H(0) = 1, there exists q0 > 16/(ε2u) such +that for any q > q0, we have H +�β+1 +qε2u +� +> 1/2 and quε2 > 4max(bj)/d. Choosing K = +6ε(2K0 + 2)q0 so that q ≥ q0 in the sum, we see that the exponent of the exponential is +bounded above by +(max(bj)/d)logn−qε2(u/2)logn ≤ −(quε2/4)logn. +Therefore, we have for n large that +P[Hn(K,u)] = O(1) ∑ +q≥q0 +cq exp(−qu(ε2/4)logn) += O(1) ∑ +q≥q0 +exp(−qu(ε2/8)logn) = O(exp(−q0u(ε2/8)logn)) = O(n−2). +The result in this case follows by applying the Borel-Cantelli lemma. +If β = ∞, then by (3.17) and the estimates of |∪j ∪x0Un,q(x0)| as previously, we have +P[Hn(K,u)] ≤ +∑ +q≥K/(6ε(2K0+2)) +O(r−max(bj) +n +cq)P[Bin(n,(q/ℓ)ε1(εrn/5)d) < k(n)]. +We have r−max(bj) +n += O(nmax(bj)/d), and by Lemma 3.1-(b), +P[Hn(K,u)] ≤ +∑ +q≥K/(6ε(2K0+2)) +cq exp +� +(max(bj)/d)logn−qε2uk(n)H( +k(n) +qε2uk(n)) +� +. +As before, we can choose K = K1 (large) so that the H(·) term in every summand is +bounded from below by 1/2. By the super-logarithmic growth of k(n), we conclude that +P[Hn(K,u)] ≤ n−2 provided n is large, so that the Borel-Cantelli lemma gives the result +in this case too. +20 + +Proof of Proposition 3.6. If β = ∞ then let u > maxj∈[ℓ](a−1 +j ) and set r(n) := u(k(n)/n)1/d. +If β < ∞ then let u > maxj∈[ℓ](a−1 +j +ˆHβ(bj/d)) and set rn := (u(logn)/n)1/d. By Lemmas +3.10 and 3.11, there exists K ∈ N such that almost surely, En(K,u) ∪Hn(K,u) occurs for +at most finitely many n. By Lemma 3.9, if Mn,k > rn then En(K,u) ∪ Hn(K,u) occurs. +Therefore Mn,k(n) ≤ rn for all large enough n, almost surely, and the result follows. +3.4 +Proof of Theorem 2.5 +In this subsection we go back to the mathematical framework in Section 2; that is, we +make the assumptions in the statement of Theorem 2.5. In particular we return to assum- +ing A is a convex polytope in Rd with d ≥ 2, and the probability measure µ has a density +f. We shall check the conditions required in order to apply Proposition 3.6. +To check these conditions, we shall use the following lemma and notation. +Lemma 3.12. [8, Lemma 6.12] Suppose ϕ,ϕ′ are faces of A with D(ϕ) > 0 and D(ϕ′) = +d −1, and with ϕ \ϕ′ ̸= ∅. Then ϕo ∩ϕ′ = ∅ and K(ϕ,ϕ′) < ∞, where we set +K(ϕ,ϕ′) := sup +x∈ϕo +dist(x,∂ϕ) +dist(x,ϕ′) . +(3.18) +Now define +K(A) := max{K(ϕ,ϕ′) : ϕ,ϕ′ ∈ Φ(A),D(ϕ) > 0,D(ϕ′) = d −1,ϕ \ϕ′ ̸= ∅}. +(3.19) +Then K(A) < ∞ since A is a finite polytope. +For j ∈ {0,1,...,d} let Φj(A) denote the collection of j-dimensional faces of A. For +any D ⊂ A and r > 0 set Dr = {x ∈ A : B(x,r)∩D ̸= ∅}. +Lemma 3.13. The restriction of Lebesgue measure to A has the doubling property. More- +over the conditions (O), (B), (T) and (G) are met. +Proof. First we verify the doubling property. By the proof of Lemma 3.3, there exists +b > 0 such that infx∈A,r∈(0,b]r−dVol(B(x,r) ∩ A) > 0. Since Vol(B(x,2r) ∩ A) is at most +2dθdrd for r ≤ b, and is at most Vol(A) for all r, the doubling property follows. +Points of A can be ordered by using the lexicographic ordering inherited from Rd, +thus (O). Since A is convex, for all x ∈ A and r > 0 the set B(x,r)∩A is convex and hence +connected, implying (B). All convex polytopes are simply connected, and therefore uni- +coherent [5, Lemma 9.1], hence (T). Condition (G) follows immediately from Proposition +3.14, which we prove below. +Proposition 3.14. Let A be a convex finite polytope in Rd. Let N(·) denote the number +of components of a set. There exists δ1 > 0 such that for any x ∈ A any r ∈ (0,δ1), we +have N(A\B(x,r)) ≤ 2. Moreover, in the case that N(A\B(x,r)) = 2, the diameter of the +smaller component is at most cr, where c is a constant depending only on A. +21 + +Proof of Proposition 3.14. Write B for B(x,r). Our first observation is that if y ∈ A \ B, +then there is at least one vertex v ∈ Φ0(A) such that the line segment [y,v] is contained +in A \ B. Indeed, if this failed then for each v ∈ Φ0(A) there would exist a point u(v) ∈ +[y,v]∩B. But then since A is convex, y would lie in the convex hull of {v : v ∈ Φ0(A)}, and +therefore also in the convex hull of {u(v) : v ∈ Φ0(A)}. Indeed, there exist αv ≥ 0 with +∑v∈Φ0(A) αv = 1 such that y = ∑v∈Φ0(A) αvv, and there exists βv ∈ [0,1] such that u(v) = +βvy + (1 − βv)v. Substituting v by u(v) and rearranging terms shows that y = ∑v α′ +vu(v) +with some nonnegative α′ +v and ∑vα′ +v = 1, thus the claim. But then since B is convex we +would have y ∈ B, a contradiction. +We refer to the one-dimensional faces ϕ ∈ Φ1(A) as edges of A. Our second observa- +tion is that if the number of edges of A that intersect B is at most 1, then A\B is connected. +Indeed, in this case, for any distinct v,v′ ∈ Φ0(A) there is a path along edges of A from v +to v′ that avoids B. For example, if v,v′ lie in the same two-dimensional face ϕ of A then +since B intersects at most one edge of the polygon ϕ, there is a path from v to v′ along the +edges of ϕ avoiding B. Therefore all v ∈ Φ0(A) lie in the same component of A \ B, so +using the first observation we deduce that A\B is connected. +Recall the definition of K(A) at (3.19). Our third observation is that if dist(v,B) ≥ +3rK(A) for all v ∈ Φ0(A) then A \ B is connected. Indeed, suppose dist(v,B) ≥ 3rK(A) +for all v ∈ Φ0(A). Suppose ϕ,ϕ′ are distinct edges of A with B∩ϕ ̸= ∅, and pick y ∈ B∩ϕ. +Then dist(y,∂ϕ) ≥ 3rK(A) so that by (3.18), dist(y,ϕ′) ≥ 3rK(A)/K(ϕ,ϕ′) ≥ 3r. Hence +by the triangle inequality dist(B,ϕ′) ≥ 3r−2r = r, so that B∩ϕ′ = ∅. Hence B intersects +at most one edge of A, and by our second observation A\B is connected. +Suppose dist(v,B) ≤ 3rK(A) for some v ∈ Φ0(A). Provided r is small enough, this +cannot happen for more than one v ∈ Φ0(A). If u,u′ ∈ Φ0(A) \ {v}, then v /∈ [u,u′] so +dist(v,[u,u′]) > 0. Therefore provided r is small enough, [u,u′] ⊂ A\B. Thus provided r +is small enough, all vertices u ∈ Φ0(A)\{v} lie in the same component of A\B. If also v +lies in this component, then (by our first observation) A\B is connected. +Thus A\B is disconnected only if v lies in a different component of A\B than all the +other vertices. In that case, for y ∈ A\B, if [y,v] ⊂ A\B then y is in the same component +as v; otherwise (by our first observation) y lies in the same component as all of the other +vertices, and thus A\B has exactly two components. +If A \ B has two components, and y ∈ A \ B with ∥y − v∥ > (3K(A) + 2)r, then we +claim [y,v] ∩B ̸= ∅. Indeed, for each u ∈ Φ0(A) \ {v} the ray from v in the direction of +u passes through B. But then by an argument based on the convexity of both A and B, +the ray from v in the direction of y must also pass through B. Since dist(v,B) ≤ 3rK(A) +and diam(B) = 2r, this ray must pass through B at a distance at most (3K(A) +2)r from +v, i.e. before it reaches y, and the claim follows. Therefore y lies in the component of +A \ B that does not contain v, and thus the component containing v has diameter at most +(3K(A)+2)r. +22 + +To apply Proposition 3.6, we need to define a partition of A for each small r > 0, then +estimate the corresponding covering numbers and µ-measures in (3.12). +Taking into account a variety of boundary effects near ∂A, one should consider sepa- +rately regions near different faces of A. It is however not trivial to construct this partition +in such a way that we can obtain tight µ-measure estimates in (3.12). The matter is com- +plicated by the fact that the set G in (3.12) that intersects a region near ϕ is potentially +close to a lower dimensional face lying inside ∂ϕ. We can avoid the boundary complica- +tions by constructing inductively from regions near to the highest dimensional face to the +lowest, with increasing ’thickness’. The partition made of T(ϕ,r)’s defined below and +the left-over interior region is defined for this purpose. +Let (Kj) j∈N be an increasing sequence with K1 = 1, and with Kj+1 > (2K(A)+1)Kj +for each j ∈ N. For instance, we could take Kj = (2K(A)+2) j−1. +Now for each r > 0 and ϕ ∈ Φ(A), define the set +T(ϕ,r) := ϕrKd−D(ϕ) \∪ϕ′∈Φ(A):ϕ′⊊ϕ(ϕ′)rKd−D(ϕ′), +where the T stands for ‘territory’. Also define T(A,r) := A \ ∪ϕ∈Φ(A)ϕrKd−D(ϕ) For each +ϕ ∈ Φ∗(A), we have T(ϕ,r) ̸= ∅ for all r sufficiently small. Hence, there exists r0 > 0 +such that for all ϕ and all r < r0, T(ϕ,r) ̸= ∅. Moreover, territories of distinct faces are +disjoint, as we show in the following lemma. +Lemma 3.15. There exists r0 > 0 such that for all r ∈ (0,r0), and any distinct ϕ,ϕ′ ∈ +Φ∗(A), it holds that T(ϕ,r) ∩T(ϕ′,r) = ∅. Moreover, if ϕ,ϕ′ ∈ Φ(A) with ϕ \ ϕ′ ̸= ∅, +and y ∈ T(ϕ,r), then B(y,r) does not intersect ϕ′. +Proof. We can (and do) assume without loss of generality that ϕ \ϕ′ ̸= ∅ and ϕ′ \ϕ ̸= ∅. +Indeed, if ϕ ⊂ ϕ′, then by construction T(ϕ′,r)∩T(ϕ,r) = ∅. +If ϕ is a vertex, then dist(ϕ,ϕ′) > 0 so that T(ϕ,r)∩T(ϕ′,r) = ∅ for all r small. So +it suffices to consider the case where D(ϕ) > 0 and D(ϕ′) > 0. +Let j := d −D(ϕ) and j′ := d −D(ϕ′). We can and do assume j′ ≤ j ≤ d −1. +If there exists x ∈ T(ϕ,r)∩T(ϕ′,r), then we can find z ∈ ϕ,z′ ∈ ϕ′ such that ∥x−z∥ ≤ +rKj and ∥x−z′∥ ≤ rKj′. Therefore dist(z,ϕ′) ≤ r(Kj +Kj′) ≤ 2rKj. +On the other hand, since x ∈ T(ϕ,r), dist(x,∂ϕ) ≥ rKj+1, and so by the triangle in- +equality, rKj+1 − rKj ≤ dist(z,∂ϕ) ≤ K(A)dist(z,ϕ′), where the last inequality comes +from (3.18). Combining the estimates leads to Kj+1 ≤ (2K(A)+1)Kj, which is a contra- +diction. The first claim follows. +Moving to the second claim, let ϕ,ϕ′ ∈ Φ(A) with ϕ \ϕ′ ̸= ∅. Suppose y ∈ ϕ′ +r. Set +˜Φ := {ψ ∈ Φ(A) : ψ ⊊ ϕ′,y ∈ ψKD−d(ψ)}. +If ˜Φ = ∅ then y ∈ T(ϕ′,r). Otherwise, choose ψ ∈ ˜Φ of minimal dimension. Then +y ∈ T(ψ,r). Either way, y /∈ T(ϕ,r) by the first claim. Therefore T(ϕ,r)∩ϕ′ +r = ∅. +23 + +As a last ingredient for applying Proposition 3.6, for each J > 1 and r ∈ (0,1), we con- +struct a partition of A and show (3.12) for all G with diameter at most Jr. The coefficients +aj depend on the location of G in relation to faces of A. +Lemma 3.16. Let J ∈ N and ε > 0. Then the following hold: +(i) For each ϕ ∈ Φ(A) we have κ(T(ϕ,2Jr),r) = O(r−D(ϕ)) as r ↓ 0. Moreover we +have κ(A\∪ϕ∈Φ(A)T(ϕ,2Jr),r) = O(r−d) as r ↓ 0. +(ii) For all small r > 0 and any G ⊂ A with diam(G) ≤ Jr, if it intersects T(ϕ,2Jr) for +some ϕ ∈ Φ∗(A), then +µ(Gr \G) ≥ (1−ε) fϕρϕrd. +(3.20) +Proof. Item (i) follows by the definition of T(ϕ,r). Indeed, ϕ is contained in a bounded +region within a D(ϕ)-dimensional affine space, and therefore can be covered by O(r−D(ϕ)) +balls of radius r. If we then take balls of radius r(1 +2JKd−D(ϕ)) with the same centres, +they will cover T(ϕ,2Jr), and one can then cover each of the larger balls with a fixed +number of balls of radius r. +For (ii), let G ⊂ A with diam(G) ≤ Jr. Suppose first that G∩T(ϕ,2Jr) ̸= ∅ for some +ϕ ∈ Φ(A). Let x0 ∈ G ∩T(ϕ,2Jr). Then Gr ⊂ B(x0,2Jr). By Lemma 3.15, we see that +B(x0,2Jr) does not intersect any ϕ′ ∈ Φ(A) with ϕ \ϕ′ ̸= ∅. It follows that +B(x0,2Jr)∩A = B(x0,2Jr)∩(z0 +Kϕ) +(3.21) +where Kϕ is the cone determined by ϕ and z0 is the point of ϕ closest to x0. +Set D(x,r) := B(x,r) ∩ (x + Kϕ). We claim that for any x ∈ G, we have D(x,r) ⊂ +A. Indeed, given y ∈ D(x,r), we can write y = z0 + (x − z0) + (y − x) =: z0 + θ1 + θ2. +Here θ1,θ2 ∈ Kϕ. By convexity and scale invariance of Kϕ, we have θ1 + θ2 ∈ Kϕ so +y ∈ z0 + Kϕ. Also ∥y − x0∥ ≤ ∥y − x∥ + ∥x − x0∥ ≤ 2Jr, and hence y ∈ A by (3.21), as +claimed. +It follows that (with ⊕ denoting Minkowski addition) +µ(Gr \G) ≥ µ((G⊕D(o,r))\G) ≥ Vol((G⊕D(o,r))\G) +inf +x∈G⊕D(o,r) f(x). +By the Brunn-Minkowski inequality [5, Section 5.3], we have Vol(G⊕D(o,r)) ≥ Vol(G)+ +Vol(D(o,r)) = Vol(G)+ρϕrd. The claim (3.20) follows by the continuity of f on ∂A. +As for the case ϕ = A, suppose now that G∩T(A,2Jr) ̸= ∅. Taking x ∈ G∩T(A,2Jr) +we have dist(x,∂A) ≥ 2Jr, and hence dist(G,∂A) ≥ 2Jr −Jr = Jr. Therefore Gr ⊂ A, so +by the Brunn-Minkowski inequality +µ(Gr \G) ≥ f0Vol((G⊕B(o,r))\G) ≥ f0θdrd. +In this case fϕ = f0 and ρϕ = θd, and the claim (3.20) follows in this case too, completing +the proof of (ii). +24 + +Proof of Theorem 2.5. By (2.9), and Theorem 2.1, it suffices to prove the upper bound. +We shall do this by applying Proposition 3.6 in the situation of Theorem 2.5. +By Lemma 3.13, the restriction to A of Lebesgue measure has the doubling property, +and conditions (O), (B), (T) and (G) are satisfied +To apply Proposition 3.6, we need to define (for each K ∈ N and each r ∈ (0,r0(K))) +a finite partition {T( j,K,r)}. For this we take the sets T(ϕ,2Kr),ϕ ∈ Φ∗(A). By Lemma +3.15, and the definition of T(A,r), for each K ∈ N there exists r0(K) > 0 such that for +r ∈ (0,r0(K)) the sets T(ϕ,2Kr), ϕ ∈ Φ∗(A), do indeed partition A. +For each ϕ ∈ Φ∗(A), using Lemma 3.16-(i) we have the condition (3.11) in Proposition +3.6, where the constant denoted bj there is equal to D(ϕ). Also, using Lemma 3.16-(ii) +we have the condition (3.12) in proposition 3.6, where the constant denoted aj there is +equal to (1−ε) fϕρϕ. +Suppose β < ∞. By applying Proposition 3.6 in the manner described above we see +that for ε > 0, we have +limsup +n→∞ +n(Mn,k(n))d/logn ≤ +max +ϕ∈Φ∗(A) +� ˆHβ(D(ϕ)/d) +(1−ε) fϕρϕ +� +, +and the result follows. If β = ∞, using corresponding part of Proposition 3.6 gives the +result in this case too. +References +[1] Baccelli, F. and Błaszczyszyn, B. (2009). Stochastic geometry and wireless networks +I: Theory. Foundations and Trends in Networking 4, 1–312. +[2] Bobrowski, O. (2022). Homological connectivity in random ˇCech complexes. +Probab. Theory Relat. Fields 183, 715–788. +[3] Bobrowski, O. and Kahle, M. (2018) Topology of random geometric complexes: a +survey. J. Appl. Comput. Topol. 1, 331–364. +[4] Last, G. and Penrose, M. (2018). Lectures on the Poisson Process. Cambridge Uni- +versity Press, Cambridge. +[5] Penrose, M. (2003) Random Geometric Graphs. Oxford University Press. +[6] Penrose, M. D. (1999). A strong law for the largest nearest-neighbour link between +random points. J. London Math. Soc. (2) 60, 951–960. +[7] Penrose, M. D. (1999) A strong law for the longest edge of the minimal spanning +tree. Ann. Probab. 27, 246–260. +25 + +[8] Penrose, M.D. (2022+) Random Euclidean coverage from within. arXiv:2101.06306, +to appear in Probab. Theory Relat. Fields. +26 + diff --git a/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/load_file.txt b/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83231ce3d4b1f222b1f602cc59a81b2f197a2162 --- /dev/null +++ b/ZdE0T4oBgHgl3EQfnAFi/content/tmp_files/load_file.txt @@ -0,0 +1,931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf,len=930 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='02506v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='PR] 6 Jan 2023 Largest nearest-neighbour link and connectivity threshold in a polytopal random sample † Mathew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Penrose ‡ Xiaochuan Yang § January 9, 2023 Abstract Let X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' be independent identically distributed random points in a convex polytopal domain A ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Define the largest nearest neighbour link Ln to be the smallest r such that every point of Xn := {X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',Xn} has another such point within distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We obtain a strong law of large numbers for Ln in the large-n limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' A related threshold, the connectivity threshold Mn, is the smallest r such that the random geometric graph G(Xn,r) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We show that as n → ∞, almost surely nLd n/logn tends to a limit that depends on the geometry of A, and nMd n/logn tends to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 1 Introduction This paper is primarily concerned with the connectivity threshold and largest nearest- neighbour link for a random sample Xn of n points specified compact region A in a d- dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The connectivity threshold, here denoted Mn, is defined to be the smallest r such that the random geometric graph G(Xn,r) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For any finite X ⊂ Rd the graph G(X ,r) is defined to have vertex set X with edges between those pairs of vertices x,y such that ∥x −y∥ ≤ r, where ∥ · ∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' More generally, for k ∈ N, the k-connectivity threshold Mn,k is the smallest r such that G(Xn,r) is k-connected (see the definition in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' †Supported by EPSRC grant EP/T028653/1 ‡Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='penrose@bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='uk §Department of Mathematics, Brunel University London, Uxbridge, UB83PH, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' xiaochuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='yang@brunel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='uk ORCID:0000-0003-2435-4615 1 The largest nearest neighbour link, here denoted Ln, is defined to be the the smallest r such that every vertex in G(Xn,r) has degree at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' More generally, for k ∈ N with k < n, the largest k-nearest neighbour link Ln,k is the smallest r such that every vertex in G(Xn,r) has degree at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' These thresholds are random variables, because the locations of the centres are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We investigate their probabilistic behaviour as n becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We shall derive strong laws of large numbers showing that that nLd n,k/logn converges almost surely (as n → ∞) to a finite positive limit, and establishing the value of the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover we show that nMd n,k/logn converges to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' These strong laws carry over to more general cases where k may vary with n, and the distribution of points may be non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We give results of this type for A a convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Previous results of this type (both for Ln,k and for Mn,k) were obtained for A having a smooth boundary, and for A a d-dimensional hypercube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It is perhaps not obvious from the earlier results, however, how the limiting constant depends on the geometry of ∂A, the topological boundary of A, for general polytopal A, which is quite subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It turns out, for example, that when d = 3 and the points are uniformly distributed over a polyhedron, the limiting behaviour of Ln is determined by the angle of the sharpest edge if this angle is less than π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We believe (but do not formally prove here) that if this angle exceeds π/2 then the point of Xn furthest from the rest of Xn is asymptotically uniformly distributed over ∂A, but if this angle is less than π/2 the location of this point in is asymptotically uniformly distributed over the union of those edges which are sharpest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Our motivation for this study is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First, understanding the connectivity thresh- old in dimension two is vital in telecommunications, for example, in 5G wireless net- work design, with the nodes of Xn representing mobile transceivers (see for example [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Second, detecting connectivity is a fundamental step for detecting all other higher dimensional topological features in modern topological data analysis (TDA), where the dimension of the ambient space may be very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' See [2, 3] for discussion of issues related to the one considered here, in relation to TDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' General motivation for considering random geometric graphs is discussed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' While our main results are presented (in Section 2) in the concrete setting of a poly- topal sample in Rd, our proofs proceed via general lower and upper bounds (Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6) that are presented in the more general setting of a random sample of points in a metric space satisfying certain regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This could be useful in pos- sible future work dealing with similar problems for random samples in, for example, a Riemannian manifold with boundary, a setting of importance in TDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 2 2 Statement of results Throughout this paper, we work within the following mathematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose we have the following ingredients: A finite compact convex polytope A ⊂ Rd (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=', one with finitely many faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' A Borel probability measure µ on A with probability density function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' On a common probability space (S,F,P), a sequence X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' of independent identically distributed random d-vectors with common probability distribution µ, and also a unit rate Poisson counting process (Zt,t ≥ 0), independent of (X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=') (so Zt is Poisson distributed with mean t for each t > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N, t > 0, let Xn := {X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',Xn}, and let Pt := {X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',XZt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' These are the point processes that concern us here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Observe that Pt is a Poisson point process in Rd with intensity measure tµ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For x ∈ Rd and r > 0 set B(x,r) := {y ∈ Rd : ∥y−x∥ ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For r > 0, let A(r) := {x ∈ A : B(x,r) ⊂ Ao}, the ‘r-interior’ of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For any point set X ⊂ Rd and any D ⊂ Rd we write X (D) for the number of points of X in D, and we use below the convention inf(∅) := +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given n,k ∈ N, and t ∈ (0,∞), define the largest k-nearest neighbour link Ln,k by Ln,k := inf({r > 0 : Xn(B(x,r)) ≥ k +1 ∀x ∈ Xn}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) Set Ln := Ln,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then Ln is the largest nearest-neighbour link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We are chiefly interested in the asymptotic behaviour of Ln for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' More generally, we consider Ln,k where k may vary with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let θd := πd/2/Γ(1 + d/2), the volume of the unit ball in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given x,y ∈ Rd, we denote by [x,y] the line segment from x to y, that is, the convex hull of the set {x,y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given m ∈ N and functions f : N∩[m,∞) → R and g : N∩[m,∞) → (0,∞), we write f(n) = O(g(n)) as n → ∞, if limsupn→∞ | f(n)|/g(n) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We write f(n) = Ω(g(n)) as n → ∞ if liminfn→∞( f(n)/g(n)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given s > 0 and functions f : (0,s) → R and g : (0,s) → (0,∞), we write f(r) = O(g(r)) as r ↓ 0 if limsupr↓0 | f(r)|/g(r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We write f(r) = Ω(g(r)) as r ↓ 0, if liminfr↓0( f(r)/g(r)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Throughout this section, assume we are given a constant β ∈ [0,∞] and a sequence k : N → N with lim n→∞(k(n)/logn) = β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' lim n→∞(k(n)/n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) 3 We make use of the following notation throughout: f0 := ess infx∈A f(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' f1 := inf x∈∂A f(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3) H(t) := � 1−t +t logt, if t > 0 1, if t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4) Observe that −H(·) is unimodal with a maximum value of 0 at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given a ∈ [0,∞), we define the function ˆHa : [0,∞) → [a,∞) by y = ˆHa(x) ⇐⇒ yH(a/y) = x, y ≥ a, with ˆH0(0) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Note that ˆHa(x) is increasing in x, and that ˆH0(x) = x and ˆHa(0) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Throughout this paper, the phrase ‘almost surely’ or ‘a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='s.’ means ‘except on a set of P-measure zero’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N, we use [n] to denote {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We write f|A for the restriction of f to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let Φ(A) denote the set of all faces of the polytope A (of all dimensions up to d −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also, let Φ∗(A) := Φ(A)∪{A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' it is sometimes useful for us to think of A itself as a face, of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given a face ϕ ∈ Φ∗(A), denote the dimension of this face by D(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then 0 ≤ D(ϕ) ≤ d, and ϕ is a D(ϕ)-dimensional polytope embedded in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ϕo denote the relative interior of ϕ, and set ∂ϕ := ϕ \\ϕo (if D(ϕ) = 0 we take ϕo := ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If D(ϕ) < d then set fϕ := infx∈ϕ f(x), and if ϕ = A then set fϕ := f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then there is a cone Kϕ in Rd such that every x ∈ ϕo has a neighbourhood Ux such that A∩Ux = (x+Kϕ)∩Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Define the angular volume ρϕ of ϕ to be the d-dimensional Lebesgue measure of Kϕ ∩B(o,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For example, if ϕ = A then ρϕ = θd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If D(ϕ) = d −1 then ρϕ = θd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If D(ϕ) = 0 then ϕ = {v} for some vertex v ∈ ∂A, and ρϕ equals the volume of B(v,r) ∩ A, divided by rd, for all sufficiently small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If d = 2, D(ϕ) = 0 and ωϕ denotes the angle subtended by A at the vertex ϕ, then ρϕ = ωϕ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If d = 3 and D(ϕ) = 1, and αϕ denotes the angle subtended by A at the edge ϕ (which is the angle between the two boundary planes of A meeting at ϕ), then ρϕ = 2αϕ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose A is a compact convex finite polytope in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume that f|A is continuous at x for all x ∈ ∂A, and that f0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume k(·) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, almost surely, lim n→∞nLd n,k(n)/k(n) = max ϕ∈Φ∗(A) � 1 fϕρϕ � if β = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5) lim n→∞nLd n,k(n)/logn = max ϕ∈Φ∗(A) � ˆHβ(D(ϕ)/d) fϕρϕ � if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6) 4 In the next three results, we spell out some special cases of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose that d = 2, A is a convex polygon and f|A is continuous at x for all x ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let V denote the set of vertices of A, and for v ∈ V let ωv denote the angle subtended by A at vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) holds with β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, almost surely, lim n→∞ �nL2 n,k(n) logn � = max � ˆHβ(1) π f0 , 2 ˆHβ(1/2) π f1 ,max v∈V � 2β ωv f(v) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7) In particular, for any constant k ∈ N, limn→∞ � nπL2 n,k logn � = 1 f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose d = 3 (so θd = 4π/3), A is a convex polyhedron and f|A is continuous at x for all x ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let V denote the set of vertices of A, and E the set of edges of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For e ∈ E, let αe denote the angle subtended by A at edge e, and fe the infimum of f over e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For v ∈ V let ρv denote the angular volume of vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) holds with β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, almost surely, lim n→∞ �nL3 n,k(n) logn � = max � ˆHβ(1) θ3 f0 , 2 ˆHβ(2/3) θ3 f1 , 3 ˆHβ(1/3) 2mine∈E(αe fe),max v∈V � β ρv f(v) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In particular, if β = 0 the above limit comes to max � 3 4π f0, 1 π f1,maxe∈E � 1 2αe fe �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4 ([5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose A = [0,1]d, and f|A is continuous at x for all x ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For 1 ≤ j ≤ d let ∂j denote the union of all (d − j)-dimensional faces of A, and let f j denote the infimum of f over ∂j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) with β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then lim n→∞ �nLd n,k(n) logn � = max 0≤j≤d � 2j ˆHβ(1− j/d) θd f j � , a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8) It is perhaps worth spelling out what the preceding results mean in the special case where β = 0 (for example, if k(n) is a constant) and also µ is the uniform distribu- tion on A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' f(x) ≡ f0 on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this case, the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6) comes to maxϕ∈Φ∗(A) D(ϕ) (d f0ρϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7) comes to 1/(π f0), while the limit in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3 comes to f −1 0 max[1/π,maxe(1/(2αe))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' So far we have only presented results for the largest k-nearest neighbor link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' A closely related threshold is the k-connectivity threshold defined by Mn,k := inf{r > 0 : G(Xn,r) is k-connected}, 5 where a graph G of order n is said to be k-connected (k < n) if G cannot be disconnected by the removal of at most k − 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set Mn,1 = Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then Mn is the connectivity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Notice that for all k,n with k < n we have Ln,k ≤ Mn,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9) Indeed, if r < Ln,k, then there exists i ∈ [n] such that degXi < k in G(Xn,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then the removal of all vertices adjacent to Xi disconnects G(Xn,r), implying that r < Mn,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Our second main result shows that (Mn,k/Ln,k) → 1 almost surely as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For this result we need d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose A is a compact convex finite polytope in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume that f|A is continuous at x for all x ∈ ∂A, and that f0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume k(·) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) Then, almost surely, lim n→∞nMd n,k(n)/k(n) = max ϕ∈Φ∗(A) � 1 fϕρϕ � if β = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10) lim n→∞nMd n,k(n)/logn = max ϕ∈Φ∗(A) � ˆHβ(D(ϕ)/d) fϕρϕ � if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' One can spell out consequences of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5 in dimensions d = 2,3 and the case of [0,1]d with exactly the same statement as in Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5 extend earlier work found in [5] on the case where A is the unit cube, to more general polytopal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The case where A has a smooth boundary is also considered in [5] (in this case with also k(n) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=', the result was first given in [6] for Ln,k and in [7] for Mn,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In [8], similar results are given for the k-coverage threshold Rn,k, which is given by Rn,k := inf{r > 0 : Xn(B(x,r)) ≥ k ∀x ∈ A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' n,k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) Our results here, together with [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2], show that both Ln,k(n) and Mn,k(n) are asymptotic to Rn,k(n) almost surely, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 3 Proofs In this section we prove the results stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Throughout this section we are assuming we are given a constant β ∈ [0,∞] and a sequence (k(n))n∈N satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 6 Recall that µ denotes the distribution of X1, and this has a density f with support A, and that Ln,k is defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall that ˆHβ(x) is defined to be the y ≥ β such that yH(β/y) = x, where H(·) was defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N and p ∈ [0,1] let Bin(n, p) denote a binomial random variable with param- eters n, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall that H(·) was defined at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4), and Zt is a Poisson(t) variable for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The proofs in this section rely heavily on the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1 (Chernoff bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose n ∈ N, p ∈ (0,1), t > 0 and 0 ≤ k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (a) If k ≥ np then P[Bin(n, p) ≥ k] ≤ exp(−npH(k/(np))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (b) If k ≤ np then P[Bin(n, p) ≤ k] ≤ exp(−npH(k/(np))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (c) If k ≥ e2np then P[Bin(n, p) ≥ k] ≤ exp(−(k/2)log(k/(np))) ≤ e−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (d) If k < t then P[Zt ≤ k] ≤ exp(−tH(k/t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (e) If k ∈ N then P[Zt = k] ≥ (2πk)−1/2e−1/(12k) exp(−tH(k/t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' [5, Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1 A general lower bound In this subsection we present an asymptotic lower bound on Ln,k(n), not requiring any extra assumptions on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In fact, A here can be any metric space endowed with a Borel probability measure µ which satisfies the following for some ε′ > 0 and some d > 0: µ(B(x,r)) ≥ ε′rd, ∀ r ∈ (0,1),x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) The definition of Ln,k at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) carries over in an obvious way to this general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Later, we shall derive the results stated in Section 2 by applying the results of this sub- section to the different regions within A (namely interior, boundary, and lower-dimensional faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given r > 0,a > 0, define the ‘packing number’ ν(r,a) be the largest number m such that there exists a collection of m disjoint closed balls of radius r centred on points of A, each with µ-measure at most a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 (General lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) with d,ε′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let a > 0,b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose ν(r,ard) = Ω(r−b) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then almost surely, if β = ∞ then liminfn→∞ � nLd n,k(n)/k(n) � ≥ 1/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β < ∞ then liminfn→∞ � nLd n,k(n)/logn � ≥ a−1 ˆHβ(b/d), almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First suppose β = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let u ∈ (0,1/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set rn := (uk(n)/n)1/d, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2), rn → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, given n sufficiently large, we have ν(rn,ard n) > 0 so we can find yn ∈ A such that µ(B(yn,rn)) ≤ ard n, and hence nµ(B(yn,rn)) ≤ auk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If 7 k(n) ≤ e2nµ(B(yn,rn)) (and hence nµ(B(yn,rn)) ≥ e−2k(n)), then since Xn(B(yn,rn)) is binomial with parameters n and µ(B(yn,rn)), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(a) we have that P[Xn(B(yn,rn)) ≥ k(n)] ≤ exp � −nµ(B(yn,rn))H � k(n) nµ(B(yn,rn)) �� ≤ exp � −e−2k(n)H � (au)−1�� , while if k(n) > e2nµ(B(yn,rn)) then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(c), P[Xn(B(yn,rn)) ≥ k(n)] ≤ e−k(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore P[Xn(B(yn,rn)) ≥ k(n)] is summable in n because k(n)/logn → ∞ as n → ∞ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let δ0 ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) µ(B(yn,δ0rn) ≥ ε′δ d 0 uk(n)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(b), P[Xn(B(yn,δ0rn)) = 0] ≤ exp(−ε′δ d 0 uk(n)), which is summable in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus by the Borel-Cantelli lemma, almost surely event Fn := {Xn(B(yn,rn)) < k(n)}∩ {Xn(B(yn,δ0rn)) > 0} occurs for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' But if Fn occurs then Ln,k(n) ≥ (1−δ0)rn so that nLd n,k(n)/k(n) ≥ (1−δ0)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This gives the result for β = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now suppose instead that β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose first that b = 0, so that ˆHβ(b/d) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume that β > 0 (otherwise the result is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Choose β ′ ∈ (0,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let δ > 0 with β ′ < β −2δ and with β ′H � β−2δ β ′ � > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This is possible because H(β/β ′) > 0 and H(·) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N, set rn := ((β ′ logn)/(an))1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also set k′(n) = ⌈(β −δ)logn⌉, and k′′(n) = ⌈(β −2δ)logn⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By assumption ν(rn,ard n) = Ω(1), so for all n large enough, we can (and do) choose xn ∈ A such that nµ(B(xn,rn)) ≤ nard n = β ′logn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then by a simple coupling, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(a), P[Xn(B(xn,rn)) ≥ k′′(n)] ≤ P � Bin � n,(β ′logn)/n) � ≥ k′′(n) � ≤ exp � − � β ′logn � H �β −2δ β ′ �� ≤ n−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let δ ′ ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1), for n large enough and all x ∈ A, nµ(B(x,δ ′rn)) ≥ nε′(δ ′rn)d = ε′(δ ′)d(β ′/a)logn so that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(b), P[Xn(B(x,δ ′rn)) = 0] ≤ n−ε′(δ ′)dβ ′/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now choose K ∈ N such that δK > 1 and Kε′(δ ′)dβ ′/a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N set z(n) := nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For all large enough n we have k′(z(n)) ≥ k′′(z(n+1)), so by the preceding estimates, P[Xz(n+1)(B(xz(n+1),rz(n+1))) ≥ k′(z(n))] ≤ P[Xz(n+1)(B(xz(n+1),rz(n+1))) ≥ k′′(z(n+1))] ≤ (n+1)−δK, and since xz(n+1) ∈ A, also P[Xz(n)(B(xz(n+1),δ ′rz(n))) = 0] ≤ n−ε′(δ ′)dβ ′K/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Both of these upper bounds are summable in n, so by the Borel-Cantelli lemma, almost surely for all 8 large enough n we have the event {Xz(n+1)(B(xz(n+1),rz(n+1))) < k′(z(n))}∩{Xz(n)(B(xz(n+1),δ ′rz(n))) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose the above event occurs and suppose m ∈ N with z(n) ≤ m ≤ z(n +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Note that rz(n+1)/rz(n) → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, provided n is large enough, Lm,k′(z(n)) ≥ rz(n+1) −δ ′rz(n) ≥ (1−δ ′)2rm, and moreover k′(z(n)) ≤ k(m) so that Lm,k(m) ≥ (1−δ ′)2rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence it is almost surely the case that liminf m→∞ (mLd m,k(m)/logm) ≥ (1−δ ′)2d liminf m→∞ (mrd m/logm) = (1−δ ′)2da−1β ′, and this yields the result for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now suppose instead that β < ∞ and b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let u ∈ (a−1β,a−1 ˆHβ(b/d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' note that this implies uaH(β/(ua)) < b/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Choose ε > 0 such that (1+ε)uaH(β/(ua)) < (b/d)− 9ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also let δ ′ ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For each n ∈ N set rn = (u(logn)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let mn := ν(rn,ard n), and choose xn,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=', xn,mn ∈ A such that the balls B(xn,1,rn),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',B(xn,mn,rn) are pairwise disjoint and each have µ-measure at most ard n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set λ(n) := n + n3/4 and λ −(n) := n − n3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For 1 ≤ i ≤ mn, if k(n) ≥ 1 then by a simple coupling, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(e), P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ P[Zλ(n)ardn ≤ k(n)] ≥ � e−1/(12k(n)) � 2πk(n) � exp � −λ(n)ard nH � k(n) λ(n)ardn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now λ(n)rd n/logn → u so by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2), k(n)/(λ(n)ard n) → β/(ua) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus by the continuity of H(·), provided n is large enough, for 1 ≤ i ≤ mn, P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ � e−1/12 � 2π(β +1)logn � exp � −(1+ε)auH � β au � logn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence, by our choice of ε, there is a constant c > 0 such that for all large enough n and all i ∈ [mn] we have P[Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ c(logn)−1/2n9ε−b/d ≥ n8ε−b/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) Since xn,i ∈ A, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1), for n large enough and 1 ≤ i ≤ mn we have µ(B(xn,i,δ ′rn)) ≥ ε′(δ ′rn)d (as well as µ(B(xn,i,rn)) ≤ ard n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus, given the value of Pλ(n)(B(xn,i,rn)), 9 the value of Pλ −(n)(B(xn,i,δ ′rn)) is binomially distributed with probability parameter bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also max1≤i≤mn E[Pλ(n)(B(xn,i,rn))] tends to infinity as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore there exists η > 0 such that for all large enough n, defining the event En,i := {Pλ(n)(B(xn,i,rn)) ≤ k(n)}∩{Pλ −(n)(B(xn,i,δ ′rn) ≥ 1}, we have for all large enough n that inf 1≤i≤mn P[En,i|Pλ(n)(B(xn,i,rn)) ≤ k(n)] ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence, setting En := ∪mn i=1En,i, for all large enough n we have P[Ec n] ≤ (1−ηn8ε−b/d)mn ≤ exp(−ηmnn8ε−b/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By assumption mn = ν(rn,ard n) = Ω(r−b n ) so that for large enough n we have mn ≥ n(b/d)−ε, and therefore P[Ec n] is is summable in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(d), and Taylor expansion of H(x) about x = 1 (see the print version of [5, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4] for details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' there may be a typo in the electronic version), for all n large enough P[Zλ(n) < n] ≤ exp(−1 9n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Similarly P[Zλ −(n) > n] ≤ exp(−1 9n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If En occurs, and Zλ −(n) ≤ n, and Zλ(n) ≥ n, then for some i ≤ mn there is at least one point of Xn in B(xn,i,δ ′rn) and at most k(n) points of Xn in B(xn,i,rn), and hence Ln,k(n) > (1−δ ′)rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence by the union bound P[Ln,k(n) ≤ rn(1−δ ′)] ≤ P[Ec n]+P[Zλ(n) < n]+P[Zλ −(n) > n], which is summable in n by the preceding estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore by the Borel-Cantelli lemma, P[liminf(nLd n,k(n)/logn) ≥ u(1−δ ′)d] = 1, u < a−1 ˆHβ(b/d),δ ′ ∈ (0,1), so the result follows for this case too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1 In this subsection we assume, as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1, that A is a compact convex finite poly- tope in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We also assume that the probability measure µ has density f with respect to Lebesgue measure on Rd, and that f|A is continuous at x for all x ∈ ∂A, and that f0 > 0, recalling from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3) that f0 := ess infx∈A f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also we let k(n) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) for some β ∈ [0,∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let Vol denote d-dimensional Lebesgue measure Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There exists ε′ > 0 depending only on f0 and A, such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let B0 be a (fixed) ball contained in A, and let b denote the radius of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For x ∈ A, let Sx denote the convex hull of B0 ∪{x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then Sx ⊂ A since A is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If x /∈ B0, then for r < b the set B(x,r)∩Sx is the intersection of B(x,r) with a cone having vertex x, and since A is bounded the angular volume of this cone is bounded away from zero, uniformly over x ∈ A \\ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore r−dVol(B(x,r) ∩ A) is bounded away from zero uniformly over r ∈ (0,b) and x ∈ A \\ B0 (and hence over x ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since we assume f0 > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall that ν(r,a) was defined just before Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall that for each face ϕ ∈ Φ∗(A) we denote the angular volume of A at ϕ by ρϕ, and set fϕ := infϕ f(·) (if ϕ ∈ Φ(A)) or fϕ = f0 (if ϕ = A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ϕ ∈ Φ∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume f|A is continuous at x for all x ∈ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, almost surely: liminf n→∞ � nLd n,k(n)/k(n) � ≥ (ρϕ fϕ)−1 if β = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3) liminf n→∞ � nLd n,k(n)/logn � ≥ (ρϕ fϕ)−1 ˆHβ(D(ϕ)/d) if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let a > fϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Take x0 ∈ ϕ such that f(x0) < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If D(ϕ) > 0, assume also that x0 ∈ ϕo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the assumed continuity of f|A at x0, for all small enough r > 0 we have µ(B(x0,r)) ≤ aρϕrd, so that ν(r,aρϕrd) = Ω(1) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 (taking b = 0), if β = ∞ then almost surely liminfn→∞ nLd n,k(n)/k(n) ≥ 1/(aρϕ), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β < ∞ and if D(ϕ) = 0, then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 (with b = 0), almost surely liminfn→∞(nLd n,k(n)/logn) ≥ ˆHβ(0)/(aρϕ), and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now suppose β < ∞ and D(ϕ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Take δ > 0 such that f(x) < a for all x ∈ B(x0,2δ) ∩ A, and such that moreover B(x0,2δ) ∩ A = B(x0,2δ) ∩ (x0 + Kϕ) (the cone Kϕ was defined in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then for all x ∈ B(x0,δ) ∩ ϕ and all r ∈ (0,δ), we have µ(B(x,r)) ≤ aρϕrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There is a constant c > 0 such that for small enough r > 0 we can find at least cr−D(ϕ) points xi ∈ B(x0,δ)∩ϕ that are all at a distance more than 2r from each other, and there- fore ν(r,aρϕrd) = Ω(r−D(ϕ)) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 we have liminf n→∞ � nLd n,k(n)/k(n) � ≥ (aρϕ)−1 ˆHβ(D(ϕ)/d), almost surely, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If we assumed f|A to be continuous on all of A, we would not need the next lemma because we could instead use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4 for ϕ = A as well as for lower-dimensional faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' However, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1 we make the weaker assumption that f|A is continuous at x only for x ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this situation, we also require the following lemma to deal with ϕ = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 11 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It is the case that P[liminf(nLd n,k(n)/k(n)) ≥ 1/(θd f0)] = 1 if β = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5) P[liminf n→∞ (nLd n,k(n)/logn) ≥ ˆHβ(1)/(θd f0)] = 1 if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let α > f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then by taking B = A in [8, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4], liminf r↓0 rdν(r,αθdrd) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7) Set rn := (k(n)/(nθdα))1/d if β = ∞, and set rn := ( ˆHβ(1)(logn)/(nθdα))1/d if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7) we can apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2 (taking a = αθd and b = 0) to deduce that liminfn→∞ nLd n,k(n)/k(n) ≥ (θdα)−1, almost surely, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose instead that β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7), ν(r,αθdrd) = Ω(r−d) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2, almost surely liminfn→∞ � nLd n,k(n)/logn � ≥ (αθd)−1 ˆHβ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The result follows by letting α ↓ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First suppose β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) that Ln,k ≤ Rn,k+1 for all n,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2) we have (k(n)+1)/logn → β as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore using [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2] for the second inequality below, we obtain almost surely that limsup n→∞ �nLd n,k(n) logn � ≤ limsup n→∞ �nRd n,k(n)+1 logn � ≤ max ϕ∈Φ∗(A) � ˆHβ(D(ϕ)/d) fϕρϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8) Alternatively, this upper bound could be derived using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9) and the asymptotic upper bound on Mn that we shall derive in the next section for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4, we have a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' that liminf n→∞ � nLd n,k(n)/logn � ≥ max ϕ∈Φ∗(A) � ˆHβ(D(ϕ)/d) fϕρϕ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9) and combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8) yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now suppose β = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this case, again using the inequality Ln,k ≤ Rn,k+1 and [8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2], we obtain instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8) that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' limsup n→∞ � nLd n,k(n)/k(n) � ≤ max ϕ∈Φ∗(A) � 1 fϕρϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10) Also by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4, instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9) we have a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' that liminf n→∞ � nLd n,k(n)/k(n) � ≥ max ϕ∈Φ∗(A) � 1 fϕρϕ � , and combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10) yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3 A general upper bound In this subsection we present an asymptotic upper bound for Mn,k(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' As we did for the lower bound in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1, we shall give our result (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 below) in a more general setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' we assume that A is a general metric space endowed with two Borel mea- sures µ and µ∗ (possibly the same measure, possibly not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume that µ is a probability measure and that µ∗ is a doubling measure, meaning that there is a constant c∗ (called a doubling constant for µ∗) such that µ∗(B(x,2r)) ≤ c∗µ∗(B(x,r)) for all x ∈ A and r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We shall require further conditions on A: an ordering condition (O), a condition on balls (B), a topological condition (T) and a geometrical condition (G) as follows: (O) There is a total ordering of the elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (B) For all x ∈ A and r > 0, the ball B(x,r) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (T) The space A is unicoherent (see [5, Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1]), and also connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (G) There exists δ1 > 0, and K0 ∈ (1,∞), such that for all r < δ1 and any x ∈ A, the number of components of A\\B(x,r) is at most two, and if there are two components, at least one of these components has diameter at most K0r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given D ⊂ A and r > 0, we write Dr for {y ∈ A : dist(y,D) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also, let κ(D,r) be the r-covering number of D, that is, the minimal m ∈ N such that D can be covered by m balls centred in D with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' As before, given µ we assume X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' to be independent µ-distributed random elements of A with the k-connectivity threshold Mn,k defined to be the minimal r such that G(Xn,r) is k-connected, with Xn := {X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 (General upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose that (A,µ,µ∗) are as described above and A satisfies conditions (O), (B), (T), (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ℓ ∈ N and let d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For each j ∈ [ℓ] let aj > 0,bj ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose that for each K ∈ N, there exists r0(K) > 0 such that for all r ∈ (0,r0(K)), there is a partition {T( j,K,r), j ∈ [ℓ]} of A with the following two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Firstly for each fixed K ∈ N, j ∈ [ℓ], we have κ(T( j,K,r),r) = O(r−bj) as r ↓ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11) and secondly, for all K ∈ N, j ∈ [ℓ], r ∈ (0,r0(K)) and any G ⊂ A intersecting T( j,K,r) with diam(G) ≤ Kr, we have µ(Gr \\G) ≥ ajrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, almost surely, limsup n→∞ � nMd n,k(n)/k(n) � ≤ max j∈[ℓ](a−1 j ) if β = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' limsup n→∞ � nMd n,k(n)/logn � ≤ max j∈[ℓ](a−1 j ˆHβ(bj/d)) if β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 13 Later we shall use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 in the case where A is a convex polytope in Rd to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5, taking µ to be the measure with density f and taking µ∗ to be the restriction of Lebesgue measure to A (in fact, if f is bounded above then we could take µ∗ = µ instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The sets in the partition each represent a region near to a particular face ϕ ∈ Φ∗(A) (if ϕ = A the corresponding set in the partition is an interior region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this case, coefficients aj in the measure lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) depend heavily on the geometry of the determining cone near a particular face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' As a first step towards proving Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, we spell out some useful consequences of the measure doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this result (and again later) we use |·| to denote the cardinality (number of elements) of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let µ∗ be a doubling measure on the metric space A, with doubling constant c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (i) For any ε ∈ (0,1), there exists ρ(ε) ∈ N such that κ(B(x,r),εr) ≤ ρ(ε) for all x ∈ A,r ∈ (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (ii) For all r ∈ (0,1) and all D ⊂ A, we can find L ⊂ D with |L | ≤ κ(D,r/5), such that D ⊂ ∪x∈L B(x,r), and moreover the balls B(x,r/5), x ∈ L , are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' To prove (i), let x ∈ A,r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the Vitali covering lemma, we can find a set U ⊂ B(x,r) such that balls B(y,εr/5),y ∈ U are disjoint and that B(x,r) ⊂ ∪y∈U B(y,εr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set ρ(ε) := ⌈c⌈log2(15/ε)⌉ ∗ ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then by using the doubling property of µ∗ repeatedly, we have µ∗(B(y,3r)) ≤ ρ(ε)µ∗(B(y,r/5)) for all y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover B(x,2r) ⊂ B(y,3r) for all y ∈ U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also ∪y∈U B(y,εr/5) ⊂ B(x,2r) and the union is disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus |U |µ∗(B(x,2r)) ≤ ∑ y∈U µ∗(B(y,3r)) ≤ ρ(ε) ∑ y∈U µ∗(B(y,εr/5)) ≤ ρ(ε)µ∗(B(x,2r)), and therefore |U | ≤ ρ(ε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' the claim about κ(B(x,r),εr) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now we prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let L 0 ⊂ D with |L 0| = κ(D,r/5) and with B ⊂ ∪x∈L B(x,r/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the Vitali covering lemma, we can find L ⊂ L 0 such that D ⊂ ∪x∈L B(x,r) and the balls B(x,r/5),x ∈ L , are disjoint, and (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Given countable σ ⊂ A, r > 0 and k ∈ N, we say that σ is (r,k)-connected if the geometric graph G(σ,r) is k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assuming condition (B) holds, we see that σ is (r,1) connected if and only if σr/2 is a connected subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8 (Peierls argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ℓ ∈ N, a ∈ [1,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let r ∈ (0,1/a) and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let L ⊂ A with the property that |L ∩B(x,r)| ≤ ℓ for all x ∈ A, and let x0 ∈ Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then the number of (ar,1)-connected subsets of L containing x0 with cardinality n is at most cn, where c depends only on ℓ, a and c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First we claim that |L ∩ B(x,ar)| ≤ ℓρ(1/a) for all x ∈ A, where ρ(1/a) is as given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, we can cover B(x,ar) by ρ(1/a) balls of radius r, and each of these balls contains at most ℓ points of L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There is a standard algorithm (of constructing a non-decreasing sequence of lists) for counting the connected sets of Zd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' see [5, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3] for details of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The algorithm remains valid in this general setting, with the lexicographical ordering replaced by the total ordering of A (using assumption (O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This algorithm has to stop at time n (cardinality of the set), and at each step the number of possibilities for the set of the added elements is bounded by 2ℓρ(1/a) (all possible subsets of the set of points of L within distance ar from a fixed point);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' hence the number of ar-connected sets of cardinality n is at most 2ℓρ(1/a)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Preparing for a proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, we recall a condition that is equivalent to k-connectedness of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We say that non-empty sets U,W ⊂ V in a graph G with vertex set V form a k-separating pair if (i) the subgraph of G induced by U is connected, and likewise for W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (ii) no element of U is adjacent to any element of W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (iii) the number of vertices of V \\ (U ∪W) lying adjacent to U ∪W is at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We say that U is a k- separating set for G if (i) the subgraph of G induced by U is connected, and (ii) at most k vertices of V \\U lie adjacent to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The relevance of these definitions is presented in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' [5, Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1] Let G be a graph with more than k+1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then G is either (k +1)-connected, or it has k separating pair, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9, to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 it suffices to prove, for arbitrary u > maxj a−1 j ˆHβ(bj/d), the non-existence of (k(n) − 1)-separating pairs in G(Xn,rn) with rn = (ulogn/n)1/d, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Notice that, for any fixed K ∈ N, if (U,W) is a (k − 1)- separating pair, then either both U and W have diameter at least Krn, or one of them, say U, is a (k − 1)-separating set of diameter at most Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Here by the diameter of a a non-empty set U ⊂ A we mean the number diam(U) := supu,v∈U dist(u,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The goal is to prove that neither outcome is possible when n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let us first eliminate the existence of a small separating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose the assumptions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞, let u > maxj a−1 j and for n ∈ N, set rn = (uk(n)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β < ∞, let u > maxj∈[ℓ]a−1 j ˆHβ(bj/d), and for n ∈ N set rn = (u(logn)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For K ∈ N, let En(K,u) be the event that there exists a (k(n)−1)-separating set for G(Xn,rn) of diameter at most Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then, given any K ∈ N, almost surely En(K,u) occurs for only finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First assume β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The condition on u implies that uaj > β and uajH(β/(uaj)) > bj/d, for each j ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then we can and do choose β ′ > β and ε ∈ (0,1/4) such that for 15 each j ∈ [ℓ], (1−3ε)duaj > β ′ and (1−3ε)duajH � β ′ (1−3ε)duaj � > bj d +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ∈ N define k′(n) = ⌈β ′ logn⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let K ∈ N, and for r ∈ (0,r0(K)) let T( j,K,r) be as in the assumptions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For j ∈ [ℓ], we claim that κ(T( j,K,rn),εrn/5) = O(r−bj n ) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, κ(T( j,K,rn),εrn/5) ≤ κ(T( j,K,rn),rn)sup x∈A κ(B(x,rn),εrn/5) ≤ ρκ(T( j,K,rn),rn), where ρ = ρ(ε/5) is the constant in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The claim follows from the assump- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Choose n0 ∈ N such that rn < r0(k) for all n ∈ N with n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7-(i), for each j ∈ [ℓ] and n ∈ N we can find a set L j n ⊂ T( j,K,rn), with |L j n | ≤ κ(T( j,K,rn),εrn/5) = O(r−bj n ), such that T( j,K,rn) ⊂ ∪x∈L j n B(x,εrn) and that the balls B(x,rnε/5), x ∈ L j n , are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set Ln := ∪ℓ i=1L j n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='13) For n ≥ n0, j ∈ [ℓ] let T j n = {σ ⊂ Ln : diam(σ) ≤ 2Krn,σ ∩ T( j,K,rn) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that the cardinality of T j n is O(|L j n |) = O(r−bj n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, σ ∩T( j,K,rn) ̸= ∅ means σ ∩L j n ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover, as explained below, limsup n→∞ sup x∈Ln |B(x,2Krn)∩Ln| < ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='14) and diam(σ) ≤ 2Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The claim about cardinality follows from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now we show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='7-(i), for n large and for all x ∈ A, we can cover B(x,2Krn) by ρ(ε/(10K)) balls of radius rnε/5, and each of these balls contains at most ℓ points of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For n ≥ n0 and σ ⊂ Ln, set Dσ,n := σ(1−2ε)rn \\σεrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='15) Let J ∈ N with J > 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For m ∈ N, define z(m) := mJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For σ ⊂ Lz(m), define Fm(σ) = {Xz(m)(Dσ,z(m)) < k′(z(m))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now let n ∈ N and choose m = m(n) such that z(m) ≤ n < z(m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Assume z(m) ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose that En(K,u) occurs and let U be a (k(n) − 1)-separating set of G(Xn,rn) with diam(U) ≤ Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We define its ‘pixel version’ σ(U) := Lz(m(n)) ∩Uεrz(m(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 16 Since σ(U) ⊂ A, there exists j ∈ [ℓ] such that σ(U) ∩ T( j,K,rz(m(n))) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By our choice of ε, provided n is large enough we have diam(σ(U)) ≤ 2Krz(m(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore σ(U) ∈ ∪[ℓ] j=1T j z(m(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since U is (k(n) − 1)-separating for G(Xn,rn), we have Xn(Urn \\U) < k(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that Xn(Dσ(U),z(m(n))) < k(n) provided n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, by the triangle inequality σ(U)(1−2ε)rz(m(n)) ⊂ U(1−ε)rz(m(n)) ⊂ Urn (for n large), while U ⊂ σ(U)εrz(n(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus Dσ(U),z(m(n)) ⊂ Urn \\U, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also, provided n is large enough, we have k(n) ≤ k′(z(m(n))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus we have the event inclusions En(K,u) ⊂ ∪ℓ j=1 ∪σ∈T j z(m(n)) {Xn(Dσ,z(m(n))) < k(n)} ⊂ ∪ℓ j=1 ∪σ∈T j z(m(n)) Fm(n)(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='15), for any n ∈ N and σ ⊂ Ln we have Dσ,n ⊃ (σεrn)(1−3ε)rn \\σεrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12), for all large enough n and all σ ∈ ∪j∈[ℓ]T j n we have µ(Dσ,n) ≥ aj(1−3ε)drd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' A simple coupling shows that, provided m is large, we have P[∪j∈[ℓ] ∪σ∈T j z(m) Fm(σ)] = ℓ ∑ j=1 O(r−bj z(m))P[Bin(z(m),(1−3ε)dajrd z(m)) < k′(z(m))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(b) and our choice of rn and ε, provided m is large, we have P[∪j∈[ℓ] ∪σ∈T j z(m) Fm(σ)] = O(1) ℓ ∑ j=1 exp � (bj/d)logz(m)−(1−3ε)duajH � β ′ (1−3ε)duaj � logz(m) � = O(m−Jε), which is summable in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It follows from the Borel-Cantelli lemma that almost surely ∪j∈[ℓ] ∪σ∈T j z(m) Fm(σ) occurs only for finitely many m which implies that En(K,u) occurs for only finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This completes the proof of the case β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now assume β = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For the rest of the proof assume also that ε ∈ (0,1) is such that uaj(1 −ε)d > 1 for all j ∈ [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We do not have to go through the subsequence argu- ment as before because the growth of k(n) is super-logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now redefine Fn(σ) := {Xn(Dσ,n) < k(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If En(K,u) happens then we now redefine the pixel version of the separating set U as σ(U) := Ln ∩Uεrn, and enumerate the possible shapes σ of the pixel version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus we have En(K,u) ⊂ ∪ℓ j=1 ∪σ∈T j n Fn(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 17 Using estimates of |T j n |, we have P[En(K,u)] = ℓ ∑ j=1 O(r−bj n )P[Bin(n,(1−3ε)dajrd n) < k(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Noticing r−1 n = O(n1/d), and applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1-(b) leads to P[En(K,u)] = O(nbj/d) ℓ ∑ j=1 exp � −(1−3ε)dajuk(n)H � k(n) (1−3ε)dajuk(n) �� which is summable in n, and the claim follows by the Borel-Cantelli lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The following lemma eliminates the existence of a (k(n) − 1)-separating pair with both diameters larger than Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let the assumptions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞, let u > maxj a−1 j and for n ∈ N, set rn = (uk(n)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β < ∞, let u > maxj∈[ℓ]a−1 j ˆHβ(bj/d), and for n ∈ N set rn = (u(logn)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For K ∈ N let Hn(K,u) denote the event that there exists a (k(n)−1)-separating pair (U,W) in G(Xn,rn) such that min(diam(U),diam(W)) ≥ Krn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then there exists K1 ∈ N such that almost surely Hn(K1,u) occurs for only finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose Hn(K,u) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then Urn/2 and Wrn/2 are disjoint and connected in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' One of the components of A \\Urn/2 contains W, denoted by W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set U′ = A \\W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then U ⊂ U′, W ⊂ W ′ and A = W ′ ∪U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ∂WU := W ′ ∩U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then ∂WU is connected by the unicoherence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover, any continuous path in A connecting U and W must pass through ∂WU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall δ1 and K0 in the assumption (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim (and show in the next few para- graphs) that diam(∂WU) ≥ 1 2K0 +2 min(δ1/3,diam(W)/3,diam(U)/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16) Suppose the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Setting b = diam(∂WU), we can find x ∈ A such that ∂WU ⊂ B(x,b), and we can find X ∈ U \\ B(x,b),Y ∈ W \\ B(x,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since b < δ1/3, the number of compo- nents of A \\ B(x,b) is at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There have to be two components because otherwise X and Y can be connected by a path in A disjoint from ∂U, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose that X lies in the component of A \\ B(x,b) having diameter at most K0b, denoted by QX, and Y lies in the other component, denoted by QY (if it is the other way round we reverse the roles of X and Y in the rest of this argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that there exists X′ ∈ U such that dist(X,X′) > (2K0 + 2)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If not, then for any X1,X2 ∈ U, 18 we have by triangle inequality that dist(X1,X2) ≤ 2(2K0 +2)b, yielding that diam(U) ≤ 2(2K0 +2)b, contradicting diam(U) > 3(2K0 +2)b by the negation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that dist(X,B(x,b)) ≤ K0b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' To see this, using the assumed connectivity of A, take a continuous path in A from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The first exit point of this path from QX lies in B(x,b) (else it would not be an exit point from QX) but also in the closure of QX, and hence in B(X,K0b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This yields the latest claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We show that X′ and Y have to be in the same component of A \\ B(x,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' To this end, notice first that X′ cannot be in QX, because for any z ∈ QX, dist(X,z) ≤ K0b < (2K0 +2)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Secondly, X′ cannot be in B(x,b) either because for any z ∈ B(x,b), we have dist(z,X) ≤ dist(X,B(x,b))+2b ≤ (K0 +2)b < (2K0 +2)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore, X′ has to be in QY, and we reach again to a contradiction that X′ and Y can be connected by a path in A disjoint from ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We have thus proved (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let ε ∈ (0,1/9) and let Ln be as defined at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='13) (the ε does not have to be the same as it was there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall that Ln has the covering property that for every x ∈ A we have Ln ∩B(x,rnε) ̸= ∅ and the spacing property that |Ln ∩B(x,rnε/3)| ≤ ℓ for all such x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Define DWU = {x ∈ Ln : B(x,εrn) ∩ ∂WU ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then by the covering property of Ln, (DWU)εrn is connected and covers ∂WU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' That is, DWU, as a subset of the metric space A, is (2εrn,1)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16) and the occurrence of Hn(K,u), we have 2εrn|DWU| ≥ diam(∂WU) ≥ min(δ1/3,Krn/3)/(2K0 +2) Therefore, provided n is large, we have |DWU| ≥ K/(6ε(2K0 +2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that there is a constant c ∈ (0,∞), independent of n, such that for all q ∈ N, if |DWU| = q then DWU can take at most O(r−max(bj) n cq) possible ’shapes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, given x0 ∈ Ln, set Un,q(x0) := {σ ⊂ Ln : |σ| = q,σ is (2εrn,1)-connected,x0 ∈ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then DWU ∈ ∪j∈[ℓ] ∪x0∈T(j,K,rn)∩Ln Un,q(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='8, we have |Un,q(x0)| ≤ cq for some finite constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10 that |T( j,K,rn)∩Ln| = O(r−bj n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For all n ∈ N, if x ∈ ∂WU then dist(x,U) = rn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore by the triangle inequal- ity, (DWU)εrn/5 ⊂ Ur, while U ∩ (DWU)εrn/5 = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' hence Xn ∩ (DWU)εrn/5 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' This, together with the the union bound, yields that P[Hn(K,u)] ≤ ∑ q≥K/(6ε(2K0+2))∑ σ P[Xn(σεrn/5) < k(n)], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='17) 19 where the second sum is over all possible shapes σ ⊂ Ln of cardinality q that are (2εrn,1)- connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since every point in A is covered at most ℓ times, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) (with G = {z}), there exists ε1 ∈ (0,1) such that µ(σεrn/5) ≥ (1/ℓ) ∑ z∈σ µ(B(z,εrn/5)) ≥ (q/ℓ)ε1(εrn/5)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set ε2 := (ε1/ℓ)(ε/5)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='17) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1(b), provided n is large, P[Hn(K,u)] ≤ ∑ q≥K/(6ε(2K0+2)) O(r−max(bj) n cq)P[Bin(n,ε2qrd n) < (β +1)logn] = O(1) ∑ q≥K/(6ε(2K0+2)) cq exp � (max(bj)/d)logn−ε2quH �β +1 ε2qu � logn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the continuity of H(·) and the fact that H(0) = 1, there exists q0 > 16/(ε2u) such that for any q > q0, we have H �β+1 qε2u � > 1/2 and quε2 > 4max(bj)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Choosing K = 6ε(2K0 + 2)q0 so that q ≥ q0 in the sum, we see that the exponent of the exponential is bounded above by (max(bj)/d)logn−qε2(u/2)logn ≤ −(quε2/4)logn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore, we have for n large that P[Hn(K,u)] = O(1) ∑ q≥q0 cq exp(−qu(ε2/4)logn) = O(1) ∑ q≥q0 exp(−qu(ε2/8)logn) = O(exp(−q0u(ε2/8)logn)) = O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The result in this case follows by applying the Borel-Cantelli lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='17) and the estimates of |∪j ∪x0Un,q(x0)| as previously, we have P[Hn(K,u)] ≤ ∑ q≥K/(6ε(2K0+2)) O(r−max(bj) n cq)P[Bin(n,(q/ℓ)ε1(εrn/5)d) < k(n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We have r−max(bj) n = O(nmax(bj)/d), and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1-(b), P[Hn(K,u)] ≤ ∑ q≥K/(6ε(2K0+2)) cq exp � (max(bj)/d)logn−qε2uk(n)H( k(n) qε2uk(n)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' As before, we can choose K = K1 (large) so that the H(·) term in every summand is bounded from below by 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the super-logarithmic growth of k(n), we conclude that P[Hn(K,u)] ≤ n−2 provided n is large, so that the Borel-Cantelli lemma gives the result in this case too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 20 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞ then let u > maxj∈[ℓ](a−1 j ) and set r(n) := u(k(n)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β < ∞ then let u > maxj∈[ℓ](a−1 j ˆHβ(bj/d)) and set rn := (u(logn)/n)1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11, there exists K ∈ N such that almost surely, En(K,u) ∪Hn(K,u) occurs for at most finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9, if Mn,k > rn then En(K,u) ∪ Hn(K,u) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore Mn,k(n) ≤ rn for all large enough n, almost surely, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='4 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5 In this subsection we go back to the mathematical framework in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' that is, we make the assumptions in the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In particular we return to assum- ing A is a convex polytope in Rd with d ≥ 2, and the probability measure µ has a density f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We shall check the conditions required in order to apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' To check these conditions, we shall use the following lemma and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' [8, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12] Suppose ϕ,ϕ′ are faces of A with D(ϕ) > 0 and D(ϕ′) = d −1, and with ϕ \\ϕ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then ϕo ∩ϕ′ = ∅ and K(ϕ,ϕ′) < ∞, where we set K(ϕ,ϕ′) := sup x∈ϕo dist(x,∂ϕ) dist(x,ϕ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='18) Now define K(A) := max{K(ϕ,ϕ′) : ϕ,ϕ′ ∈ Φ(A),D(ϕ) > 0,D(ϕ′) = d −1,ϕ \\ϕ′ ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='19) Then K(A) < ∞ since A is a finite polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For j ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=',d} let Φj(A) denote the collection of j-dimensional faces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For any D ⊂ A and r > 0 set Dr = {x ∈ A : B(x,r)∩D ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The restriction of Lebesgue measure to A has the doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' More- over the conditions (O), (B), (T) and (G) are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' First we verify the doubling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3, there exists b > 0 such that infx∈A,r∈(0,b]r−dVol(B(x,r) ∩ A) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since Vol(B(x,2r) ∩ A) is at most 2dθdrd for r ≤ b, and is at most Vol(A) for all r, the doubling property follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Points of A can be ordered by using the lexicographic ordering inherited from Rd, thus (O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since A is convex, for all x ∈ A and r > 0 the set B(x,r)∩A is convex and hence connected, implying (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' All convex polytopes are simply connected, and therefore uni- coherent [5, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1], hence (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Condition (G) follows immediately from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='14, which we prove below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let A be a convex finite polytope in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let N(·) denote the number of components of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There exists δ1 > 0 such that for any x ∈ A any r ∈ (0,δ1), we have N(A\\B(x,r)) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover, in the case that N(A\\B(x,r)) = 2, the diameter of the smaller component is at most cr, where c is a constant depending only on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 21 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Write B for B(x,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Our first observation is that if y ∈ A \\ B, then there is at least one vertex v ∈ Φ0(A) such that the line segment [y,v] is contained in A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, if this failed then for each v ∈ Φ0(A) there would exist a point u(v) ∈ [y,v]∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' But then since A is convex, y would lie in the convex hull of {v : v ∈ Φ0(A)}, and therefore also in the convex hull of {u(v) : v ∈ Φ0(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, there exist αv ≥ 0 with ∑v∈Φ0(A) αv = 1 such that y = ∑v∈Φ0(A) αvv, and there exists βv ∈ [0,1] such that u(v) = βvy + (1 − βv)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Substituting v by u(v) and rearranging terms shows that y = ∑v α′ vu(v) with some nonnegative α′ v and ∑vα′ v = 1, thus the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' But then since B is convex we would have y ∈ B, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We refer to the one-dimensional faces ϕ ∈ Φ1(A) as edges of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Our second observa- tion is that if the number of edges of A that intersect B is at most 1, then A\\B is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, in this case, for any distinct v,v′ ∈ Φ0(A) there is a path along edges of A from v to v′ that avoids B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For example, if v,v′ lie in the same two-dimensional face ϕ of A then since B intersects at most one edge of the polygon ϕ, there is a path from v to v′ along the edges of ϕ avoiding B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore all v ∈ Φ0(A) lie in the same component of A \\ B, so using the first observation we deduce that A\\B is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Recall the definition of K(A) at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Our third observation is that if dist(v,B) ≥ 3rK(A) for all v ∈ Φ0(A) then A \\ B is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, suppose dist(v,B) ≥ 3rK(A) for all v ∈ Φ0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose ϕ,ϕ′ are distinct edges of A with B∩ϕ ̸= ∅, and pick y ∈ B∩ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then dist(y,∂ϕ) ≥ 3rK(A) so that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='18), dist(y,ϕ′) ≥ 3rK(A)/K(ϕ,ϕ′) ≥ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence by the triangle inequality dist(B,ϕ′) ≥ 3r−2r = r, so that B∩ϕ′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence B intersects at most one edge of A, and by our second observation A\\B is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose dist(v,B) ≤ 3rK(A) for some v ∈ Φ0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Provided r is small enough, this cannot happen for more than one v ∈ Φ0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If u,u′ ∈ Φ0(A) \\ {v}, then v /∈ [u,u′] so dist(v,[u,u′]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore provided r is small enough, [u,u′] ⊂ A\\B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus provided r is small enough, all vertices u ∈ Φ0(A)\\{v} lie in the same component of A\\B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If also v lies in this component, then (by our first observation) A\\B is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Thus A\\B is disconnected only if v lies in a different component of A\\B than all the other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In that case, for y ∈ A\\B, if [y,v] ⊂ A\\B then y is in the same component as v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' otherwise (by our first observation) y lies in the same component as all of the other vertices, and thus A\\B has exactly two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If A \\ B has two components, and y ∈ A \\ B with ∥y − v∥ > (3K(A) + 2)r, then we claim [y,v] ∩B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, for each u ∈ Φ0(A) \\ {v} the ray from v in the direction of u passes through B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' But then by an argument based on the convexity of both A and B, the ray from v in the direction of y must also pass through B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Since dist(v,B) ≤ 3rK(A) and diam(B) = 2r, this ray must pass through B at a distance at most (3K(A) +2)r from v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' before it reaches y, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore y lies in the component of A \\ B that does not contain v, and thus the component containing v has diameter at most (3K(A)+2)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 22 To apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, we need to define a partition of A for each small r > 0, then estimate the corresponding covering numbers and µ-measures in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Taking into account a variety of boundary effects near ∂A, one should consider sepa- rately regions near different faces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It is however not trivial to construct this partition in such a way that we can obtain tight µ-measure estimates in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The matter is com- plicated by the fact that the set G in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) that intersects a region near ϕ is potentially close to a lower dimensional face lying inside ∂ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We can avoid the boundary complica- tions by constructing inductively from regions near to the highest dimensional face to the lowest, with increasing ’thickness’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The partition made of T(ϕ,r)’s defined below and the left-over interior region is defined for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let (Kj) j∈N be an increasing sequence with K1 = 1, and with Kj+1 > (2K(A)+1)Kj for each j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For instance, we could take Kj = (2K(A)+2) j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Now for each r > 0 and ϕ ∈ Φ(A), define the set T(ϕ,r) := ϕrKd−D(ϕ) \\∪ϕ′∈Φ(A):ϕ′⊊ϕ(ϕ′)rKd−D(ϕ′), where the T stands for ‘territory’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also define T(A,r) := A \\ ∪ϕ∈Φ(A)ϕrKd−D(ϕ) For each ϕ ∈ Φ∗(A), we have T(ϕ,r) ̸= ∅ for all r sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Hence, there exists r0 > 0 such that for all ϕ and all r < r0, T(ϕ,r) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover, territories of distinct faces are disjoint, as we show in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' There exists r0 > 0 such that for all r ∈ (0,r0), and any distinct ϕ,ϕ′ ∈ Φ∗(A), it holds that T(ϕ,r) ∩T(ϕ′,r) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover, if ϕ,ϕ′ ∈ Φ(A) with ϕ \\ ϕ′ ̸= ∅, and y ∈ T(ϕ,r), then B(y,r) does not intersect ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We can (and do) assume without loss of generality that ϕ \\ϕ′ ̸= ∅ and ϕ′ \\ϕ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, if ϕ ⊂ ϕ′, then by construction T(ϕ′,r)∩T(ϕ,r) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If ϕ is a vertex, then dist(ϕ,ϕ′) > 0 so that T(ϕ,r)∩T(ϕ′,r) = ∅ for all r small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' So it suffices to consider the case where D(ϕ) > 0 and D(ϕ′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let j := d −D(ϕ) and j′ := d −D(ϕ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We can and do assume j′ ≤ j ≤ d −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If there exists x ∈ T(ϕ,r)∩T(ϕ′,r), then we can find z ∈ ϕ,z′ ∈ ϕ′ such that ∥x−z∥ ≤ rKj and ∥x−z′∥ ≤ rKj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore dist(z,ϕ′) ≤ r(Kj +Kj′) ≤ 2rKj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' On the other hand, since x ∈ T(ϕ,r), dist(x,∂ϕ) ≥ rKj+1, and so by the triangle in- equality, rKj+1 − rKj ≤ dist(z,∂ϕ) ≤ K(A)dist(z,ϕ′), where the last inequality comes from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Combining the estimates leads to Kj+1 ≤ (2K(A)+1)Kj, which is a contra- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The first claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moving to the second claim, let ϕ,ϕ′ ∈ Φ(A) with ϕ \\ϕ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose y ∈ ϕ′ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set ˜Φ := {ψ ∈ Φ(A) : ψ ⊊ ϕ′,y ∈ ψKD−d(ψ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If ˜Φ = ∅ then y ∈ T(ϕ′,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Otherwise, choose ψ ∈ ˜Φ of minimal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then y ∈ T(ψ,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Either way, y /∈ T(ϕ,r) by the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore T(ϕ,r)∩ϕ′ r = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 23 As a last ingredient for applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, for each J > 1 and r ∈ (0,1), we con- struct a partition of A and show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) for all G with diameter at most Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The coefficients aj depend on the location of G in relation to faces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let J ∈ N and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then the following hold: (i) For each ϕ ∈ Φ(A) we have κ(T(ϕ,2Jr),r) = O(r−D(ϕ)) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Moreover we have κ(A\\∪ϕ∈Φ(A)T(ϕ,2Jr),r) = O(r−d) as r ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (ii) For all small r > 0 and any G ⊂ A with diam(G) ≤ Jr, if it intersects T(ϕ,2Jr) for some ϕ ∈ Φ∗(A), then µ(Gr \\G) ≥ (1−ε) fϕρϕrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Item (i) follows by the definition of T(ϕ,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, ϕ is contained in a bounded region within a D(ϕ)-dimensional affine space, and therefore can be covered by O(r−D(ϕ)) balls of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If we then take balls of radius r(1 +2JKd−D(ϕ)) with the same centres, they will cover T(ϕ,2Jr), and one can then cover each of the larger balls with a fixed number of balls of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For (ii), let G ⊂ A with diam(G) ≤ Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose first that G∩T(ϕ,2Jr) ̸= ∅ for some ϕ ∈ Φ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Let x0 ∈ G ∩T(ϕ,2Jr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Then Gr ⊂ B(x0,2Jr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='15, we see that B(x0,2Jr) does not intersect any ϕ′ ∈ Φ(A) with ϕ \\ϕ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It follows that B(x0,2Jr)∩A = B(x0,2Jr)∩(z0 +Kϕ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='21) where Kϕ is the cone determined by ϕ and z0 is the point of ϕ closest to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Set D(x,r) := B(x,r) ∩ (x + Kϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We claim that for any x ∈ G, we have D(x,r) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Indeed, given y ∈ D(x,r), we can write y = z0 + (x − z0) + (y − x) =: z0 + θ1 + θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Here θ1,θ2 ∈ Kϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By convexity and scale invariance of Kϕ, we have θ1 + θ2 ∈ Kϕ so y ∈ z0 + Kϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also ∥y − x0∥ ≤ ∥y − x∥ + ∥x − x0∥ ≤ 2Jr, and hence y ∈ A by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='21), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' It follows that (with ⊕ denoting Minkowski addition) µ(Gr \\G) ≥ µ((G⊕D(o,r))\\G) ≥ Vol((G⊕D(o,r))\\G) inf x∈G⊕D(o,r) f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By the Brunn-Minkowski inequality [5, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='3], we have Vol(G⊕D(o,r)) ≥ Vol(G)+ Vol(D(o,r)) = Vol(G)+ρϕrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' The claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='20) follows by the continuity of f on ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' As for the case ϕ = A, suppose now that G∩T(A,2Jr) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Taking x ∈ G∩T(A,2Jr) we have dist(x,∂A) ≥ 2Jr, and hence dist(G,∂A) ≥ 2Jr −Jr = Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Therefore Gr ⊂ A, so by the Brunn-Minkowski inequality µ(Gr \\G) ≥ f0Vol((G⊕B(o,r))\\G) ≥ f0θdrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' In this case fϕ = f0 and ρϕ = θd, and the claim (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='20) follows in this case too, completing the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' 24 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='9), and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='1, it suffices to prove the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' We shall do this by applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 in the situation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='13, the restriction to A of Lebesgue measure has the doubling property, and conditions (O), (B), (T) and (G) are satisfied To apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, we need to define (for each K ∈ N and each r ∈ (0,r0(K))) a finite partition {T( j,K,r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For this we take the sets T(ϕ,2Kr),ϕ ∈ Φ∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='15, and the definition of T(A,r), for each K ∈ N there exists r0(K) > 0 such that for r ∈ (0,r0(K)) the sets T(ϕ,2Kr), ϕ ∈ Φ∗(A), do indeed partition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' For each ϕ ∈ Φ∗(A), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16-(i) we have the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='11) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, where the constant denoted bj there is equal to D(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Also, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='16-(ii) we have the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='12) in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6, where the constant denoted aj there is equal to (1−ε) fϕρϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' Suppose β < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' By applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 in the manner described above we see that for ε > 0, we have limsup n→∞ n(Mn,k(n))d/logn ≤ max ϕ∈Φ∗(A) � ˆHβ(D(ϕ)/d) (1−ε) fϕρϕ � , and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content=' If β = ∞, using corresponding part of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE0T4oBgHgl3EQfnAFi/content/2301.02506v1.pdf'} +page_content='6 gives the result in this case too.' metadata={'source': 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Rastegar∗ +Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran and +Molecular Simulation Laboratory (MSL), Azarbaijan Shahid Madani University, Tabriz, Iran +A. Phirouznia +Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran and +Condensed Matter Computational Research Lab., +Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran +(Dated: January 3, 2023) +Edelstein and spin Hall effect response functions for a two-dimensional (2D) system of pseudospin- +1 particles is investigated. +These two response functions denoted by σSH and σEE have been +analyzed in a pseudospin-1 particle system in the presence of the Rashba-type spin-orbit interaction +using the Kubo-Streda technique and vertex corrections with non-magnetic impurities. Then, for a +given range of the Rashba coupling, response functions of the spin Hall effect (SHE) and Edelstein +effect (EE) have been estimated at various energy gaps and Fermi energies. Results indicate that +in this type of the two-dimensional materials, SHE and EE are essentially induced by the vertex +corrections and without considering these corrections the amount of these effects are really negligible. +It has also been realized that SHE and EE conductivities can be modulated by the Rashba coupling +strength at low and intermediate level of this spin-orbit type interaction. +I. +INTRODUCTION +Pseudospin-1 particle compounds have recently at- +tracted a lot of interest in condensed matter physics. +Low-energy gapless pseudospin-1 fermions have an en- +ergy spectrum that is linear in momentum, except for a +flat band at zero energy, just like electrons in graphene. +One of the most prevalent two-dimensional realizations +of pseudospin-1 fermions (2D) is the T3 lattice. Atoms +at the centers and vertices of the hexagonal lattice could +be used to create a simple tight-binding model for this +type of materials. Due to the placement of three sites per +unit cell, the electron states in this model are given by +three-component fermions with three bands of the energy +spectrum. Such a system is shown schematically in Fig. +1. +The spin Hall effect (SHE) has received much attention +because of its prospective applications in spintronic tech- +nologies. Hirsch [1] explored the spin Hall effect (SHE), +which Dyakonov and Perel [2, 3] studied in 1971. The +spin Hall effect (SHE) and the inverse spin Hall effect +(ISHE) [4], a group of transport phenomena [5, 6], greatly +aided the development of experimental spintronic devices +[1, 3, 7]. One may list examples of its use in memory, +logic, and sensing devices. Kane and Mele discovered the +quantum spin Hall effect in graphene in 2005 [8]. The +low-energy electronic structure of a single layer graphene +with spin-orbit interactions was investigated. They found +that graphene changes from its ideal two-dimensional +semimetallic state at low temperatures to a quantum spin +Hall insulator due to the symmetry-enabled spin-orbit +∗ Corresponding author’s Email: s.rastegar@azaruniv.ac.ir +FIG. 1. +A schematic representation of the T3 lattice for +pseudospin-1 fermion sample in the xy plane. +Translation +vectors are V1 = (3/2; − +√ +3/2)a and V2 = (3/2; +√ +3/2)a, +with lattice constant a. Sites A and B that showed by solid +circles and squares, making a hexagonal lattice, respectively. +whereas, solid circles mark the hub sites C making a triangu- +lar lattice. +potential. +After initial findings about these phenomena in semi- +conductors and metals, it was known that the spin Hall +effect might offer a new strategy for the interconversion +of spin and charge indications [5, 6, 9, 10]. SHE, which +results from strong spin-orbit coupling (SOC), may pave +the way for generating pure spin current from charge cur- +arXiv:2301.00550v1 [cond-mat.mes-hall] 2 Jan 2023 + +B2 +FIG. 2. +Schematic representation of the SHE in a two- +dimensional material. Both the charge and spin Hall currents +are driven by an external electrical field. +rent. In the spin Hall regime, a spin current is generated +perpendicular to an applied electric field. Additionally, +unlike SHE within the inverse SHE regime, spin currents +can be converted into electric signals [11, 12]. +Meanwhile, spin Seebeck effect [13–15] demonstrates +that the temperature gradient can generate spin current +which confirms that the spin-orbit interactions are also +crucial in the field of ”spin caloritronics.” +The strength of this effect is given by spin Hall angle +γ that measures the material’s effectiveness according to +relation γ = J⊥/J∥, which is defined as the ratio be- +tween steady-state longitudinal charge-current, J∥ and z- +polarized transverse spin currents given by J⊥ as shown +in Fig. 2. The material’s SHE efficiency is determined by +the spin Hall angle. The charge efficiency in the present +spin transition can be determined using materials with a +large spin Hall angle (SHA). +The anomalous effect is another well-known phe- +nomenon in this field. The nonmagnetic spin Hall effect +[16] and the well-known anomalous Hall effect (AHE) [16] +in ferromagnetic metals are closely linked. The magnetic +field Hz and magnetization Mz both affect the Hall resis- +tance ρH, through the relation ρH = R0Hz + RsMz[16]. +Where, the normal and anomalous Hall resistances are R0 +and Rs, respectively [17, 18]. The anomalous Hall effect +has become one of the most urgent issues in solid-state +physics because it is challenging to forecast the carrier +density in ferromagnets [19–22]. +The Edelstein effect (EE) [23] and the inverse Edel- +stein effects (IEE) [24, 25] are two additional effects that +have recently attracted a lot of interest. In the regime +of EE a continuous non-equilibrium spin polarization Sy +is produced by a constant current Jx driven by an elec- +tric field Ex. It was also demonstrated that the EE and +the SHE are connected [23]. The inverse Edelstein effect +is the process underlying the spin-to-charge transforma- +tion. The Edelstein effect is a promising phenomena for +spintronics applications since it can produce spin polar- +ization in nonmagnetic materials just electrically. There- +fore, it is highly desirable to find materials that are highly +efficient at ”converting” the electric current into spin po- +larization. Meanwhile, it should be noted that the spin +and charge quantum numbers can also be converted to +pseudospin index in graphene-like materials [26, 27]. +Edelstein effect of enormous topological-insulator- +graphene heterostructures has been investigated by +Rodriguez-Vega et al. +[28], where they show that +the nonequilibrium uniform spin-density accumulation +caused by a charge current in magnetic TI-graphene het- +erostructures can be 10 − 100 times larger than in TIs +alone, resulting in a massive Edelstein effect. +Valley Edelstein effect (VEE) has been discovered +in the monolayer transition-metal dichalcogenides by +Taguchi et al [29]. They found that in gated monolayer +transition-metal dichalcogenides (MTMDs), Berry curva- +tures resulting from coexisting Rashba and Ising SOCs +combined with traditional Edelstein effects lead to valley- +contrasting spin polarization parallel to the applied elec- +tric field. +The ability to introduce SOC with multiple symme- +tries [30, 31] and varying spatial extent is a fascinating +property of two-dimensional materials. Recent theoreti- +cal investigations have clarified the significant role that +the SOC symmetry plays in the resonant scattering do- +main [31, 32]. +There is a microscopic approach for the spin Hall effect +in amorphous materials using vertex corrections. This +approach has been employed in amorphous graphene +[33] where the nonperturbative quantum diagrammatic +method which has been applied in this approach can also +be used to obtain the response function in materials re- +sembling graphene [33]. +For non-magnetic impurities, the vertex coefficient +vanishes when the Fermi level is in the upper band (both +bands are occupied). This problem leads to the disap- +pearance of the spin-Hall effect in the thermodynamic +limit for any degree of disorder [34]. +It has also been +shown that the bare vertex is a good approximation and +that the bare bubble diagram is adequate for calculating +the spin-Hall conductivity in the presence of magnetic +impurities [35]. +The SHA rises linearly with the SOC impurity den- +sity in disordered samples where other variables restrict +charge mobility [36]. The spin Hall current in graphene +identically vanishes when the spin-orbit interaction (SOI) +is absent even in the presence of the magnetic impurities +[37]. +Therefore, it can be inferred that the SOI is of +crucial importance for SHE and it cannot be generated +merely by the spin-flip scatterings [37]. It has been shown +that the spin Hall conductivity (SHC) are generally finite +in the presence of SOI and magnetic impurities under a +zero external magnetic field [37]. +When the magnetic impurities are combined with +the SOI, the magnetic scattering centers act as spin- +dependent barriers, causing a charge imbalance at the +boundaries. +Changing the chemical potential close to +the gap should also reveal charge and spin Hall effects +in graphene. Several investigations have been performed +on the charge and spin Hall phenomena in magnetically +impure graphene. +Using the Kubo formula, analytical +formulations of the charge and spin Hall conductivities +has been developed theoretically. [37]. + +1 +Jc +X3 +The SHE and EE can be described by the Edelstein +conductivity (EC) and the SHC that are defined through +the following relations Jz +y = σSHE +xy +Ex, sy = σEE +xy Ex re- +spectively. In this study spin Hall and Edelstein conduc- +tivities for pseudospin-1 particles with vertex corrections +have been investigated. Using the Kubo-Streda method +and vertex corrections in the presence of non-magnetic +impurities, these two response functions, σSHE and σEE, +have been evaluated in a pseudospin-1 particle system +with Rashba-type spin-orbit interaction. Numerical cal- +culations provide the normalized operators within a self- +consistent approach. Results show that the bare Kubo +response function of both EE and SHE are absolutely +small and the main contribution comes the vertex cor- +rections. Therefore, in these type of materials we cannot +expect high SHA. Meanwhile, EC and SHC could reveal +the different physics behind these novel materials. +II. +METHODOLOGY +Within a low energy approximation, effective Hamil- +tonian of massive Dirac-like pseudospin-1 particles in +the presence of the Rashba spin-orbit interaction can be +stated as follows: +H = ℏvF (Sxkx ⊗ Iσ + Syky ⊗ Iσ) ++ λR(kxIδ ⊗ σy − kyIδ ⊗ σx) + mSz ⊗ Iσ +(1) +where vF is the pseudospin-1 particle’s Fermi velocity, λR +is the Rashba coefficient and S = (Sx, Sy, Sz) is a vector +of matrices with pseudospin components [38]: +Sx = +1 +√ +2 +� +� +0 1 0 +1 0 1 +0 1 0 +� +� , +Sy = +1 +√ +2 +� +� +0 −i +0 +i +0 +−i +0 +i +0 +� +� , +Sz = +� +� +1 0 +0 +0 0 +0 +0 0 −1 +� +� . +The three matrices represent all pseudospin-1 particles +that fulfill the angular momentum commutation relations +[Sl, Sm] = iϵlmnSn with three eigenvalues, s = ±1, 0, +where ϵlmn is the Levi-Civita symbol. +However, they +do not form a Clifford algebra, as opposed to Pauli +matrices; that is, {Sn, Sm} ̸= 2δn,mI3×3. +These three +dimensional matrices represent the contribution of three +A, B and C sublattices. +After the Fourier transformation +H = +� +k +Ψ† +kH(k)Ψk, +(2) +where Ψ† +k += +(cA,k,↑, cB,k,↑, cC,k,↑, cA,k,↓, cB,k,↓, cC,k,↓) +and c† +ikσ denotes the creation operator of electron on the +i sublattice with spin of σ and wave-number of k. +Then in presence of Rashba coupling, the Hamilto- +nian is given by +H = ℏvF +√ +2 +� +� +0 +k− +0 +k+ +0 +k− +0 +k+ +0 +� +� ⊗ Iσ ++ λR +� +� +kx +0 +0 +0 +kx +0 +0 +0 +kx +� +� ⊗ σy − λR +� +� +ky +0 +0 +0 +ky +0 +0 +0 +ky +� +� ⊗ σx ++ m +� +� +1 0 +0 +0 0 +0 +0 0 −1 +� +� ⊗ Iσ +(3) +The energy spectrum can be obtained as +E1,2(k) = ±|k|λR +E3,4(k) = ± +� +m′(k) + k2λ2 +R − 2 +� +k2λ2 +Rm′(k) +E5,6(k) = ± +� +m′(k) + k2λ2 +R + 2 +� +k2λ2 +Rm′(k), +(4) +where m′(k) = m2 + k2ℏ2v2 +F . +Therefore, electrons in the T3 lattice as seen in Fig. 1, +can behave as massless Dirac fermions with pseudospin +S = 1, rather than S = 1/2 for those trapped in the +hexagonal lattice. Each unit cell in this T3 lattice has +three inequivalent sublattices. Two lattice sites, A and +B, are triply coordinated, while site C connects six of its +closest neighbors. +III. +THE KUBO-STREDA APPROACH +The density operators for charge and spin-current can +be defined as follows [39–41]: +J = −eΨ†(x)vΨ(x), +(5) +J α +β = −eΨ†(x)1 +2{sα, vβ}Ψ(x), +(6) +where {., .} represents the anticommutator, v represents +the velocity operator, −e represents the electron,s charge, +and sz ≡ s3 represents the diagonal Pauli matrix with +eigenvalues ±1. Using the Kubo-Streda method the con- +ductivity tensor in the presence of SOC [42] is: +σz +ij = +ℏ +2πΩTr +� +Ji(GR−GA)JiGA−Jj(GR−GA)JiGR� +dis, +(7) +GR(A) denotes the retarded (advanced) Green,s function, +which is defined as follows: +GR(A) = +1 +ϵ − H0 − V ± i0+ , +(8) + +4 +Where H stands for the total Hamiltonian, H = H0 +V , +with H0 for the non-perturbed Hamiltonian, and V repre- +sents the perturbation. Cartesian elements of the charge- +and spin-current density operators’ are denoted by Ji and +Ji respectively. +� +... +� +dis indicates the configurational dis- +order average compared to the conventional version. +� +O +� +dis = +lim +N,Ω→∞ +� N +� +i=1 +� +Ω +d2xi +Ω +� +O(x1, ..., xN) +���� N +Ω =n +, (9) +Where i represents impurities, Ω represents the sample +area, and Tr represents the trace over the entire Hilbert +space. The conductivity of SH is equal to +σSHE +xy += +1 +πΩTr[ +� +GRJxGA� +disJy] +(10) +and also EE conductivity is given as follows +σEE +xy = +1 +πΩTr[ +� +GRSyGA� +disJx] +(11) +For short-range impurities, calculations were made using +the weak-scattering regime. +Within the Kubo formal- +ism, the main focus has been devoted on analyzing the +electric dc Edelstein and Hall conductivities in a typi- +cal pseudospin-1 particle system with nonmagnetic dis- +orders. +The linear response Kubo formalism has been used +to explain the fully quantum mechanical response func- +tions by taking the vertex correction into account. The +response functions were obtained at zero temperature, +which may be specified using Green’s time-ordered func- +tions. Because the calculations were performed in the dc +regime, it can be achieved by ω −→ 0 limit at the end of +the calculations. In the meantime, this method has been +necessary for all ac-based calculations in this approach. +III.0.1. +Disordered Vertex Correction +A randomly distributed disorder potential has been as- +sumed as follows +ˆV (r) = V0 +N +� +i=1 +δ(r − Ri), +(12) +weak, and short-range Gaussian correlation can be con- +sidered by imposing following relations +� ˆV (r) +� +imp = 0, +� ˆV (r1) ˆV (r2) +� +imp = nV 2 +0 δ(r1 − r2), +(13) +The impurity concentration and scattering potential are +indicated by n = 5 × 1010cm−2 and V0 = 0.1eV , respec- +tively. This method has been used extensively to study +the properties of disordered systems. The Kubo relation +for dc electric current conductivity is: +σxν = +ℏ +2πL2 Tr +�ˆjx ˆGRˆjν ˆGA� +imp, +(14) +where L2 represents the system area and ˆGR(A)(ϵ) = +(ϵ ± i0 − ˆH − ˆV )−1 refers to the retarded (advanced) +Green,s function in the Pauli space. The current operator +is expressed as ˆj = −e(−i/ℏ)[ˆr, ˆH] = −evF ˆz × ˆσ. Mean- +while, +� +... +� +imp shows an ensemble average over random +nonmagnetic impurity configurations. The conductivity +then is given as +σxν ≈ +ℏ +2πL2 Tr[ˆjx +� ˆGR�ˆjν +� ˆGA� +] + Vertex correction +(15) +≡ +ℏ +2πL2 Tr[ˆjx +� ˆGR�ˆ˜jν +� ˆGA� +], +Where +� ˆGR(A)� +is the averaged Green,s function and ˆjν +is the bare charge current density operators. The ladder +diagram correction to the bare charge current density op- +erator can be evaluated using the following relation[43], +ˆ˜jν(εF ) ≡ ˆjν + δˆ˜jν(εF ), +(16) +where ˜jν is the normalized charge current operator within +the ladder vertex correction and δˆ˜jν is the charge cur- +rent vertex correction. The averaged Green function is +expressed using the Dyson equation and the Born ap- +proximation, as shown in Fig. 3. +� ˆGR(A)� += +� +(z − ˆH − ˆV )−1� +imp +(17) += ˆGR(A) +0 ++ ˆGR(A) +0 +ˆΣR(A)� ˆGR(A)� +, +with z = ϵ ± i0 in which ϵ is small positive. Here, the +solution may then be written as: +� ˆG +� += (( ˆGR(A) +0 +)−1 − ˆΣR(A))−1, +(18) +in which the self-energy in the Born approximation is +ˆΣR(A) = +� ˆV +� +imp + +� ˆV ˆGR(A) +0 +ˆV +� +imp, +(19) +in which the impurity configuration average can be ob- +tained as +� ˆV ˆGR(A) +0 +ˆV +� +imp = nV 2 +0 +� +d2q +(2π)2 ˆGR(A) +0 +. +(20) +The total of all ladder diagrams in Fig. 3 depicts the +vertex function in the Born approximation. The ladder +vertex-normalized charge current, ˆjν, is followed by the +integral (Bethe-Salpeter) equation as shown in Fig. +3 +which can be expressed as following relations: +ˆ˜jν = ˆjν + nV 2 +0 +� +d2q +(2π)2 +� ˆGR�ˆ˜jν +� ˆGA� +. +(21) +σSHE +xy += +1 +πΩTr[GRJxGAJy] ++ 1 +πΩTr[GAJyGR� +T RGRJxGAT A� +dis], +(22) + +5 +and similarly Bethe-Salpter equation for the EC results +in: +σEC +xy = +1 +πΩTr[GRJxGASy] ++ 1 +πΩTr[GASyGR� +T RGRJxGAT A� +dis]. +(23) +FIG. 3. +(a) Ladder diagram of the response function suc- +cessive interactions of an electron-hole pair with impurities. +(b) Iterative equation of vertex corrections (Bethe-Salpeter +equation) is presented using the diagrammatic approach. +IV. +RESULTS AND DISCUSSION +It can be realized that both the EE and SHE are ob- +tained as a result of the vertex corrections, whereas in the +absence of these corrections, the magnitudes of the EE +and SHE response function are negligible. Accordingly, +vertex correction plays a significant role in the realization +of SHE and EE in pseudospin-1 systems. As a result, the +EE and SHE in pseudospin-1 particle systems cannot be +considered first order effects that could be captured by +a linear response function or band broadening given by +the relaxation time broadening of self-energies within the +Born approximation. Unlike the pseudospin-1 particles, +it has been shown that the spin Hall conductivity of two- +dimensional electron gas systems with isotropic short- +range defects identically vanishes with the vertex correc- +tions [44]. This is due to the fact that the dressed current +operator, which is the renormalized vertex-modified cur- +rent, is corrected in such a way that the spin dependent +part of the electric current operator, that comes from +the Rashba coupling contribution in the current opera- +tor, is suppressed by the vertex corrections [44]. In a two- +dimensional electron gas system, this means that vertex +corrections decrease the correlation between electric cur- +rent and spin current. This makes the SH conductivity go +away. However, As mentioned before, vertex corrections +could effectively generate accountable SH conductivity in +pseudospin-1 systems. +In the absence of the vertex corrections, we have re- +alized that the magnitude of SH and EE conductivities +is negligible, while the vertex corrections enhance these +conductivities significantly. Therefore, it can be inferred +that there is a deep difference between the influence of +the vertex corrections in the electron gas and pseudospin- +1 systems. On the other hand, it has been reported that +the vertex corrections enhance the SHE in graphene, this +can be obtained within the Dirac cone approximation by +taking into account chiral bands [45]. +Accordingly, this similar influence of the vertex cor- +rections in graphene and pseudospin-1 systems suggests +that the linear nature of the band energies in both the +graphene and pseudospin-1 samples may provide this +deep difference with the results of the electron gas sys- +tems. It should be noted that the pseudospin-1 systems +with the Rashba spin-orbit coupling show linear disper- +sion as indicated by Eq. (2.5). However, due to the low +Rashba coupling strength these linear bands cannot re- +sult in metallic sample within the Brillouin zone at m ̸= 0 +as shown in Fig. 4. +FIG. 4. Band structures of pseudospin-1 systems for m = 1.0, +λR = 0.05 and Ef = 0.25. +SHC and EC have been obtained as a function of the +Rashba coupling strength at different gap values as shown +in figures 5 and 6. +It can be seen that the SHC and +EC can be manipulated by the Rashba coupling. It has +also been realized that, at high Rashba couplings, the +SHE and EE conductivities decrease rapidly. However, in +the gapless pseudospin-1 systems, vertex corrections are +very effective at low and intermediate Rashba couplings, +where both SHE and EE conductivities are significantly +increased as a result of the vertex corrections. At some +especial Rashba coupling strength, SHE and EE conduc- +tivities abruptly fall in gapped pseudospin-1 systems. +As it can be verified numerically that scattering rate +decreases by increasing the Rashba couplings. It seems +that this simple picture could explain the suppression +of the SHE and EE conductivities at high Rashba cou- +plings. Meanwhile, it should be noted that the effect of +the Rashba coupling should be considered beyond the + +(a) +Jy +(b) +Soz +Oz ++0z6 +4 +2 +代 +E +0 +-2 +-4 +-6 +-4 +-2 +0 +2 +4 +ka6 +first order linear response regime, since in the case of +the pseudospin-1 systems, only the vertex corrections can +contribute to the SHE and EE response functions, and +as mentioned before, there is no first order contribution. +Vertex correction takes into account the multiple inde- +pendent scatterings in which both retarded and advanced +branches of the electron propagator are interacting with +impurities, as shown in Fig 3. Accordingly, we expect +some different behavior when the single scatterings are +dominant and the response function can be described +within the first-order approximations. Physically, these +multiple, consecutive scatterings that don’t cross each +other create a series of independent events called a lad- +der diagram. These diagrams represent the quantum side +jump (QSJ) contribution to the SHE [33]. +Ladder diagrams should be used to explain the role of +Rashba coupling in the framework of independent mul- +tiple scatterings. The ladder diagrams essentially mea- +sure the side-jump contribution, which corresponds to +the transverse coordinate shift of the carriers after scat- +tering process. Therefore, at the intermediate level of the +Rashba coupling strength it seems that this shift is satu- +rated [43] and further increasing of the Rashba coupling +strength just randomizes the final spin state of the car- +riers, due to the increased spin-mixing provided by the +increased Rashba coupling. Accordingly, the suppression +of the EE and SHE is expected as a result of the Rashba +coupling increment after this saturation limit. +Mean- +while, there are other reports confirming that the SH +conductivity of 2DEG is suppressed by scattering from +short range nonmagnetic impurities with a linear Rashba +interaction [44, 46–49]. +As shown in Fig. 5 and Fig. 6, EE is suppressed in +the gapless sample, while SHE can be increased by an +order of magnitude in that type of systems. Meanwhile, +in gapped systems, it should be considered that both EE +and SHE are increased by decreasing the gap value. This +can be explained if we consider that increasing the gap +value decreases spin-band mixing. +As shown in Fig. +7 and Fig. +8, the SHE and EE +do not have a monotonic dependence on Fermi energy. +However, it has been mentioned that the SH conductiv- +ity in graphene is inversely proportional to the chemical +potential [37]. +V. +CONCLUSIONS +The spin Hall and Edelstein conductivities are stud- +ied numerically in the current work for the new type +two-dimensional materials called pseudospin-1 systems. +These two response functions have been evaluated in a +pseudospin-1 particle system with Rashba-type interac- +tion using the Kubo-Streda technique and vertex cor- +rections with non-magnetic impurities. +Numerical cal- +culations were carried out to derive the normalized op- +erators self-consistently. +It has been shown that the +vertex corrections play a significant role on the realiza- +tion of the SHE and EE in the pseudospin-1 systems, +where in the absence of these corrections SHE and EE +response functions are essentially negligible. 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Niu, Physical +Review B 77, 075304 (2008). +[42] A. Cr´epieux, Phys. Rev. B 64, 014416 (2001). +[43] N. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. Dugaev, +and J. Sinova, Physical Review B 75, 045315 (2007). +[44] J.-i. Inoue, G. E. Bauer, and L. W. Molenkamp, Physical +Review B 70, 041303 (2004). +[45] N. Sinitsyn, J. Hill, H. Min, J. Sinova, and A. MacDon- +ald, Physical review letters 97, 106804 (2006). +[46] E. G. Mishchenko, A. V. Shytov, +and B. I. Halperin, +Physical review letters 93, 226602 (2004). +[47] J.-i. Inoue, G. E. Bauer, and L. W. Molenkamp, Physical +Review B 67, 033104 (2003). +[48] R. Raimondi and P. Schwab, Physical Review B 71, +033311 (2005). +[49] V. D. Ol’ga, Physical Review B 71, 245327 (2005). + +8 +FIG. 5. The EE conductivity in (a) gapped systems for m = 0.5 and m = 1.0, and (b) gappless sample (m = 0). +FIG. 6. The SHE conductivity in (a) gapped systems for m = 0.5 and m = 1.0, and (b) gappless sample (m = 0). +FIG. 7. +The EE conductivity in systems with Ef = 0.15 +(black line), Ef = 0.25 (red line) and Ef = 0.35 (green line). + +(q) +(a) +0.4 +m/t=0.5 +2 +m=0 +2 +(eUF) +(na) +0.3 +-m/t=1.0 +0.1 +0.2 +-2 +0.05 +ha +0.1 +6 +b6 +0 +0 +0 +0.05 +0.1 +0 +0.02 +0.04 +ΛR/t +ΛR/tX10~5 +X10~3 +2.5 +12 +(a) +m/t=0.5 +2 +(b) +2 +-m=0 +(Hna) +10 +2 +-m/t=1.0 +8 +1.5 +n12 +2 +6 +HE +4 +ha +0.5 +6 +2 +0 +0 +0.020.040.060.08 +0 +0.02 +0.04 +^R/t0.15 +E,/t=0.15 +2 +E/t=0.25 +0.1 +/t=0.35 +h-2 +E +0.05 +E +6 +0 +0 +0.01 +0.02 +0.03 +0.04 +ΛR/t9 +FIG. 8. The SHE conductivity in systems with Ef = 0.15 +(black line), Ef = 0.25 (red line) and Ef = 0.35 (orange +line). + +×10-5 +A +E/t=0.15 +(Haa) +3 +.E./t=0.25 +E/t=0.35 +2 +ha +0 +0 +0.02 +0.04 +ΛR/t \ No newline at end of file diff --git a/_9AyT4oBgHgl3EQfqvjk/content/tmp_files/load_file.txt b/_9AyT4oBgHgl3EQfqvjk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13154ff3052ffab89ef1f7d296259351022efdc1 --- /dev/null +++ b/_9AyT4oBgHgl3EQfqvjk/content/tmp_files/load_file.txt @@ -0,0 +1,505 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf,len=504 +page_content='Spin Hall and Edelstein Effects in pseudospin-1 systems: significant contribution of vertex corrections S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Rastegar∗ Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran and Molecular Simulation Laboratory (MSL), Azarbaijan Shahid Madani University, Tabriz, Iran A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Phirouznia Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran and Condensed Matter Computational Research Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=', Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran (Dated: January 3, 2023) Edelstein and spin Hall effect response functions for a two-dimensional (2D) system of pseudospin- 1 particles is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' These two response functions denoted by σSH and σEE have been analyzed in a pseudospin-1 particle system in the presence of the Rashba-type spin-orbit interaction using the Kubo-Streda technique and vertex corrections with non-magnetic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Then, for a given range of the Rashba coupling, response functions of the spin Hall effect (SHE) and Edelstein effect (EE) have been estimated at various energy gaps and Fermi energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Results indicate that in this type of the two-dimensional materials, SHE and EE are essentially induced by the vertex corrections and without considering these corrections the amount of these effects are really negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It has also been realized that SHE and EE conductivities can be modulated by the Rashba coupling strength at low and intermediate level of this spin-orbit type interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' INTRODUCTION Pseudospin-1 particle compounds have recently at- tracted a lot of interest in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Low-energy gapless pseudospin-1 fermions have an en- ergy spectrum that is linear in momentum, except for a flat band at zero energy, just like electrons in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' One of the most prevalent two-dimensional realizations of pseudospin-1 fermions (2D) is the T3 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Atoms at the centers and vertices of the hexagonal lattice could be used to create a simple tight-binding model for this type of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Due to the placement of three sites per unit cell, the electron states in this model are given by three-component fermions with three bands of the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Such a system is shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The spin Hall effect (SHE) has received much attention because of its prospective applications in spintronic tech- nologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Hirsch [1] explored the spin Hall effect (SHE), which Dyakonov and Perel [2, 3] studied in 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The spin Hall effect (SHE) and the inverse spin Hall effect (ISHE) [4], a group of transport phenomena [5, 6], greatly aided the development of experimental spintronic devices [1, 3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' One may list examples of its use in memory, logic, and sensing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Kane and Mele discovered the quantum spin Hall effect in graphene in 2005 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The low-energy electronic structure of a single layer graphene with spin-orbit interactions was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' They found that graphene changes from its ideal two-dimensional semimetallic state at low temperatures to a quantum spin Hall insulator due to the symmetry-enabled spin-orbit ∗ Corresponding author’s Email: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='rastegar@azaruniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='ir FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' A schematic representation of the T3 lattice for pseudospin-1 fermion sample in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Translation vectors are V1 = (3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' − √ 3/2)a and V2 = (3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' √ 3/2)a, with lattice constant a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sites A and B that showed by solid circles and squares, making a hexagonal lattice, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' whereas, solid circles mark the hub sites C making a triangu- lar lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' After initial findings about these phenomena in semi- conductors and metals, it was known that the spin Hall effect might offer a new strategy for the interconversion of spin and charge indications [5, 6, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' SHE, which results from strong spin-orbit coupling (SOC), may pave the way for generating pure spin current from charge cur- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='00550v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='mes-hall] 2 Jan 2023 B2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Schematic representation of the SHE in a two- dimensional material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Both the charge and spin Hall currents are driven by an external electrical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In the spin Hall regime, a spin current is generated perpendicular to an applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Additionally, unlike SHE within the inverse SHE regime, spin currents can be converted into electric signals [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' spin Seebeck effect [13–15] demonstrates that the temperature gradient can generate spin current which confirms that the spin-orbit interactions are also crucial in the field of ”spin caloritronics.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The strength of this effect is given by spin Hall angle γ that measures the material’s effectiveness according to relation γ = J⊥/J∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' which is defined as the ratio be- tween steady-state longitudinal charge-current,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' J∥ and z- polarized transverse spin currents given by J⊥ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The material’s SHE efficiency is determined by the spin Hall angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The charge efficiency in the present spin transition can be determined using materials with a large spin Hall angle (SHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The anomalous effect is another well-known phe- nomenon in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The nonmagnetic spin Hall effect [16] and the well-known anomalous Hall effect (AHE) [16] in ferromagnetic metals are closely linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The magnetic field Hz and magnetization Mz both affect the Hall resis- tance ρH, through the relation ρH = R0Hz + RsMz[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Where, the normal and anomalous Hall resistances are R0 and Rs, respectively [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The anomalous Hall effect has become one of the most urgent issues in solid-state physics because it is challenging to forecast the carrier density in ferromagnets [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The Edelstein effect (EE) [23] and the inverse Edel- stein effects (IEE) [24, 25] are two additional effects that have recently attracted a lot of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In the regime of EE a continuous non-equilibrium spin polarization Sy is produced by a constant current Jx driven by an elec- tric field Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It was also demonstrated that the EE and the SHE are connected [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The inverse Edelstein effect is the process underlying the spin-to-charge transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The Edelstein effect is a promising phenomena for spintronics applications since it can produce spin polar- ization in nonmagnetic materials just electrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' There- fore, it is highly desirable to find materials that are highly efficient at ”converting” the electric current into spin po- larization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile, it should be noted that the spin and charge quantum numbers can also be converted to pseudospin index in graphene-like materials [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Edelstein effect of enormous topological-insulator- graphene heterostructures has been investigated by Rodriguez-Vega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [28], where they show that the nonequilibrium uniform spin-density accumulation caused by a charge current in magnetic TI-graphene het- erostructures can be 10 − 100 times larger than in TIs alone, resulting in a massive Edelstein effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Valley Edelstein effect (VEE) has been discovered in the monolayer transition-metal dichalcogenides by Taguchi et al [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' They found that in gated monolayer transition-metal dichalcogenides (MTMDs), Berry curva- tures resulting from coexisting Rashba and Ising SOCs combined with traditional Edelstein effects lead to valley- contrasting spin polarization parallel to the applied elec- tric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The ability to introduce SOC with multiple symme- tries [30, 31] and varying spatial extent is a fascinating property of two-dimensional materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Recent theoreti- cal investigations have clarified the significant role that the SOC symmetry plays in the resonant scattering do- main [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' There is a microscopic approach for the spin Hall effect in amorphous materials using vertex corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This approach has been employed in amorphous graphene [33] where the nonperturbative quantum diagrammatic method which has been applied in this approach can also be used to obtain the response function in materials re- sembling graphene [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' For non-magnetic impurities, the vertex coefficient vanishes when the Fermi level is in the upper band (both bands are occupied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This problem leads to the disap- pearance of the spin-Hall effect in the thermodynamic limit for any degree of disorder [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It has also been shown that the bare vertex is a good approximation and that the bare bubble diagram is adequate for calculating the spin-Hall conductivity in the presence of magnetic impurities [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The SHA rises linearly with the SOC impurity den- sity in disordered samples where other variables restrict charge mobility [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The spin Hall current in graphene identically vanishes when the spin-orbit interaction (SOI) is absent even in the presence of the magnetic impurities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefore, it can be inferred that the SOI is of crucial importance for SHE and it cannot be generated merely by the spin-flip scatterings [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It has been shown that the spin Hall conductivity (SHC) are generally finite in the presence of SOI and magnetic impurities under a zero external magnetic field [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' When the magnetic impurities are combined with the SOI, the magnetic scattering centers act as spin- dependent barriers, causing a charge imbalance at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Changing the chemical potential close to the gap should also reveal charge and spin Hall effects in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Several investigations have been performed on the charge and spin Hall phenomena in magnetically impure graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Using the Kubo formula, analytical formulations of the charge and spin Hall conductivities has been developed theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 1 Jc X3 The SHE and EE can be described by the Edelstein conductivity (EC) and the SHC that are defined through the following relations Jz y = σSHE xy Ex, sy = σEE xy Ex re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In this study spin Hall and Edelstein conduc- tivities for pseudospin-1 particles with vertex corrections have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Using the Kubo-Streda method and vertex corrections in the presence of non-magnetic impurities, these two response functions, σSHE and σEE, have been evaluated in a pseudospin-1 particle system with Rashba-type spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Numerical cal- culations provide the normalized operators within a self- consistent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Results show that the bare Kubo response function of both EE and SHE are absolutely small and the main contribution comes the vertex cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefore, in these type of materials we cannot expect high SHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile, EC and SHC could reveal the different physics behind these novel materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' METHODOLOGY Within a low energy approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' effective Hamil- tonian of massive Dirac-like pseudospin-1 particles in the presence of the Rashba spin-orbit interaction can be stated as follows: H = ℏvF (Sxkx ⊗ Iσ + Syky ⊗ Iσ) + λR(kxIδ ⊗ σy − kyIδ ⊗ σx) + mSz ⊗ Iσ (1) where vF is the pseudospin-1 particle’s Fermi velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' λR is the Rashba coefficient and S = (Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sz) is a vector of matrices with pseudospin components [38]: Sx = 1 √ 2 � � 0 1 0 1 0 1 0 1 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sy = 1 √ 2 � � 0 −i 0 i 0 −i 0 i 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sz = � � 1 0 0 0 0 0 0 0 −1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The three matrices represent all pseudospin-1 particles that fulfill the angular momentum commutation relations [Sl, Sm] = iϵlmnSn with three eigenvalues, s = ±1, 0, where ϵlmn is the Levi-Civita symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' However, they do not form a Clifford algebra, as opposed to Pauli matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' that is, {Sn, Sm} ̸= 2δn,mI3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' These three dimensional matrices represent the contribution of three A, B and C sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' After the Fourier transformation H = � k Ψ† kH(k)Ψk, (2) where Ψ† k = (cA,k,↑, cB,k,↑, cC,k,↑, cA,k,↓, cB,k,↓, cC,k,↓) and c† ikσ denotes the creation operator of electron on the i sublattice with spin of σ and wave-number of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Then in presence of Rashba coupling, the Hamilto- nian is given by H = ℏvF √ 2 � � 0 k− 0 k+ 0 k− 0 k+ 0 � � ⊗ Iσ + λR � � kx 0 0 0 kx 0 0 0 kx � � ⊗ σy − λR � � ky 0 0 0 ky 0 0 0 ky � � ⊗ σx + m � � 1 0 0 0 0 0 0 0 −1 � � ⊗ Iσ (3) The energy spectrum can be obtained as E1,2(k) = ±|k|λR E3,4(k) = ± � m′(k) + k2λ2 R − 2 � k2λ2 Rm′(k) E5,6(k) = ± � m′(k) + k2λ2 R + 2 � k2λ2 Rm′(k), (4) where m′(k) = m2 + k2ℏ2v2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefore, electrons in the T3 lattice as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 1, can behave as massless Dirac fermions with pseudospin S = 1, rather than S = 1/2 for those trapped in the hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Each unit cell in this T3 lattice has three inequivalent sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Two lattice sites, A and B, are triply coordinated, while site C connects six of its closest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' THE KUBO-STREDA APPROACH The density operators for charge and spin-current can be defined as follows [39–41]: J = −eΨ†(x)vΨ(x), (5) J α β = −eΨ†(x)1 2{sα, vβ}Ψ(x), (6) where {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='} represents the anticommutator, v represents the velocity operator, −e represents the electron,s charge, and sz ≡ s3 represents the diagonal Pauli matrix with eigenvalues ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Using the Kubo-Streda method the con- ductivity tensor in the presence of SOC [42] is: σz ij = ℏ 2πΩTr � Ji(GR−GA)JiGA−Jj(GR−GA)JiGR� dis, (7) GR(A) denotes the retarded (advanced) Green,s function, which is defined as follows: GR(A) = 1 ϵ − H0 − V ± i0+ , (8) 4 Where H stands for the total Hamiltonian, H = H0 +V , with H0 for the non-perturbed Hamiltonian, and V repre- sents the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Cartesian elements of the charge- and spin-current density operators’ are denoted by Ji and Ji respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' � dis indicates the configurational dis- order average compared to the conventional version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' � O � dis = lim N,Ω→∞ � N � i=1 � Ω d2xi Ω � O(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=', xN) ���� N Ω =n , (9) Where i represents impurities, Ω represents the sample area, and Tr represents the trace over the entire Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The conductivity of SH is equal to σSHE xy = 1 πΩTr[ � GRJxGA� disJy] (10) and also EE conductivity is given as follows σEE xy = 1 πΩTr[ � GRSyGA� disJx] (11) For short-range impurities, calculations were made using the weak-scattering regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Within the Kubo formal- ism, the main focus has been devoted on analyzing the electric dc Edelstein and Hall conductivities in a typi- cal pseudospin-1 particle system with nonmagnetic dis- orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The linear response Kubo formalism has been used to explain the fully quantum mechanical response func- tions by taking the vertex correction into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The response functions were obtained at zero temperature, which may be specified using Green’s time-ordered func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Because the calculations were performed in the dc regime, it can be achieved by ω −→ 0 limit at the end of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In the meantime, this method has been necessary for all ac-based calculations in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Disordered Vertex Correction A randomly distributed disorder potential has been as- sumed as follows ˆV (r) = V0 N � i=1 δ(r − Ri), (12) weak, and short-range Gaussian correlation can be con- sidered by imposing following relations � ˆV (r) � imp = 0, � ˆV (r1) ˆV (r2) � imp = nV 2 0 δ(r1 − r2), (13) The impurity concentration and scattering potential are indicated by n = 5 × 1010cm−2 and V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1eV , respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This method has been used extensively to study the properties of disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The Kubo relation for dc electric current conductivity is: σxν = ℏ 2πL2 Tr �ˆjx ˆGRˆjν ˆGA� imp, (14) where L2 represents the system area and ˆGR(A)(ϵ) = (ϵ ± i0 − ˆH − ˆV )−1 refers to the retarded (advanced) Green,s function in the Pauli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The current operator is expressed as ˆj = −e(−i/ℏ)[ˆr, ˆH] = −evF ˆz × ˆσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Mean- while, � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' � imp shows an ensemble average over random nonmagnetic impurity configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The conductivity then is given as σxν ≈ ℏ 2πL2 Tr[ˆjx � ˆGR�ˆjν � ˆGA� ] + Vertex correction (15) ≡ ℏ 2πL2 Tr[ˆjx � ˆGR�ˆ˜jν � ˆGA� ], Where � ˆGR(A)� is the averaged Green,s function and ˆjν is the bare charge current density operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The ladder diagram correction to the bare charge current density op- erator can be evaluated using the following relation[43], ˆ˜jν(εF ) ≡ ˆjν + δˆ˜jν(εF ), (16) where ˜jν is the normalized charge current operator within the ladder vertex correction and δˆ˜jν is the charge cur- rent vertex correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The averaged Green function is expressed using the Dyson equation and the Born ap- proximation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' � ˆGR(A)� = � (z − ˆH − ˆV )−1� imp (17) = ˆGR(A) 0 + ˆGR(A) 0 ˆΣR(A)� ˆGR(A)� , with z = ϵ ± i0 in which ϵ is small positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Here, the solution may then be written as: � ˆG � = (( ˆGR(A) 0 )−1 − ˆΣR(A))−1, (18) in which the self-energy in the Born approximation is ˆΣR(A) = � ˆV � imp + � ˆV ˆGR(A) 0 ˆV � imp, (19) in which the impurity configuration average can be ob- tained as � ˆV ˆGR(A) 0 ˆV � imp = nV 2 0 � d2q (2π)2 ˆGR(A) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (20) The total of all ladder diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 3 depicts the vertex function in the Born approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The ladder vertex-normalized charge current, ˆjν, is followed by the integral (Bethe-Salpeter) equation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 3 which can be expressed as following relations: ˆ˜jν = ˆjν + nV 2 0 � d2q (2π)2 � ˆGR�ˆ˜jν � ˆGA� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (21) σSHE xy = 1 πΩTr[GRJxGAJy] + 1 πΩTr[GAJyGR� T RGRJxGAT A� dis], (22) 5 and similarly Bethe-Salpter equation for the EC results in: σEC xy = 1 πΩTr[GRJxGASy] + 1 πΩTr[GASyGR� T RGRJxGAT A� dis].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (23) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (a) Ladder diagram of the response function suc- cessive interactions of an electron-hole pair with impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (b) Iterative equation of vertex corrections (Bethe-Salpeter equation) is presented using the diagrammatic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' RESULTS AND DISCUSSION It can be realized that both the EE and SHE are ob- tained as a result of the vertex corrections, whereas in the absence of these corrections, the magnitudes of the EE and SHE response function are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Accordingly, vertex correction plays a significant role in the realization of SHE and EE in pseudospin-1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' As a result, the EE and SHE in pseudospin-1 particle systems cannot be considered first order effects that could be captured by a linear response function or band broadening given by the relaxation time broadening of self-energies within the Born approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Unlike the pseudospin-1 particles, it has been shown that the spin Hall conductivity of two- dimensional electron gas systems with isotropic short- range defects identically vanishes with the vertex correc- tions [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This is due to the fact that the dressed current operator, which is the renormalized vertex-modified cur- rent, is corrected in such a way that the spin dependent part of the electric current operator, that comes from the Rashba coupling contribution in the current opera- tor, is suppressed by the vertex corrections [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In a two- dimensional electron gas system, this means that vertex corrections decrease the correlation between electric cur- rent and spin current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This makes the SH conductivity go away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' However, As mentioned before, vertex corrections could effectively generate accountable SH conductivity in pseudospin-1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' In the absence of the vertex corrections, we have re- alized that the magnitude of SH and EE conductivities is negligible, while the vertex corrections enhance these conductivities significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefore, it can be inferred that there is a deep difference between the influence of the vertex corrections in the electron gas and pseudospin- 1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' On the other hand, it has been reported that the vertex corrections enhance the SHE in graphene, this can be obtained within the Dirac cone approximation by taking into account chiral bands [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Accordingly, this similar influence of the vertex cor- rections in graphene and pseudospin-1 systems suggests that the linear nature of the band energies in both the graphene and pseudospin-1 samples may provide this deep difference with the results of the electron gas sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It should be noted that the pseudospin-1 systems with the Rashba spin-orbit coupling show linear disper- sion as indicated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' However, due to the low Rashba coupling strength these linear bands cannot re- sult in metallic sample within the Brillouin zone at m ̸= 0 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Band structures of pseudospin-1 systems for m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0, λR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='05 and Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' SHC and EC have been obtained as a function of the Rashba coupling strength at different gap values as shown in figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It can be seen that the SHC and EC can be manipulated by the Rashba coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It has also been realized that, at high Rashba couplings, the SHE and EE conductivities decrease rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' However, in the gapless pseudospin-1 systems, vertex corrections are very effective at low and intermediate Rashba couplings, where both SHE and EE conductivities are significantly increased as a result of the vertex corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' At some especial Rashba coupling strength, SHE and EE conduc- tivities abruptly fall in gapped pseudospin-1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' As it can be verified numerically that scattering rate decreases by increasing the Rashba couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It seems that this simple picture could explain the suppression of the SHE and EE conductivities at high Rashba cou- plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile, it should be noted that the effect of the Rashba coupling should be considered beyond the (a) Jy (b) Soz Oz +0z6 4 2 代 E 0 2 4 6 4 2 0 2 4 ka6 first order linear response regime, since in the case of the pseudospin-1 systems, only the vertex corrections can contribute to the SHE and EE response functions, and as mentioned before, there is no first order contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Vertex correction takes into account the multiple inde- pendent scatterings in which both retarded and advanced branches of the electron propagator are interacting with impurities, as shown in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Accordingly, we expect some different behavior when the single scatterings are dominant and the response function can be described within the first-order approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Physically, these multiple, consecutive scatterings that don’t cross each other create a series of independent events called a lad- der diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' These diagrams represent the quantum side jump (QSJ) contribution to the SHE [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Ladder diagrams should be used to explain the role of Rashba coupling in the framework of independent mul- tiple scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The ladder diagrams essentially mea- sure the side-jump contribution, which corresponds to the transverse coordinate shift of the carriers after scat- tering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefore, at the intermediate level of the Rashba coupling strength it seems that this shift is satu- rated [43] and further increasing of the Rashba coupling strength just randomizes the final spin state of the car- riers, due to the increased spin-mixing provided by the increased Rashba coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Accordingly, the suppression of the EE and SHE is expected as a result of the Rashba coupling increment after this saturation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Mean- while, there are other reports confirming that the SH conductivity of 2DEG is suppressed by scattering from short range nonmagnetic impurities with a linear Rashba interaction [44, 46–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 6, EE is suppressed in the gapless sample, while SHE can be increased by an order of magnitude in that type of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile, in gapped systems, it should be considered that both EE and SHE are increased by decreasing the gap value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' This can be explained if we consider that increasing the gap value decreases spin-band mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 8, the SHE and EE do not have a monotonic dependence on Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' However, it has been mentioned that the SH conductiv- ity in graphene is inversely proportional to the chemical potential [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' CONCLUSIONS The spin Hall and Edelstein conductivities are stud- ied numerically in the current work for the new type two-dimensional materials called pseudospin-1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' These two response functions have been evaluated in a pseudospin-1 particle system with Rashba-type interac- tion using the Kubo-Streda technique and vertex cor- rections with non-magnetic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Numerical cal- culations were carried out to derive the normalized op- erators self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' It has been shown that the vertex corrections play a significant role on the realiza- tion of the SHE and EE in the pseudospin-1 systems, where in the absence of these corrections SHE and EE response functions are essentially negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The ladder diagrams mainly capture the side jump effect contribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Therefor, it can be inferred that the side jump effect is one of the main effects which can induce SHE and EE in pseudospin-1 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Meanwhile, the con- 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+page_content=' Zhu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Berakdar, EPL (Euro- physics Letters) 88, 58001 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Bercioux, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Urban, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Grabert, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' H¨ausler, Physical Review A 80, 063603 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [39] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Rashba, Physical Review B 68, 241315 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Zhang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Yang, Physical review letters 94, 066602 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [41] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Shi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Xiao, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Niu, Physical Review B 77, 075304 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Cr´epieux, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' B 64, 014416 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [43] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sinitsyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' MacDonald, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Jungwirth, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Dugaev, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sinova, Physical Review B 75, 045315 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Inoue, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Bauer, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Molenkamp, Physical Review B 70, 041303 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [45] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sinitsyn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Hill, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Min, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Sinova, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' MacDon- ald, Physical review letters 97, 106804 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [46] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Mishchenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Shytov, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Halperin, Physical review letters 93, 226602 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Inoue, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Bauer, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Molenkamp, Physical Review B 67, 033104 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [48] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Raimondi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Schwab, Physical Review B 71, 033311 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' [49] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' Ol’ga, Physical Review B 71, 245327 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The EE conductivity in (a) gapped systems for m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 and m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0, and (b) gappless sample (m = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The SHE conductivity in (a) gapped systems for m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 and m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0, and (b) gappless sample (m = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The EE conductivity in systems with Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='15 (black line), Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='25 (red line) and Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='35 (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' (q) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='4 m/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 2 m=0 2 (eUF) (na) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='3 m/t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='05 ha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1 6 b6 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='04 ΛR/t ΛR/tX10~5 X10~3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 12 (a) m/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 2 (b) 2 m=0 (Hna) 10 2 m/t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='0 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 n12 2 6 HE 4 ha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='5 6 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='08 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='04 ^R/t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='15 E,/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='15 2 E/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='1 /t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='35 h-2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='05 E 6 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='04 ΛR/t9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' The SHE conductivity in systems with Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='15 (black line), Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='25 (red line) and Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='35 (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content=' ×10-5 A E/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='15 (Haa) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='25 E/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfqvjk/content/2301.00550v1.pdf'} +page_content='35 2 ha 0 0 0.' 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Progreso Tizapán, C.P. 01080, CDMX +México +E-mail: jfdez@itam.mx +∗ Corresponding Author: +E-mail: mercedes@itam.mx +Abstract +Fernández-Durán and Gregorio-Domínguez (2014) defined a family of probability +distributions for a vector of circular random variables by considering multiple non- +negative trigonometric sums. These distributions are highly flexible and can present +many modes and skewness. Several operations on these multivariate distributions are +translated into operations on the vector of parameters; for example, marginalization +involves the calculation of the eigenvectors and eigenvalues of a matrix and, indepen- +dence among subsets of the elements of the vector of circular variables translates to a +case in which the vector of parameters is the Kronecker product of the corresponding +subsets of the vector of parameters. An alternative parameter estimation algorithm +for high-dimensional circular data is presented and applied to a real dataset on wind +directions. +Keywords: Vector of circular random variables, Independence, Marginalization, Condi- +tional distribution, Kronecker product +1 + +1 +Introduction +The main objective of this study is to consider the properties of the multivariate nonnegative +trigonometric sums (MNNTS) distributions developed by Fernández-Durán and Gregorio- +Domínguez (2014), such as conditional and marginal distributions and the conditions for +independence among subsets of the vector of circular random variables. A circular (angular) +random variable is defined as one where the support of its probability density function is +the unit circle, and it must be a function of period 2π, i.e., if f(θ) represents the density +function of a circular random variable, θ, then, f(θ + 2kπ) = f(θ), where k is an integer. A +multivariate circular random vector is a vector in which each component is a circular random +variable, θ = (θ1, θ2, . . . , θn)⊤, where θ1, θ2, . . . , θn are circular random variables. +Multivariate circular data are available in many disciplines. The dihedral angles in a +protein, wind directions recorded at different monitoring stations, time of occurrence of dif- +ferent diseases, and time of flowering of different plant species are some examples among +many others. +Note that in many applications, multivariate circular data consist of the +time of occurrence of different events. The support of the domain of the distribution of a +multivariate circular random vector is a hypertorus. The bivariate von Mises model was +developed by Mardia (1975a; see also Mardia and Jupp, 2000). Johnson and Wehrly (1977) +and Wehrly and Johnson (1980) presented bivariate circular models based on a decompo- +sition of the bivariate cumulative distribution that is equivalent to a copula (Nelsen, 1999) +used by Fernández-Durán (2007) to construct bivariate circular and circular-linear proba- +bility models based on univariate nonnegative trigonometric sums (Fernández-Durán, 2004 +and Fernández-Durán and Gregorio-Domínguez, 2012). Other bivariate circular models are +presented by Singh et al. (2002), Lennox et al. (2009), Mardia et al. (2007) and Shieh and +Johnson (2005). In the multivariate circular case, Kim et al. (2016) extended the bivari- +ate copula decomposition of Johnson and Wehrly (1977) and Wehrly and Johnson (1980) +2 + +to the multivariate case. Mardia et al. (2012) considered mixtures of multivariate circu- +lar distributions that are extensions of the univariate von Mises model. +Another model +that is an extension of the univariate von Mises model is presented by Mardia and Voss +(2014). Fernández-Durán and Gregorio-Domínguez (2014) extended the univariate nonneg- +ative trigonometric model to the multivariate case in which the parameters are estimated +by maximum likelihood (see Fernández-Durán and Gregorio-Domínguez, 2010) and com- +putationally implemented using the R software (R Development Core Team, 2020) in the +CircNNTSR library (Fernández-Durán and Gregorio-Domínguez, 2012 and 2016). We refer +to this family of multivariate circular distributions as the MNNTS. The MNNTS probability +density function for multivariate circular random vectors is defined as +f12···n(θ) = ||cHe||2 += +cHeeHc +(1) += +M1 +� +k1=0 +M2 +� +k2=0 +· · · +Mn +� +kn=0 +M1 +� +m1=0 +M2 +� +m2=0 +· · · +Mn +� +mn=0 +ck1k2···kn¯cm1m2···mne +�n +s=1 i(ks−ms)θs, +where i = √−1, c = cR+icI, cR and cI are the real and imaginary parts of complex number c, +and ¯c = cR − icI is the conjugate of complex number c. The complex vector, c, is the vector +of parameters. +The complex vector, e, contains the multivariate trigonometric moments +defined as e +�n +s=1 rsθs for integer values r1, r2, . . . , rn. +Both vectors have the dimension of +�n +s=1(Ms + 1). The vector of parameters, c, must satisfy the following constraint: +||c||2 = +M1 +� +k1=0 +M2 +� +k2=0 +· · · +Mn +� +kn=0 +||ck1k2···kn||2 = +1 +(2π)n. +(2) +Given this constraint, the first element of the c vector, c00···0, is a nonnegative real number, +and the parameter space is a complex hypersphere having the dimension of �s +s=1(Mn + 1), +which is isomorphic to a real hypersphere having the dimension of 2 �s +s=1(Mn + 1) − 1 by +taking the real and imaginary parts of the complex numbers in c. +Given the definition of the multivariate density function in Equation 2, it is easy to obtain +the expression for the marginal distributions; however, it is not clear what the c parameters +3 + +of the marginal distributions of any dimension are. In addition, the conditional densities of +a subset of the circular variables, given other subsets of the joint vector, are not clear at the +first instance. +The remainder of this paper is organized as follows. Section 2 presents alternative ways to +write the density function in Equation 2 that are more adequate for obtaining the parameters +of marginal and conditional distributions. Section 3 describes the development of marginal +distributions. Section 4 provides the conditions for independence among the elements of the +multivariate circular vector, and Section 5 details the derivation of the conditional distribu- +tions. The derivation of the marginal and conditional distributions is developed for the joint +bivariate cases, although their generalization to any dimension is direct. In Section 6, we +propose an efficient algorithm to estimate the parameters of high-dimensional circular data +as an alternative to maximum likelihood estimation that, in the case of high-dimensional +circular data, could have the limitations of slow or non-convergence. +Section 7 presents +the application of the results described in the previous sections to a real dataset on wind +directions for a monitoring network in Mexico. Finally, Section 8 concludes this paper. +2 +MNNTS Distributions +Consider a vector of circular random variables, θ = (θ1, θ2, . . . , θn)⊤, that is distributed +as an MNNTS distribution (Fernández-Durán, 2007 and Fernández-Durán and Gregorio- +Domínguez, 2014). Then, to perform efficient (numerical) calculations, the density function +of θ in Equation 2 can be written in terms of Kronecker products as +f12···n(θ) = cHeeHc = cH +� +n +� +s=1 +es +n +� +s=1 +eH +s +� +c = cH +� +n +� +s=1 +eseH +s +� +c, +(3) +where es = (1, eiθs, e2iθs, . . . , eMsiθs)⊤ is the vector of the trigonometric moments of the s-th +circular random variable in the random vector, θ. This result is based on the successive +4 + +applications of the following property of Kronecker products: +(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD), +(4) +where A, B, C, and D are general matrices for which the products are defined. Note that this +expression can also be used in computational programs to perform numerical calculations +more efficiently. Furthermore, the set of indexes of the vector of parameters, c, is obtained +using the Kronecker product, (0, 1, . . . , M1) �(0, 1, . . . , M2) � · · ·�(0, 1, . . . , Mn), for an n- +dimensional MNNTS model. The order of the elements of circular vector θ can be modified +to simplify the derivations presented in the following sections. +3 +Marginal Distributions +The expression for the marginal distribution of any dimension of an MNNTS distribution +is obtained by integrating the marginalizing components. For the bivariate joint MNNTS +distribution in terms of the sum, +f1(θ1) = +� 2π +0 +f12(θ1, θ2)dθ2 += +� 2π +0 +M1 +� +k1=0 +M2 +� +k2=0 +M1 +� +m1=0 +M2 +� +m2=0 +c(12) +k1k2¯c(12) +m1m2ei(k1−m1)θ1+i(k2−m2)θ2dθ2 += +M1 +� +k1=0 +M2 +� +k2=0 +M1 +� +m1=0 +M2 +� +m2=0 +c(12) +k1k2¯c(12) +m1m2ei(k1−m1)θ1 +� 2π +0 +ei(k2−m2)θ2dθ2 += +2π +M1 +� +k1=0 +M1 +� +m1=0 +� M2 +� +m2=0 +c(12) +k1m2¯c(12) +m1m2 +� +ei(k1−m1)θ1. +(5) +Then, if the general sum form of the marginal distribution of θ1 is +f1(θ1) = +M1 +� +k1=1 +M1 +� +m1=1 +c(1) +k1 ¯c(1) +m1ei(k1−m1)θ1, +(6) +we can obtain +c(1) +k1 ¯c(1) +m1 = 2π +M2 +� +m2=0 +c(12) +k1m2¯c(12) +m1m2, +(7) +where c(1) is the vector of the parameters of marginal density f1. The expression for f1(θ1) +is easy to obtain; however, obtaining the expression of the vector of the parameters of the +5 + +univariate distribution, c(1) = (c(1) +0 , c(1) +1 , . . . , c(1) +M1)⊤, in terms of the bivariate distribution vec- +tor of parameters, c(12) = (c(12) +00 , c(12) +01 , . . . , c(12) +0M2, . . . , c(12) +M10, . . . , c(12) +M1M2)⊤, is a more difficult task +that requires the use of matrix algebra. Considering the joint bivariate MNNTS distribution +for vector θ = (θ1, θ2)⊤, we obtain the marginal distribution of θ1. +f1(θ1) = +� 2π +0 +f12(θ1, θ2)dθ2 += +� 2π +0 +c(12)H � +(e1eH +1 ) ⊗ (e2eH +2 ) +� +c(12)dθ2 += +c(12)H +� +(e1eH +1 ) ⊗ +�� 2π +0 +e2eH +2 dθ2 +�� +c(12) +(8) +Thus, +f1(θ1) = 2πc(12)H � +(e1eH +1 ) ⊗ IM2+1 +� +c(12) = 2πc(12)H ((e1 ⊗ IM2+1) +� +eH +1 ⊗ IM2+1 +� +c(12). +(9) +If the bivariate MNNTS vector parameter is written as +c(12) = (c(12) +•0 , c(12) +•1 , . . . , c(12) +•M2)⊤ +(10) +with c(12) +•m2 = (c(12) +0m2, c(12) +1m2, . . . , c(12) +M1m2)⊤ for m2 = 0, 1, . . . , M2, then +f1(θ1) += +2πc(12)H ((e1 ⊗ IM2+1) +� +eH +1 ⊗ IM2+1 +� +c(12) += +2π +M2 +� +m2=0 +c(12)H +•m2 e1eH +1 c(12) +•m2 += +2πeH +1 +� M2 +� +m2=0 +c(12) +•m2c(12)H +•m2 +� +e1. +(11) +The spectral decomposition of C•2 = 2π �M2 +m2=0 c(12) +•m2c(12)H +•m2 +satisfies +C•2 = +M2 +� +m2=0 +pm2c∗(12) +•m2 c∗(12)H +•m2 +. +(12) +Then, the marginal distribution of θ1 is a mixture of the univariate NNTS densities: +f1(θ1) = eH +1 +� M2 +� +m2=0 +pm2c∗(12) +•m2 c∗(12)H +•m2 +� +e1 = +M2 +� +m2=0 +pm2c∗(12)H +•m2 +e1eH +1 c(12)∗ +•m2 +(13) +where the mixture probabilities, p0, p1, . . ., pM2, and mixture parameter vectors, c∗(12) +•0 +, c∗(12) +•1 +. . ., c∗ +•M2, correspond to the eigenvalues and eigenvectors of matrix C•2. This result gener- +alizes to the marginals of any dimension of an MNNTS distribution, i.e., if θ = (θR, θRc)⊤, +6 + +where R is the set of indexes of the circular random variables for which we need its marginal +MNNTS distribution and Rc is the complement set of R, then the marginal distribution of +θR is a mixture of MNNTS distributions. +4 +Independence +In the simplest case of a bivariate MNNTS distribution for θ = (θ1, θ2)⊤, the circular random +variables, θ1 and θ2, are independent if and only if the joint vector of parameters, c(12), is the +Kronecker product of the marginal vector of parameters c(1) and c(2), i.e., c(12) = c(1) ⊗ c(2). +The proof uses the result of Equation 4. If c(12) = c(1) ⊗ c(2), +f12(θ1, θ2) = c(12)H � +(e1eH +1 ) ⊗ (e2eH +2 ) +� +c(12) = (c(1)⊗c(2))H � +(e1eH +1 ) ⊗ (e2eH +2 ) +� +(c(1)⊗c(2)). (14) +Repeated applications of the property of Kronecker products in Equation 4 obtain +f12(θ1, θ2) = +� +(c(1)H(e1eH +1 )) ⊗ (c(2)H(e2eH +2 )) +� +(c(1)⊗c(2)) = +� +c(1)H(e1eH +1 )c(1)� +⊗ +� +c(2)H(e2eH +2 )c(2)� +, +(15) +which is equal to f1(θ1)f2(θ2) because the Kronecker product of the two scalars is equal to +a simple product. The proof that if θ1 and θ2 are independent, then the joint parameter +vector is the product of the marginal parameter vectors is obtained from the previous proof +starting from the end. This result can be generalized to MNNTS of any dimension. Let +θ = (θR, θRc)⊤, then θR is independent of θRc if and only if c(R � Rc) = c(R) ⊗ c(Rc). Using +this result, we can construct a likelihood ratio test for independence to determine the inde- +pendence among relevant subsets of the vector of circular random variables, θ. In terms of +the decomposition of the univariate marginal distribution as a mixture of distributions in +Equation 13, θR is independent of θRc when there is only one element in the mixture, i.e., +the eigenvalues (mixing probabilities) are equal to zero except for the first, which is equal to +one. +7 + +Based on the definition of the MNNTS density, it is clear that the elements of the vector +of circular random variables, θ = (θ1, θ2, . . . , θn)⊤, are exchangeable if all the components +of the vector of the number of terms in the sum, M = (M1, M2, . . . , Mn)⊤, are equal, i.e., +Mk = M for k = 1, 2, . . . , n. +5 +Conditional Distributions +The conditional distributions of an MNNTS distribution are also MNNTS distributions, that +is, the MNNTS family of distributions is closed under conditioning. For the bivariate case, +the conditional distribution of θ1, given θ2 = θ∗ +2, is obtained as follows: +f1|2(θ1 | θ2 = θ∗ +2) = f12(θ1, θ∗ +2) +f2(θ∗ +2) +. +(16) +If f12(θ1, θ∗ +2) for a fixed value of θ2 = θ∗ +2, +f12(θ1, θ∗ +2) = c(12)H � +(e1eH +1 ) ⊗ (e∗ +2e∗H +2 ) +� +c(12) = c(12)H � +(e1eH +1 ) ⊗ (e∗ +2e∗H +2 ) +� +c(12). +(17) +Because for an m×n A and a p×q B matrices, A⊗B = (A⊗Ip)(In ⊗B) = (Im ⊗B)(A⊗Iq) +where Is denotes the s-by-s identity matrix, +f12(θ1, θ∗ +2) += +c(12)H �� +(e1eH +1 ) ⊗ IM2+1 +� � +IM1+1 ⊗ (e∗ +2e∗H +2 ) +�� +c(12) += +c(12)H �� +(e1eH +1 ) ⊗ (IM2+1IM2+1) +� � +(IM1+1IM1+1) ⊗ (e∗ +2e∗H +2 ) +�� +c(12) += +c(12)H � +(e1 ⊗ IM2+1)(eH +1 ⊗ IM2+1)(IM1+1 ⊗ e∗ +2)(IM1+1 ⊗ e∗H +2 ) +� +c(12) += +c(12)H � +(e1 ⊗ IM2+1)(e∗ +2 ⊗ 1)(1 ⊗ eH +1 )(IM1+1 ⊗ e∗H +2 ) +� +c(12) += +c(12)H � +(IM1+1 ⊗ e∗ +2)(1 ⊗ e1)(1 ⊗ eH +1 )(IM1+1 ⊗ e∗H +2 ) +� +c(12) += +c∗(12)H(e1eH +1 )c∗(12), +(18) +where c∗(12) = (IM1+1 ⊗ e∗H +2 )c(12). Then, the vector of parameters of the conditional distri- +bution, c(1|2), satisfies +c(1|2) = +c(12)∗ +� +f2(θ∗ +2) += (IM1+1 ⊗ e∗H +2 )c(12) +� +f2(θ∗ +2) +(19) +8 + +because +� +f2(θ∗ +2) is the constant of proportionality. Geometrically, the conditional parameter +vector, c(1|2), is obtained by rotating the joint parameter vector by θ∗ +2 through e∗H +2 +and +then normalizing it. This result can be generalized to the multivariate case by considering +θ = (θCc, θC)⊤, where C is the set of indexes of the conditioning circular variables, given the +known values, θC = θ∗ +C. The conditional distribution of θCc, given θC = θ∗ +C, is equal to +fCc|C(θCc | θC = θ∗ +C) = +c(Cc � C)H �� +I� +k∈Cc(Mk+1) +� +k∈C e∗ +k +� �� +k∈Cc(ekeH +k ) +� � +I� +k∈Cc(Mk+1) +� +k∈C e∗H +k +�� +c(Cc � C) +fCc(θCc) +, (20) +and the vector of parameters of the conditional distribution, cCc|C, satisfies +c(Cc|C) = +� +I� +k∈Cc(Mk+1) +� +k∈C e∗H +k +� +c(Cc � C) +� +fCc(θCc) +. +(21) +6 +Maximum Likelihood Estimation of MNNTS and an +Alternative Estimation Method +For θ1, θ2, . . . , θn, a random sample of univariate circular observations from an NNTS model +with M terms, the likelihood function is defined as +L(θ1, θ2, . . . , θn | c) = +n +� +k=1 +f(θk | c) = +n +� +k=1 +cHeeHc, +(22) +where c = (c0, c1, . . . , cM)⊤ and ek = (1, eiθk, e2iθk, . . . , eMiθk)⊤ for k = 1, 2, . . . , n. Given the +definition of an MNNTS density in terms of the Kronecker products in Equation 3 and the +definition of θ = (θ1, θ2, . . . , θn)⊤, the likelihood function in Equation 22 can be written as +L(θ1, θ2, . . . , θn | c) = fn(θ | c∗) = c∗H +� +n +� +k=1 +ekeH +k +� +c∗ = +n +� +k=1 +cH +� +n +� +k=1 +ekeH +k +� +n +� +k=1 +c +(23) +because of the independence of the observations in the random sample, c∗ = �n +k=1 c, with +the normalizing constraint, c∗Hc∗ = +� 1 +2π +�n. If the likelihood function is maximized subject to +9 + +the normalizing constraint, the maximum likelihood estimator of parameter vector c∗, ˆc∗ +ML, +is an eigenvector of matrix �n +k=1 ekeH +k that satisfies +ˆc∗ +ML ∝ +n +� +k=1 +ek, +(24) +implying �n +k=1 ˆcML ∝ �n +k=1 ek. Thus, the maximum likelihood is proportional to the Kro- +necker product of the n trigonometric moments vectors. This result can be easily extended to +a random sample of circular random vectors, θ1, θ2, . . . , θn. Fernández-Durán and Gregorio- +Domínguez (2010) developed a numerical algorithm to obtain the maximum likelihood es- +timators in the univariate and multivariate cases. Alternatively, based on Equation 24, a +new estimator can be proposed by considering the minimization of the sum of the squared +distances of the estimator to the vectors of trigonometric moments, i.e. +ˆcMD ∝ min +c∗∗ +n +� +k=1 +||c∗∗ − ek||2. +(25) +The solution for the estimator is based on minimizing the sum of squared distances, ˆcMD: +ˆcMD ∝ 1 +n +n +� +k=1 +ek. +(26) +Thus, estimator ˆcMD is proportional to the mean resultant of the vectors of the trigonometric +moments. This result can be easily extended to the multivariate case. In many simulation +experiments, we confirmed that for large sample sizes, n, ˆcMD, and ˆcML are very similar. +Clearly, ˆcMD is significantly easier to obtain than ˆcML because it only involves the calculation +of the mean resultant of the observed vectors of trigonometric moments and its subsequent +normalization. +7 +Example +We consider a dataset of wind directions observed at seven monitoring stations in Mexico +City and its neighboring urban areas. The monitoring stations are denoted as CHO, MER, +10 + +PED, SAG, TLA, VIF, and XAL. Figure 1 shows a map of the locations in terms of the +latitude and longitude of the seven meteorological monitoring stations. The hourly directions +were taken from 1:00 to 12:00 hours for the months of November to April which correspond +to the dry season from January, 1, 2013 to April, 30, 2020. There were a total of 2017 +days with all seven stations reporting non-missing values. For this example, we modeled +the dataset as a random sample from a 7-variate MNNTS distribution with vector M = +(3, 3, 3, 3, 3, 3, 3). For simplicity, we did not consider the time correlations among the wind +directions. The estimates of the c vector of the parameters were obtained by applying the +alternative algorithm explained in Section 6. Figure 2 shows the univariate histograms of the +directions and bivariate dispersion plots. The circular correlation coefficients (Agostinelli and +Lund, 2017) are also included in Figure 2. The pattern of the circular correlation coefficients +could be related to the altitude at which the monitoring stations are located and the location +of the mountains between the monitoring stations. PED and TLA are the two monitoring +stations located at the highest altitude and have negative circular correlations with almost +all other monitoring stations. +Table 1 lists the probabilities of the elements of the mixture defining the marginal NNTS +densities, as defined in Equation 13, for each of the seven components of the MNNTS density +obtained from the eigenvalues of the matrix in Equation 12. Based on the probability values, +because the first probability corresponding to the first eigenvalue is not close to one, it can +be inferred that each wind direction is dependent on at least one of the remaining six wind +directions. This was confirmed by the likelihood ratio tests of independence. +Figure 3 depicts the marginal densities and histograms for all seven wind directions. As +shown in this figure, the majority of the seven histograms are well-fitted by the marginal +fitted (NNTS) densities with M =3; for TLA, which is the case in which the fit is not good, +it could be necessary to increase the value of M to improve the fit. Figure 4 shows the +bivariate conditional densities and their respective marginal densities for the MER and SAG +11 + +wind directions, conditional on the other five wind directions (CHO, PED, TLA, VIF, and +XAL) being fixed at their mean resultant, median, and first and third quartiles. These were +selected because their circular correlation exhibited the largest positive correlation (0.43) +(see Figure 2). The conditional marginal density of MER changes significantly in shape, +contrary to that of SAG, which maintains the same shape presenting a translation to the +left. +Figure 5 shows the same bivariate conditional and corresponding marginal densities for +the MER and PED wind directions, which have the lowest correlation of -0.33 (see Figure +2). In this case, the shape of the conditional marginal density of MER changes considerably; +however; the conditional marginal density of PED maintains its shape and does not present a +translation. Even for the cases in which the conditioning is on the median and third quartile +values of CHO, SAG, TLA, VIF, and XAL, the bivariate conditional and marginal densities +appear to be very similar. +This example shows the flexibility of MNNTS models when applied to real situations. +We presented a case with only seven wind directions, but the methodology can be applied +in the same manner to a larger number of wind directions. +8 +Conclusion +Defining probability densities for multivariate circular data is a complicated task. +The +model developed by Fernández-Durán and Gregorio-Domínguez (2014) is a very flexible +model, and the derivation of marginal and conditional densities from the joint multivariate +density is important when applying this model, for example, in time-series and spatial and +spatiotemporal datasets involving circular random variables. In this study, the necessary +algorithms for obtaining the marginal and conditional densities of any number of components +of the joint vector were specified. The algorithms showed substantially good performance +12 + +when applied to high-dimensional real circular data. +Acknowledgements +The authors wish to thank the Asociación Mexicana de Cultura, A.C. for its support. +References +[1] Agostinelli, C. and Lund, U. (2017) R package circular: Circular Statistics (version +0.4-93), https://r-forge.r-project.org/projects/circular/ +[2] Fernández-Durán, J. J. (2004) Circular distributions based on nonnegative trigonometric +sums. Biometrics, 60, 499-503. +[3] Fernández-Durán, J. J. (2007) Models for circular-linear and circular-circular data con- +structed from circular distributions based on nonnegative trigonometric sums. Biomet- +rics, 63, 579-585. +[4] Fernández-Durán, J. J. and Gregorio-Domínguez, M. M. (2010) Maximum likelihood +estimation of nonnegative trigonometric sums models using a Newton-like algorithm on +manifolds. Electron. J. Stat., 4, 1402-1410. +[5] Fernández-Durán, J. J. and Gregorio-Domínguez, M. M. (2012) CircNNTSR: An R +package for the statistical analysis of circular data using nonnegative trigonometric +sums (NNTS) models. R package version 2.0. +http://CRAN.R-project.org/package=CircNNTSR +[6] Fernández-Durán, J. J. and Gregorio-Domínguez M. M. (2014) Modeling angles in pro- +teins and circular genomes using multivariate angular distributions based on nonnegative +trigonometric sums. Stat Appl Genet Mo B, 13(1), 1-18. +13 + +[7] Fernández-Durán, J. J. and Gregorio-Domínguez M. M. (2016) CircNNTSR: An R pack- +age for the statistical analysis of circular, multivariate circular, and spherical data using +nonnegative trigonometric sums. J. Stat. Softw., 70. +[8] Johnson, R. A. and T. Wehrly (1977) Measures and models for angular correlation and +angular-linear correlation. J. Roy. Stat. Soc. B, 39, 222-229. +[9] Kim, S., SenGupta, A. and Arnold, B. C. (2016) A multivariate circular distribution +with applications to the protein structure prediction problem. J. Multivar. Anal., 143, +374-382. +[10] Lennox, K. P., Dahl, D. B., Vannucci, M. and Tsai, J. W. (2009) Density estimation +for protein conformation angles using a bivariate von Mises distribution and Bayesian +nonparametrics. J. Am. Stat. Assoc, 104(486), 586-596. +[11] Mardia, K. (1975) Statistics of directional data (with discussion). J. Roy. Statist. Soc. +Ser. B, 37, 349-393. +[12] Mardia, K. V. and Jupp, P. E. (2000) Directional Statistics. Chichester, New York: +John Wiley and Sons. +[13] Mardia, K. V., Taylor, C. C. and Subramaniam, G. K. (2007) Protein bioinformatics and +mixtures of bivariate von Mises distributions for angular data. Biometrics, 63, 505-512. +[14] Mardia, K. V., Hughes, G., Taylor, C. C. and Singh, H. (2008) A multivariate von Mises +distribution with applications to bioinformatics. Can. J. Stat., 36(1), 99-109. +[15] Mardia, K. V., Kent, J. T., Zhang, Z., Taylor, C. C. and Hamelryck, T. (2012) Mixtures +of concentrated multivariate sine distributions with applications to bioinformatics. J. +Appl. Stat., 39(11), 2475-2492. +14 + +[16] Mardia, K. V. and Voss, J. (2014). Some fundamental properties of a multivariate von +Mises distribution. Commun. Stat-Theor. M., 43(6), 1132-1144. +[17] Nelsen, R. (1999) An Introduction to Copulas, Springer Verlag, New York. +[18] R Development Core Team (2020) R: A language and environment for statistical com- +puting. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R- +project.org/. +[19] Shieh, G. S. and Johnson, R. A. (2005) Inferences based on a bivariate distribution with +von Mises marginals. Ann. I. Stat. Math., 57, 789-802. +[20] Singh, H., Hnizdo, V. and Demchuk, E. (2002) Probabilistic model for two dependent +circular variables. Biometrika, 89-3, 719-723. +[21] Wehrly, T. and Johnson, R. A. (1980). Bivariate models for dependence of angular +observations and a related Markov process. Biometrika, 67, 255-256. +15 + +CHO +MER +PED +SAG +TLA +VIF +XAL +19.3 +19.4 +19.5 +19.6 +19.7 +−99.2 +−99.1 +−99.0 +−98.9 +−98.8 +Longitude +Latitude +Map tiles by Stamen Design, under CC BY 3.0. + Data by OpenStreetMap, under CC BY SA. +Figure 1: Map showing the locations of the seven meteorological monitoring stations (CHO, +MER, PED, SAG, TLA, VIF, and XAL) at which the hourly wind directions were recorded. +16 + +CHO +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +0.23 +MER +−0.10 +−0.33 +PED +0 +2 +4 +6 +0 +2 +4 +6 +0.19 +0.43 +−0.28 +SAG +−0.11 +−0.31 +0.12 +−0.25 +TLA +0 +2 +4 +6 +0 +2 +4 +6 +0.02 +0.14 +−0.20 +0.17 +0.14 +VIF +0 +2 +4 +6 +0.08 +0.11 +0 +2 +4 +6 +−0.17 +0.20 +0 +2 +4 +6 +−0.31 +−0.03 +0 +2 +4 +6 +0 +2 +4 +6 +XAL +Figure 2: Lower diagonal half shows the bivariate dispersion plots. The histograms of each +wind direction are shown in this half. The upper diagonal half shows the values of the circular +correlation. +17 + +Mixing Probabilities +CHO +MER +PED +SAG +TLA +VIF +XAL +0.8307 +0.7009 +0.7849 +0.7674 +0.7567 +0.7818 +0.7338 +0.1027 +0.1803 +0.1177 +0.1569 +0.1722 +0.1611 +0.1684 +0.0489 +0.0817 +0.0618 +0.0563 +0.0429 +0.0415 +0.0723 +0.0177 +0.0370 +0.0356 +0.0193 +0.0282 +0.0156 +0.0254 +Table 1: Wind direction data: mixing probabilities defining the marginal densities of the +seven wind directions (Equation 13). The mixing probabilities correspond to the eigenvalues +of the matrix in Equation 12. +CHO +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +MER +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +PED +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +SAG +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +TLA +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +VIF +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +XAL +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +CHO,MER,PED,SAG, + TLA,VIF,XAL +Wind Direction +Density +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Figure 3: Wind direction data: marginal densities of the seven wind directions that corre- +spond to the mixtures of univariate NNTS densities. +18 + +SAG +MER +Density +Mean Resultant + CHO=2.66,PED=3.04,TLA=4.35, + VIF=3.81, XAL=2.14 +SAG +MER +Density +First Quartile + CHO=1.75,PED=1.33,TLA=3.75, + VIF=1.13,XAL=0.72 +SAG +MER +Density +Median + CHO=2.23,PED=3.65,TLA=4.77, + VIF=4.90,XAL=1.66 +SAG +MER +Density +Third Quartile + CHO=2.84,PED=4.01,TLA=5.59, + VIF=5.57,XAL=3.02 +Figure 4: Wind direction data: conditional bivariate and corresponding marginal univariate +densities for the MER and SAG wind directions. These two wind directions exhibit the +largest positive circular correlation (0.43). +19 + +PED +MER +Density +Mean Resultant + CHO=2.66,SAG=1.97,TLA=4.35, + VIF=3.81, XAL=2.14 +PED +MER +Density +First Quartile + CHO=1.75,SAG=0.51,TLA=3.75, + VIF=1.13,XAL=0.72 +PED +MER +Density +Median + CHO=2.23,SAG=1.22,TLA=4.77, + VIF=4.90,XAL=1.66 +PED +MER +Density +Third Quartile + CHO=2.84,SAG=3.00,TLA=5.59, + VIF=5.57,XAL=3.02 +Figure 5: Wind direction data: conditional bivariate and corresponding marginal univariate +densities for the MER and PED wind directions. These two wind directions exhibit the +largest negative circular correlation (-0.33). +20 + diff --git a/a9E2T4oBgHgl3EQfFQYC/content/tmp_files/load_file.txt b/a9E2T4oBgHgl3EQfFQYC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d245c39f98b84829976fef64596d097c0773f6 --- /dev/null +++ b/a9E2T4oBgHgl3EQfFQYC/content/tmp_files/load_file.txt @@ -0,0 +1,575 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf,len=574 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='03643v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='ME] 9 Jan 2023 Multivariate Nonnegative Trigonometric Sums Distributions for High-Dimensional Multivariate Circular Data Fernández-Durán, Juan Joséa and María Mercedes Gregorio-Domínguezb∗ abITAM Río Hondo No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 1, Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Progreso Tizapán, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 01080, CDMX México E-mail: jfdez@itam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='mx ∗ Corresponding Author: E-mail: mercedes@itam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='mx Abstract Fernández-Durán and Gregorio-Domínguez (2014) defined a family of probability distributions for a vector of circular random variables by considering multiple non- negative trigonometric sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' These distributions are highly flexible and can present many modes and skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Several operations on these multivariate distributions are translated into operations on the vector of parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' for example, marginalization involves the calculation of the eigenvectors and eigenvalues of a matrix and, indepen- dence among subsets of the elements of the vector of circular variables translates to a case in which the vector of parameters is the Kronecker product of the corresponding subsets of the vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' An alternative parameter estimation algorithm for high-dimensional circular data is presented and applied to a real dataset on wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Keywords: Vector of circular random variables, Independence, Marginalization, Condi- tional distribution, Kronecker product 1 1 Introduction The main objective of this study is to consider the properties of the multivariate nonnegative trigonometric sums (MNNTS) distributions developed by Fernández-Durán and Gregorio- Domínguez (2014), such as conditional and marginal distributions and the conditions for independence among subsets of the vector of circular random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' A circular (angular) random variable is defined as one where the support of its probability density function is the unit circle, and it must be a function of period 2π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', if f(θ) represents the density function of a circular random variable, θ, then, f(θ + 2kπ) = f(θ), where k is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' A multivariate circular random vector is a vector in which each component is a circular random variable, θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn)⊤, where θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn are circular random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Multivariate circular data are available in many disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The dihedral angles in a protein, wind directions recorded at different monitoring stations, time of occurrence of dif- ferent diseases, and time of flowering of different plant species are some examples among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Note that in many applications, multivariate circular data consist of the time of occurrence of different events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The support of the domain of the distribution of a multivariate circular random vector is a hypertorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The bivariate von Mises model was developed by Mardia (1975a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' see also Mardia and Jupp, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Johnson and Wehrly (1977) and Wehrly and Johnson (1980) presented bivariate circular models based on a decompo- sition of the bivariate cumulative distribution that is equivalent to a copula (Nelsen, 1999) used by Fernández-Durán (2007) to construct bivariate circular and circular-linear proba- bility models based on univariate nonnegative trigonometric sums (Fernández-Durán, 2004 and Fernández-Durán and Gregorio-Domínguez, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Other bivariate circular models are presented by Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2002), Lennox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2009), Mardia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2007) and Shieh and Johnson (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In the multivariate circular case, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2016) extended the bivari- ate copula decomposition of Johnson and Wehrly (1977) and Wehrly and Johnson (1980) 2 to the multivariate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Mardia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2012) considered mixtures of multivariate circu- lar distributions that are extensions of the univariate von Mises model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Another model that is an extension of the univariate von Mises model is presented by Mardia and Voss (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Fernández-Durán and Gregorio-Domínguez (2014) extended the univariate nonneg- ative trigonometric model to the multivariate case in which the parameters are estimated by maximum likelihood (see Fernández-Durán and Gregorio-Domínguez, 2010) and com- putationally implemented using the R software (R Development Core Team, 2020) in the CircNNTSR library (Fernández-Durán and Gregorio-Domínguez, 2012 and 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' We refer to this family of multivariate circular distributions as the MNNTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The MNNTS probability density function for multivariate circular random vectors is defined as f12···n(θ) = ||cHe||2 = cHeeHc (1) = M1 � k1=0 M2 � k2=0 · · Mn � kn=0 M1 � m1=0 M2 � m2=0 · · Mn � mn=0 ck1k2···kn¯cm1m2···mne �n s=1 i(ks−ms)θs, where i = √−1, c = cR+icI, cR and cI are the real and imaginary parts of complex number c, and ¯c = cR − icI is the conjugate of complex number c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The complex vector, c, is the vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The complex vector, e, contains the multivariate trigonometric moments defined as e �n s=1 rsθs for integer values r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Both vectors have the dimension of �n s=1(Ms + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The vector of parameters, c, must satisfy the following constraint: ||c||2 = M1 � k1=0 M2 � k2=0 · · Mn � kn=0 ||ck1k2···kn||2 = 1 (2π)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2) Given this constraint, the first element of the c vector, c00···0, is a nonnegative real number, and the parameter space is a complex hypersphere having the dimension of �s s=1(Mn + 1), which is isomorphic to a real hypersphere having the dimension of 2 �s s=1(Mn + 1) − 1 by taking the real and imaginary parts of the complex numbers in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Given the definition of the multivariate density function in Equation 2, it is easy to obtain the expression for the marginal distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' however, it is not clear what the c parameters 3 of the marginal distributions of any dimension are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In addition, the conditional densities of a subset of the circular variables, given other subsets of the joint vector, are not clear at the first instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Section 2 presents alternative ways to write the density function in Equation 2 that are more adequate for obtaining the parameters of marginal and conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Section 3 describes the development of marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Section 4 provides the conditions for independence among the elements of the multivariate circular vector, and Section 5 details the derivation of the conditional distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The derivation of the marginal and conditional distributions is developed for the joint bivariate cases, although their generalization to any dimension is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In Section 6, we propose an efficient algorithm to estimate the parameters of high-dimensional circular data as an alternative to maximum likelihood estimation that, in the case of high-dimensional circular data, could have the limitations of slow or non-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Section 7 presents the application of the results described in the previous sections to a real dataset on wind directions for a monitoring network in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Finally, Section 8 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 2 MNNTS Distributions Consider a vector of circular random variables, θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn)⊤, that is distributed as an MNNTS distribution (Fernández-Durán, 2007 and Fernández-Durán and Gregorio- Domínguez, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Then, to perform efficient (numerical) calculations, the density function of θ in Equation 2 can be written in terms of Kronecker products as f12···n(θ) = cHeeHc = cH � n � s=1 es n � s=1 eH s � c = cH � n � s=1 eseH s � c, (3) where es = (1, eiθs, e2iθs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , eMsiθs)⊤ is the vector of the trigonometric moments of the s-th circular random variable in the random vector, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result is based on the successive 4 applications of the following property of Kronecker products: (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD), (4) where A, B, C, and D are general matrices for which the products are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Note that this expression can also be used in computational programs to perform numerical calculations more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Furthermore, the set of indexes of the vector of parameters, c, is obtained using the Kronecker product, (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , M1) �(0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , M2) � · · ·�(0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , Mn), for an n- dimensional MNNTS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The order of the elements of circular vector θ can be modified to simplify the derivations presented in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 3 Marginal Distributions The expression for the marginal distribution of any dimension of an MNNTS distribution is obtained by integrating the marginalizing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' For the bivariate joint MNNTS distribution in terms of the sum, f1(θ1) = � 2π 0 f12(θ1, θ2)dθ2 = � 2π 0 M1 � k1=0 M2 � k2=0 M1 � m1=0 M2 � m2=0 c(12) k1k2¯c(12) m1m2ei(k1−m1)θ1+i(k2−m2)θ2dθ2 = M1 � k1=0 M2 � k2=0 M1 � m1=0 M2 � m2=0 c(12) k1k2¯c(12) m1m2ei(k1−m1)θ1 � 2π 0 ei(k2−m2)θ2dθ2 = 2π M1 � k1=0 M1 � m1=0 � M2 � m2=0 c(12) k1m2¯c(12) m1m2 � ei(k1−m1)θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (5) Then, if the general sum form of the marginal distribution of θ1 is f1(θ1) = M1 � k1=1 M1 � m1=1 c(1) k1 ¯c(1) m1ei(k1−m1)θ1, (6) we can obtain c(1) k1 ¯c(1) m1 = 2π M2 � m2=0 c(12) k1m2¯c(12) m1m2, (7) where c(1) is the vector of the parameters of marginal density f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The expression for f1(θ1) is easy to obtain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' however, obtaining the expression of the vector of the parameters of the 5 univariate distribution, c(1) = (c(1) 0 , c(1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(1) M1)⊤, in terms of the bivariate distribution vec- tor of parameters, c(12) = (c(12) 00 , c(12) 01 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(12) 0M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(12) M10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(12) M1M2)⊤, is a more difficult task that requires the use of matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Considering the joint bivariate MNNTS distribution for vector θ = (θ1, θ2)⊤, we obtain the marginal distribution of θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' f1(θ1) = � 2π 0 f12(θ1, θ2)dθ2 = � 2π 0 c(12)H � (e1eH 1 ) ⊗ (e2eH 2 ) � c(12)dθ2 = c(12)H � (e1eH 1 ) ⊗ �� 2π 0 e2eH 2 dθ2 �� c(12) (8) Thus, f1(θ1) = 2πc(12)H � (e1eH 1 ) ⊗ IM2+1 � c(12) = 2πc(12)H ((e1 ⊗ IM2+1) � eH 1 ⊗ IM2+1 � c(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (9) If the bivariate MNNTS vector parameter is written as c(12) = (c(12) 0 , c(12) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(12) M2)⊤ (10) with c(12) m2 = (c(12) 0m2, c(12) 1m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , c(12) M1m2)⊤ for m2 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , M2, then f1(θ1) = 2πc(12)H ((e1 ⊗ IM2+1) � eH 1 ⊗ IM2+1 � c(12) = 2π M2 � m2=0 c(12)H m2 e1eH 1 c(12) m2 = 2πeH 1 � M2 � m2=0 c(12) m2c(12)H m2 � e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (11) The spectral decomposition of C•2 = 2π �M2 m2=0 c(12) m2c(12)H m2 satisfies C•2 = M2 � m2=0 pm2c∗(12) m2 c∗(12)H m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (12) Then, the marginal distribution of θ1 is a mixture of the univariate NNTS densities: f1(θ1) = eH 1 � M2 � m2=0 pm2c∗(12) m2 c∗(12)H m2 � e1 = M2 � m2=0 pm2c∗(12)H m2 e1eH 1 c(12)∗ m2 (13) where the mixture probabilities, p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', pM2, and mixture parameter vectors, c∗(12) 0 , c∗(12) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', c∗ M2, correspond to the eigenvalues and eigenvectors of matrix C•2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result gener- alizes to the marginals of any dimension of an MNNTS distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', if θ = (θR, θRc)⊤, 6 where R is the set of indexes of the circular random variables for which we need its marginal MNNTS distribution and Rc is the complement set of R, then the marginal distribution of θR is a mixture of MNNTS distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 4 Independence In the simplest case of a bivariate MNNTS distribution for θ = (θ1, θ2)⊤, the circular random variables, θ1 and θ2, are independent if and only if the joint vector of parameters, c(12), is the Kronecker product of the marginal vector of parameters c(1) and c(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', c(12) = c(1) ⊗ c(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The proof uses the result of Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' If c(12) = c(1) ⊗ c(2), f12(θ1, θ2) = c(12)H � (e1eH 1 ) ⊗ (e2eH 2 ) � c(12) = (c(1)⊗c(2))H � (e1eH 1 ) ⊗ (e2eH 2 ) � (c(1)⊗c(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (14) Repeated applications of the property of Kronecker products in Equation 4 obtain f12(θ1, θ2) = � (c(1)H(e1eH 1 )) ⊗ (c(2)H(e2eH 2 )) � (c(1)⊗c(2)) = � c(1)H(e1eH 1 )c(1)� ⊗ � c(2)H(e2eH 2 )c(2)� , (15) which is equal to f1(θ1)f2(θ2) because the Kronecker product of the two scalars is equal to a simple product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The proof that if θ1 and θ2 are independent, then the joint parameter vector is the product of the marginal parameter vectors is obtained from the previous proof starting from the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result can be generalized to MNNTS of any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Let θ = (θR, θRc)⊤, then θR is independent of θRc if and only if c(R � Rc) = c(R) ⊗ c(Rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Using this result, we can construct a likelihood ratio test for independence to determine the inde- pendence among relevant subsets of the vector of circular random variables, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In terms of the decomposition of the univariate marginal distribution as a mixture of distributions in Equation 13, θR is independent of θRc when there is only one element in the mixture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', the eigenvalues (mixing probabilities) are equal to zero except for the first, which is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 7 Based on the definition of the MNNTS density, it is clear that the elements of the vector of circular random variables, θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn)⊤, are exchangeable if all the components of the vector of the number of terms in the sum, M = (M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , Mn)⊤, are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', Mk = M for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 5 Conditional Distributions The conditional distributions of an MNNTS distribution are also MNNTS distributions, that is, the MNNTS family of distributions is closed under conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' For the bivariate case, the conditional distribution of θ1, given θ2 = θ∗ 2, is obtained as follows: f1|2(θ1 | θ2 = θ∗ 2) = f12(θ1, θ∗ 2) f2(θ∗ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (16) If f12(θ1, θ∗ 2) for a fixed value of θ2 = θ∗ 2, f12(θ1, θ∗ 2) = c(12)H � (e1eH 1 ) ⊗ (e∗ 2e∗H 2 ) � c(12) = c(12)H � (e1eH 1 ) ⊗ (e∗ 2e∗H 2 ) � c(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (17) Because for an m×n A and a p×q B matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' A⊗B = (A⊗Ip)(In ⊗B) = (Im ⊗B)(A⊗Iq) where Is denotes the s-by-s identity matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' f12(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' θ∗ 2) = c(12)H �� (e1eH 1 ) ⊗ IM2+1 � � IM1+1 ⊗ (e∗ 2e∗H 2 ) �� c(12) = c(12)H �� (e1eH 1 ) ⊗ (IM2+1IM2+1) � � (IM1+1IM1+1) ⊗ (e∗ 2e∗H 2 ) �� c(12) = c(12)H � (e1 ⊗ IM2+1)(eH 1 ⊗ IM2+1)(IM1+1 ⊗ e∗ 2)(IM1+1 ⊗ e∗H 2 ) � c(12) = c(12)H � (e1 ⊗ IM2+1)(e∗ 2 ⊗ 1)(1 ⊗ eH 1 )(IM1+1 ⊗ e∗H 2 ) � c(12) = c(12)H � (IM1+1 ⊗ e∗ 2)(1 ⊗ e1)(1 ⊗ eH 1 )(IM1+1 ⊗ e∗H 2 ) � c(12) = c∗(12)H(e1eH 1 )c∗(12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (18) where c∗(12) = (IM1+1 ⊗ e∗H 2 )c(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Then, the vector of parameters of the conditional distri- bution, c(1|2), satisfies c(1|2) = c(12)∗ � f2(θ∗ 2) = (IM1+1 ⊗ e∗H 2 )c(12) � f2(θ∗ 2) (19) 8 because � f2(θ∗ 2) is the constant of proportionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Geometrically, the conditional parameter vector, c(1|2), is obtained by rotating the joint parameter vector by θ∗ 2 through e∗H 2 and then normalizing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result can be generalized to the multivariate case by considering θ = (θCc, θC)⊤, where C is the set of indexes of the conditioning circular variables, given the known values, θC = θ∗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The conditional distribution of θCc, given θC = θ∗ C, is equal to fCc|C(θCc | θC = θ∗ C) = c(Cc � C)H �� I� k∈Cc(Mk+1) � k∈C e∗ k � �� k∈Cc(ekeH k ) � � I� k∈Cc(Mk+1) � k∈C e∗H k �� c(Cc � C) fCc(θCc) , (20) and the vector of parameters of the conditional distribution, cCc|C, satisfies c(Cc|C) = � I� k∈Cc(Mk+1) � k∈C e∗H k � c(Cc � C) � fCc(θCc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (21) 6 Maximum Likelihood Estimation of MNNTS and an Alternative Estimation Method For θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn, a random sample of univariate circular observations from an NNTS model with M terms, the likelihood function is defined as L(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn | c) = n � k=1 f(θk | c) = n � k=1 cHeeHc, (22) where c = (c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , cM)⊤ and ek = (1, eiθk, e2iθk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , eMiθk)⊤ for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Given the definition of an MNNTS density in terms of the Kronecker products in Equation 3 and the definition of θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn)⊤, the likelihood function in Equation 22 can be written as L(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn | c) = fn(θ | c∗) = c∗H � n � k=1 ekeH k � c∗ = n � k=1 cH � n � k=1 ekeH k � n � k=1 c (23) because of the independence of the observations in the random sample, c∗ = �n k=1 c, with the normalizing constraint, c∗Hc∗ = � 1 2π �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' If the likelihood function is maximized subject to 9 the normalizing constraint, the maximum likelihood estimator of parameter vector c∗, ˆc∗ ML, is an eigenvector of matrix �n k=1 ekeH k that satisfies ˆc∗ ML ∝ n � k=1 ek, (24) implying �n k=1 ˆcML ∝ �n k=1 ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Thus, the maximum likelihood is proportional to the Kro- necker product of the n trigonometric moments vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result can be easily extended to a random sample of circular random vectors, θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' , θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Fernández-Durán and Gregorio- Domínguez (2010) developed a numerical algorithm to obtain the maximum likelihood es- timators in the univariate and multivariate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Alternatively, based on Equation 24, a new estimator can be proposed by considering the minimization of the sum of the squared distances of the estimator to the vectors of trigonometric moments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' ˆcMD ∝ min c∗∗ n � k=1 ||c∗∗ − ek||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (25) The solution for the estimator is based on minimizing the sum of squared distances, ˆcMD: ˆcMD ∝ 1 n n � k=1 ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (26) Thus, estimator ˆcMD is proportional to the mean resultant of the vectors of the trigonometric moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This result can be easily extended to the multivariate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In many simulation experiments, we confirmed that for large sample sizes, n, ˆcMD, and ˆcML are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Clearly, ˆcMD is significantly easier to obtain than ˆcML because it only involves the calculation of the mean resultant of the observed vectors of trigonometric moments and its subsequent normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 7 Example We consider a dataset of wind directions observed at seven monitoring stations in Mexico City and its neighboring urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The monitoring stations are denoted as CHO, MER, 10 PED, SAG, TLA, VIF, and XAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 1 shows a map of the locations in terms of the latitude and longitude of the seven meteorological monitoring stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The hourly directions were taken from 1:00 to 12:00 hours for the months of November to April which correspond to the dry season from January, 1, 2013 to April, 30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' There were a total of 2017 days with all seven stations reporting non-missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' For this example, we modeled the dataset as a random sample from a 7-variate MNNTS distribution with vector M = (3, 3, 3, 3, 3, 3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' For simplicity, we did not consider the time correlations among the wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The estimates of the c vector of the parameters were obtained by applying the alternative algorithm explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 2 shows the univariate histograms of the directions and bivariate dispersion plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The circular correlation coefficients (Agostinelli and Lund, 2017) are also included in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The pattern of the circular correlation coefficients could be related to the altitude at which the monitoring stations are located and the location of the mountains between the monitoring stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' PED and TLA are the two monitoring stations located at the highest altitude and have negative circular correlations with almost all other monitoring stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Table 1 lists the probabilities of the elements of the mixture defining the marginal NNTS densities, as defined in Equation 13, for each of the seven components of the MNNTS density obtained from the eigenvalues of the matrix in Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Based on the probability values, because the first probability corresponding to the first eigenvalue is not close to one, it can be inferred that each wind direction is dependent on at least one of the remaining six wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This was confirmed by the likelihood ratio tests of independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 3 depicts the marginal densities and histograms for all seven wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' As shown in this figure, the majority of the seven histograms are well-fitted by the marginal fitted (NNTS) densities with M =3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' for TLA, which is the case in which the fit is not good, it could be necessary to increase the value of M to improve the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 4 shows the bivariate conditional densities and their respective marginal densities for the MER and SAG 11 wind directions, conditional on the other five wind directions (CHO, PED, TLA, VIF, and XAL) being fixed at their mean resultant, median, and first and third quartiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' These were selected because their circular correlation exhibited the largest positive correlation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='43) (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The conditional marginal density of MER changes significantly in shape, contrary to that of SAG, which maintains the same shape presenting a translation to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 5 shows the same bivariate conditional and corresponding marginal densities for the MER and PED wind directions, which have the lowest correlation of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='33 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In this case, the shape of the conditional marginal density of MER changes considerably;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' the conditional marginal density of PED maintains its shape and does not present a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Even for the cases in which the conditioning is on the median and third quartile values of CHO, SAG, TLA, VIF, and XAL, the bivariate conditional and marginal densities appear to be very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' This example shows the flexibility of MNNTS models when applied to real situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' We presented a case with only seven wind directions, but the methodology can be applied in the same manner to a larger number of wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 8 Conclusion Defining probability densities for multivariate circular data is a complicated task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The model developed by Fernández-Durán and Gregorio-Domínguez (2014) is a very flexible model, and the derivation of marginal and conditional densities from the joint multivariate density is important when applying this model, for example, in time-series and spatial and spatiotemporal datasets involving circular random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' In this study, the necessary algorithms for obtaining the marginal and conditional densities of any number of components of the joint vector were specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The algorithms showed substantially good performance 12 when applied to high-dimensional real circular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Acknowledgements The authors wish to thank the Asociación Mexicana de Cultura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' for its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' References [1] Agostinelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' and Lund, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2017) R package circular: Circular Statistics (version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='4-93), https://r-forge.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', 57, 789-802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' [20] Singh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=', Hnizdo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' and Demchuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (2002) Probabilistic model for two dependent circular variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Biometrika, 89-3, 719-723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' [21] Wehrly, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' and Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Bivariate models for dependence of angular observations and a related Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Biometrika, 67, 255-256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 15 CHO MER PED SAG TLA VIF XAL 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7 −99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='2 −99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='1 −99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0 −98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='9 −98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='8 Longitude Latitude Map tiles by Stamen Design, under CC BY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Data by OpenStreetMap, under CC BY SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' Figure 1: Map showing the locations of the seven meteorological monitoring stations (CHO, MER, PED, SAG, TLA, VIF, and XAL) at which the hourly wind directions were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 16 CHO 0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='23 MER −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='33 PED 0 2 4 6 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='28 SAG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='25 TLA 0 2 4 6 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='14 VIF 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='11 0 2 4 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='20 0 2 4 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='03 0 2 4 6 0 2 4 6 XAL Figure 2: Lower diagonal half shows the bivariate dispersion plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The histograms of each wind direction are shown in this half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The upper diagonal half shows the values of the circular correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 17 Mixing Probabilities CHO MER PED SAG TLA VIF XAL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='8307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7567 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0254 Table 1: Wind direction data: mixing probabilities defining the marginal densities of the seven wind directions (Equation 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' The mixing probabilities correspond to the eigenvalues of the matrix in Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' CHO Wind Direction Density 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='0 0.' metadata={'source': 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+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='7 Figure 3: Wind direction data: marginal densities of the seven wind directions that corre- spond to the mixtures of univariate NNTS densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 18 SAG MER Density Mean Resultant CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='66,PED=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='04,TLA=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='35, VIF=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='81, XAL=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='14 SAG MER Density First Quartile CHO=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='75,PED=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='33,TLA=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='75, VIF=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='13,XAL=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='72 SAG MER Density Median CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='23,PED=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='65,TLA=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='77, VIF=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='90,XAL=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='66 SAG MER Density Third Quartile CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='84,PED=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='01,TLA=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='59, VIF=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='57,XAL=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='02 Figure 4: Wind direction data: conditional bivariate and corresponding marginal univariate densities for the MER and SAG wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' These two wind directions exhibit the largest positive circular correlation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 19 PED MER Density Mean Resultant CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='66,SAG=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='97,TLA=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='35, VIF=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='81, XAL=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='14 PED MER Density First Quartile CHO=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='75,SAG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='51,TLA=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='75, VIF=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='13,XAL=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='72 PED MER Density Median CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='23,SAG=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='22,TLA=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='77, VIF=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='90,XAL=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='66 PED MER Density Third Quartile CHO=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='84,SAG=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='00,TLA=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='59, VIF=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='57,XAL=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='02 Figure 5: Wind direction data: conditional bivariate and corresponding marginal univariate densities for the MER and PED wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' These two wind directions exhibit the largest negative circular correlation (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E2T4oBgHgl3EQfFQYC/content/2301.03643v1.pdf'} diff --git a/a9E5T4oBgHgl3EQfew9G/content/tmp_files/2301.05621v1.pdf.txt b/a9E5T4oBgHgl3EQfew9G/content/tmp_files/2301.05621v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a3223b1ab2571fdf851f243ff36ef2033acc19a --- /dev/null +++ b/a9E5T4oBgHgl3EQfew9G/content/tmp_files/2301.05621v1.pdf.txt @@ -0,0 +1,1545 @@ +arXiv:2301.05621v1 [math-ph] 13 Jan 2023 +Universality in low-dimensional BCS theory +Joscha Henheik∗ +Asbjørn Bækgaard Lauritsen† +Barbara Roos‡ +Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria +16 January 2023 +Abstract +It is a remarkable property of BCS theory that the ratio of the energy gap at zero +temperature Ξ and the critical temperature Tc is (approximately) given by a univer- +sal constant, independent of the microscopic details of the fermionic interaction. This +universality has rigorously been proven quite recently in three spatial dimensions and +three different limiting regimes: weak coupling, low density, and high density. The goal +of this short note is to extend the universal behavior to lower dimensions d = 1, 2 and +give an exemplary proof in the weak coupling limit. +Keywords: +BCS theory, energy gap, critical temperature, BCS universality. +Mathematics subject classification: +81Q10, 46N50, 82D55 +1 +Introduction +The Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [BCS57] is governed by +the BCS gap equation. For translation invariant systems without external fields the BCS gap +equation is +∆(p) = − +1 +(2π)d/2 +� +Rd +ˆV (p − q) ∆(q) +E∆(p) tanh +�E∆(p) +2T +� +dq +(1.1) +with dispersion relation E∆(p) = +� +(p2 − µ)2 + |∆(p)|2. Here, T ≥ 0 denotes the temperature +and µ > 0 the chemical potential. +We consider dimensions d ∈ { 1, 2, 3 }. +The Fourier +transform of the potential V ∈ L1(Rd) ∩ LpV (Rd) (with a d-dependent pV ≥ 1 to be specified +below), modeling their effective interaction, is denoted by ˆV (p) = (2π)−d/2 � +Rd V (x)e−ip·xdx. +According to BCS theory, a system is in a superconducting state, if there exists a non- +zero solution ∆ to the gap equation (1.1). The question of existence of such a non-trivial +solution ∆ hinges, in particular, on the temperature T. It turns out, there exists a critical +temperature Tc ≥ 0 such that for T < Tc there exists a non-trivial solution, and for T ≥ Tc it +∗joscha.henheik@ist.ac.at +†alaurits@ist.ac.at +‡barbara.roos@ist.ac.at +1 + +does not [HHSS08, Theorem 1.3 and Definition 1.4]. This critical temperature is one of the +key (physically measurable) quantities of the theory and its asymptotic behavior, in three +spatial dimensions, has been studied in three physically rather different limiting regimes: In +a weak-coupling limit (i.e. replacing V → λV and taking λ → 0) [FHNS07; HS08b], in a +low-density limit (i.e. µ → 0) [HS08a], and in a high-density limit (i.e. µ → ∞) [Hen22]. +As already indicated above, at zero temperature, the function E∆ may be interpreted as +the dispersion relation of a certain ‘approximate’ Hamiltonian of the quantum many-body +system, see [HHSS08, Appendix A]. In particular +Ξ := inf +p∈Rd E∆(p) +(1.2) +has the interpretation of an energy gap associated with the approximate BCS Hamiltonian +and as such represents a second key quantity of the theory. +Analogously to the critical +temperature, the asymptotic behavior of this energy gap, again in three spatial dimensions, +has been studied in the same three different limiting regimes: In a weak coupling limit +[HS08b], in a low density limit [Lau21], and in a high density limit [HL22]. +In this paper, we focus on a remarkable feature of BCS theory, which is well known +in the physics literature [BCS57; LTB19; NS85]: The ratio of the energy gap Ξ and criti- +cal temperature Tc tends to a universal constant, independent of the microscopic details of +the interaction between the fermions, i.e. the potential V . More precisely, in three spatial +dimension, it holds that +Ξ +Tc +≈ π +eγ ≈ 1.76 , +(1.3) +where γ ≈ 0.577 is the Euler-Mascheroni constant, in each of the three physically very +different limits mentioned above. This result follows as a limiting equality by combining +asymptotic formulas for the critical temperature Tc (see [FHNS07; HS08a; HS08b; Hen22]) +and the energy gap Ξ (see [HS08b; HL22; Lau21]) in the three different regimes. Although +these scenarios (weak coupling, low density, and high density) are physically rather different, +they all have in common that ‘superconductivity is weak’ and one can hence derive an +asymptotic formula for Tc and Ξ as they depart from being zero (in the extreme cases λ = 0, +µ = 0, µ = ∞, respectively). However, all the asymptotic expressions are not perturbative, +as they depend exponentially on the natural dimensionless small parameter in the respective +limit. We refer to the above mentioned original works for details. +The goal of this note is to prove the same universal behavior (1.3), which has already +been established in three spatial dimension, also in dimensions d = 1, 2 in the weak coupling +limit (i.e. replacing V → λV and taking λ → 0). This situation serves as a showcase for +the methods involved in the proofs of the various limits in three dimensions (see Remark 3.5 +and Remark 3.8 below). Apart from the mathematical curiosity in d = 1, 2, there have been +recent studies in lower-dimensional superconductors in the physics literature, out of which we +mention one-dimensional superconducting nanowires [NTH12] and two-dimensional ‘magic +angle’ graphene [Cao+18]. +In the remainder of this introduction, we briefly recall the mathematical formulation of +BCS theory, which has been developed mostly by Hainzl and Seiringer, but also other co- +authors [FHNS07; HHSS08; HS16]. Apart from the universality discussed here, also many +other properties of BCS theory have been shown using this formulation: Most prominently, +2 + +Ginzburg-Landau theory, as an effective theory describing superconductors close to the crit- +ical temperature, has been derived from BCS theory [DHM22a; DHM22b; FHSS12; FL16]. +More recently, it has been shown that the effect of boundary superconductivity occurs in the +BCS model [HRS22]. We refer to [HS16] for a more comprehensive review of developments in +the mathematical formulation of BCS theory. The universal behavior in the weak coupling +limit for lower dimensions d = 1, 2 is presented in Section 2. Finally, in Section 3, we provide +the proofs of the statements from Section 2. +1.1 +Mathematical formulation of BCS theory +We will now briefly recall the mathematical formulation [HHSS08; HS16] of BCS theory +[BCS57], which is an effective theory developed for describing superconductivity of a fermionic +gas. In the following, we consider these fermions in Rd, d = 1, 2, at temperature T ≥ 0 and +chemical potential µ ∈ R, interacting via a two-body potential V , for which we assume the +following. +Assumption 1.1. We have that V is real-valued, reflection symmetric, i.e. V (x) = V (−x) +for all x ∈ Rd, and it satisfies V ∈ LpV (Rd), where pV = 1 if d = 1, pV ∈ (1, ∞) if d = 2. +Moreover, we neglect external fields, in which case the system is translation invariant. +The central object in the mathematical formulation of the theory is the BCS functional, +which can naturally be viewed as a function of BCS states Γ. These states are given by a +pair of functions (γ, α) and can be conveniently represented as a 2 × 2 matrix valued Fourier +multiplier on L2(Rd) ⊕ L2(Rd) of the form +ˆΓ(p) = +�ˆγ(p) +ˆα(p) +ˆα(p) +1 − ˆγ(p) +� +(1.4) +for all p ∈ Rd. In (1.4), ˆγ(p) denotes the Fourier transform of the one particle density matrix +and ˆα(p) is the Fourier transform of the Cooper pair wave function. We require reflection +symmetry of ˆα, i.e. ˆα(−p) = ˆα(p), as well as 0 ≤ ˆΓ(p) ≤ 1 as a matrix. +The BCS free energy functional takes the form +FT[Γ] := +� +Rd(p2 − µ)ˆγ(p)dp − TS[Γ] + +� +Rd V (x)|α(x)|2dx , +Γ ∈ D, +(1.5) +D := +� +ˆΓ(p) = +�ˆγ(p) +ˆα(p) +ˆα(p) +1 − ˆγ(p) +� +: 0 ≤ ˆΓ ≤ 1 , ˆγ ∈ L1(Rd, (1 + p2)dp) , α ∈ H1 +sym(Rd) +� +, +where the entropy density is defined as +S[Γ] = − +� +Rd TrC2 +� +ˆΓ(p) log ˆΓ(p) +� +dp . +The minimization problem associated with (1.5) is well defined. In fact, the following re- +sult has only been proven for d = 3 and V ∈ L3/2(R3), but its extension to d = 1, 2 is +straightforward. +3 + +Proposition 1.2 ([HHSS08], see also [HS16]). Under Assumption 1.1 on V , the BCS free +energy is bounded below on D and attains its minimum. +The BCS gap equation (1.1) arises as the Euler–Lagrange equations of this functional [HHSS08]. +Namely by defining ∆ = −2 � +V α, the Euler–Lagrange equation for α takes the form of the +BCS gap equation (1.1). Additionally, one has the following linear criterion for the BCS gap +equation to have non-trivial solutions. Again, so far, a proof has only been given in spatial +dimension d = 3 and for V ∈ L3/2(R3), but its extension to d = 1, 2 is straightforward. +Theorem 1.3 ([HHSS08, Thm. 1]). Let V satisfy Assumption 1.1 and let µ ∈ R as well as +T ≥ 0. Then, writing FT[Γ] ≡ FT(γ, α), the following are equivalent. +1. The minimizer of FT is not attained with α = 0, i.e. +inf +(γ,α)∈D FT(γ, α) < +inf +(γ,0)∈D FT(γ, 0), +2. There exists a pair (γ, α) ∈ D with α ̸= 0 such that ∆ = −2 � +V α satisfies the BCS gap +equation (1.1), +3. The linear operator KT + V , where KT (p) = +p2−µ +tanh((p2−µ)/(2T)) has at least one negative +eigenvalue. +The third item immediately leads to the following definition of the critical temperature Tc +for the existence of non-trivial solutions of the BCS gap equation (1.1). +Definition 1.4 (Critical temperature, see [FHNS07, Def. 1]). For V satisfying Assump- +tion 1.1, we define the critical temperature Tc ≥ 0 as +Tc := inf{T > 0 : KT + V ≥ 0} . +(1.6) +By KT(p) ≥ 2T and the asymptotic behavior KT(p) ∼ p2 for |p| → ∞, Sobolev’s inequality +[LL01, Thm. 8.3] implies that the critical temperature is well defined. +The other object we study is the energy gap Ξ defined in (1.2). The energy gap depends on +the solution ∆ of the gap equation (1.1) at T = 0. A priori, ∆ may not be unique. However, +for potentials with non-positive Fourier transform, this possibility can be ruled out. +Proposition 1.5 (see [HS08b, (21)-(22) and Lemma 2]). Let V satisfy Assumption 1.1 (and +additionally V ∈ L1(R2) in case that d = 2). Moreover, we assume that ˆV ≤ 0 and ˆV (0) < 0. +Then, there exists a unique minimizer Γ of F0 (up to a constant phase in α). One can choose +the phase such that α has strictly positive Fourier transform ˆα > 0. +In particular, we conclude that ∆ is strictly positive. Moreover, by means of the gap equation +(1.1), ∆ is continuous and thus Ξ > 0. +4 + +2 +Main Results +As explained in the introduction, our main result in this short note is the extension of the +universality (1.3) from d = 3 to lower spatial dimensions d = 1, 2 in the limit of weak +coupling (i.e., replacing V → λV and taking λ → 0). We assume the following properties for +the interaction potential V . +Assumption 2.1. Let d ∈ { 1, 2 } and assume that V satisfies Assumption 1.1 as well as +ˆV ≤ 0, ˆV (0) < 0. Moreover, for d = 1 we assume that (1 + | · |ε)V ∈ L1(R1) for some ε > 0. +Finally, in case that d = 2, we suppose that V ∈ L1(R2) is radial. +By Proposition 1.5, this means that, in particular, the minimizer of F0 is unique (up to a +phase) and the associated energy gap at zero temperature (1.2) is strictly positive, Ξ > 0. +We are now ready to state our main result. +Theorem 2.2 (BCS Universality in one and two dimensions). Let V be as in Assumption 2.1. +Then the critical temperature Tc(λ) (defined in (1.6)) and the energy gap Ξ(λ) (defined in +(1.2)) are strictly positive for all λ > 0 and it holds that +lim +λ→0 +Ξ(λ) +Tc(λ) = π +eγ , +where γ ≈ 0.577 is the Euler-Mascheroni constant. +To prove the universality, we separately establish asymptotic formulas for Tc (see Theo- +rem 2.5) and Ξ (see Theorem 2.7), valid to second order, and compare them by taking their +ratio. The asymptotic formula for Tc is valid under weaker conditions on V than Assump- +tion 2.1, because we do not need uniqueness of ∆. To obtain the asymptotic formulas, we +first introduce two self-adjoint operators V(d) +µ +and W(d) +µ +mapping L2(Sd−1) → L2(Sd−1) and as +such measuring the strength of the interaction ˆV on the (rescaled) Fermi surface (see [HS08b; +Hen22; HL22]). To assure that V(d) +µ +and W(d) +µ +will be well-defined and compact, we assume +the following. +Assumption 2.3. Let V satisfy Assumption 1.1. +Additionally, assume that for d = 1, +� +1 + (ln(1 + | · |))2� +V ∈ L1(R1) and for d = 2, V ∈ L1(R2). +First, in order to capture the strength to leading order, we define V(d) +µ +via +(V(d) +µ u)(p) = +1 +(2π)d/2 +� +Sd−1 +ˆV (√µ(p − q))u(q) dω(q) , +where dω is the Lebesgue measure on Sd−1. +Since V ∈ L1(Rd), we have that ˆV is a +bounded continuous function and hence V(d) +µ +is a Hilbert-Schmidt operator (in fact, trace +class with trace being equal to (2π)−d|Sd−1| +� +Rd V (x)dx). Therefore, its lowest eigenvalue +e(d) +µ +:= inf spec V(d) +µ +satisfies e(d) +µ +≤ 0 and it is strictly negative if e.g. +� +V < 0 as in Assump- +tion 2.1. +5 + +Second, in order to capture the strength of ˆV to next to leading order, we define the +operator W(d) +µ +via its quadratic form +� +u +��W(d) +µ +��u +� += µd/2−1 +�� +|p|< +√ +2 +1 +|p2 − 1| +� +|ψ(√µp)|2 − |ψ(√µp/|p|)|2� +dp + +� +|p|> +√ +2 +1 +|p2 − 1||ψ(√µp)|2 dp +� +, +where ψ(p) = +1 +(2π)d/2 +� +Sd−1 ˆV (p−√µq)u(q) dω(q) and u ∈ L2(Sd−1). The proof of the following +proposition shall be given in Section 3.3. +Proposition 2.4. Let d ∈ { 1, 2 } and let V satisfy Assumption 2.3. The operator W(d) +µ +is +well-defined and Hilbert-Schmidt. +Next, we define the self-adjoint Hilbert-Schmidt operator +B(d) +µ (λ) := π +2 +� +λV(d) +µ +− λ2W(d) +µ +� +on L2(Sd−1) and its ground state energy +b(d) +µ (λ) := inf spec +� +B(d) +µ (λ) +� +. +(2.1) +Note that if e(d) +µ +< 0, then also b(d) +µ (λ) < 0 for small enough λ. After these preparatory +definitions, we are ready to state the separate asymptotic formulas for the critical temperature +and the energy gap in one and two dimensions, which immediately imply Theorem 2.2. +Theorem 2.5 (Critical Temperature for d = 1, 2). Let µ > 0. Let V satisfy Assumption 2.3 +and additionally e(d) +µ +< 0. Then the critical temperature Tc, given in Definition 1.4, is strictly +positive and satisfies +lim +λ→0 +� +ln +� +µ +Tc(λ) +� ++ +π +2 µd/2−1 b(d) +µ (λ) +� += −γ − ln +�2cd +π +� +, +where γ denotes the Euler-Mascheroni constant and c1 = +4 +1+ +√ +2 and c2 = 1. +Here, the Assumptions on V are weaker than Assumption 2.1, since ˆV (0) < 0 implies that +e(d) +µ +< 0. We thus have the asymptotic behavior +Tc(λ) = 2cd +eγ +π +� +1 + o(1) +� +µ eπ/(2µd/2−1b(d) +µ (λ)) +in the limit of small λ. +Remark 2.6. Theorem 2.5 is essentially a special case of [HS10, Theorem 2]. We give the +proof here for two main reasons. +(i) There is still some work required to translate the statement of [HS10, Theorem 2] into a +form in which it is comparable to that of Theorem 2.7 (in order to prove Theorem 2.2). +The main difficulty is that the operator W(d) +µ +in [HS10] is only defined via a limit, +[HS10, Equation (2.10)]. +6 + +(ii) The goal of this paper is to give an exemplary proof of Theorem 2.5 in order to compare +it to the proofs of the similar statements in the literature concerning the asymptotic +behavior of the critical temperature in various limits [HS08a; HS08b; Hen22]. +Theorem 2.5 is complemented by the following asymptotics for the energy gap. +Theorem 2.7 (Energy Gap for d = 1, 2). Let V satisfy Assumption 2.1 and let µ > 0. +Then there exists a unique radially symmetric minimizer (up to a constant phase) of the +BCS functional (1.5) at temperature T = 0. The associated energy gap Ξ, given in (1.2), is +strictly positive and satisfies +lim +λ→0 +� +ln +�µ +Ξ +� ++ +π +2 µd/2−1 b(d) +µ (λ) +� += − ln(2cd) , +where b(d) +µ +is defined in (2.1) and c1 = +4 +1+ +√ +2 and c2 = 1. +In other words, we have the asymptotic behavior +Ξ(λ) = 2cd +� +1 + o(1) +� +µ eπ/(2µd/2−1b(d) +µ (λ)) +in the limit of small λ. Now, Theorem 2.2 follows immediately from Theorems 2.5 and 2.7. +Remark 2.8 (Other limits in dimensions d = 1, 2). Similarly to the presented results, one +could also consider the limits of low and high density, i.e. µ → 0 and µ → ∞, respectively. +We expect that also here the universality +Ξ +Tc ≈ +π +eγ holds. Indeed, one would expect that the +proofs of BCS universality in dimension d = 3 should carry over to one and two dimensions +with some minor technical modifications. Note that, even for the (technically less demanding) +case of a weak coupling limit, which we consider here, there are still some technical details +that are different in dimensions d = 1, 2 compared to dimension d = 3. Hence, it is not a +trivial matter to generalize the arguments of [HS08a; Hen22; HL22; Lau21] to one and two +dimensions. Moreover, for the case of low density, there is even an issue of what exactly low +density means in dimensions one and two: In three spatial dimensions [HS08a; Lau21], the +asymptotic formulas for Tc and Ξ were obtained for potentials V with negative scattering +length but not creating bound states for the Laplacian. This latter condition ensures that +µ → 0 actually corresponds to the limit of low density. +However, in spatial dimensions +one and two, attractive potentials, no matter how weak, always give rise to bound states of +−∇2 + V , see [Sim76]. Thus for µ = 0 the particle density is non-zero. We will not deal with +the low- and high-density limits here. +The rest of the paper is devoted to proving Theorem 2.5 and Theorem 2.7. +3 +Proofs +The overall structure of our proofs is as follows: The principal idea is to derive two different +formulas for each of the two integrals +m(d) +µ (T) := +1 +|Sd−1| +� +|p|<√2µ +1 +KT(p) dp +(3.1) +7 + +and +m(d) +µ (∆) := +1 +|Sd−1| +� +|p|<√2µ +1 +E∆(p) dp. +(3.2) +The first set of formulas is derived by studying the Birman-Schwinger operators +B(d) +T +:= λV 1/2K−1 +T |V |1/2 +and +B(d) +∆ := λV 1/2E−1 +∆ |V |1/2 , +associated to the Schr¨odinger type operators KT +λV and E∆ +λV , respectively. In particu- +lar, spectral properties of these unbounded Schr¨odinger type operators naturally translate to +their associated Birman-Schwinger operators, which are compact and as such much simpler +to analyze. The second set of formulas is obtained by just calculating the integrals m(d) +µ +directly. +Indeed, for the critical temperature we obtain the following asymptotics, which, by com- +bining them, immediately prove Theorem 2.5. +Proposition 3.1. Let µ > 0. Let V satisfy Assumption 2.3 and additionally e(d) +µ +< 0. Then, +the critical temperature Tc is positive and, as λ → 0, we have that +m(d) +µ (Tc) = − +π +2b(d) +µ (λ) ++ o(1) , +m(d) +µ (Tc) = µd/2−1 +� +ln +� µ +Tc +� ++ γ + ln +�2cd +π +� ++ o(1) +� +. +For the energy gap we obtain the following asymptotics, which, again by combining them, +immediately prove Theorem 2.7. +Proposition 3.2. Let V satisfy Assumption 2.1 and let µ > 0. Then (by Proposition 1.5) +we have a strictly positive radially symmetric gap function ∆ and associated energy gap Ξ, +which, as λ → 0, satisfy the asymptotics +Ξ = ∆(√µ) +� +1 + o(1) +� +m(d) +µ (∆) = − +π +2b(d) +µ (λ) ++ o(1) +m(d) +µ (∆) = µd/2−1 +� +ln +� +µ +∆(√µ) +� ++ ln(2cd) + o(1) +� +With a slight abuse of notation, using radiality of ∆, we wrote ∆(√µ) instead of ∆(√µˆp) +for some ˆp ∈ Sd−1. +In the remainder of this paper, where we give the proofs of Propositions 3.1 and 3.2, we +shall frequently use the notation F(d) +µ +: L1(Rd) → L2(Sd−1) for the (scaled) Fourier transform +restricted to the (rescaled) Fermi sphere, +� +F(d) +µ ψ +� +(p) := +1 +(2π)d/2 +� +Rd ψ(x)e−i√µp·x dx . +Note that for an L1-function, pointwise values of its Fourier transform are well-defined by +the Riemann–Lebesgue lemma. (In particular the restriction to a co–dimension 1 manifold +of a sphere is well-defined.) +Remark 3.3. In [CM21], Cuenin and Merz use the Tomas-Stein theorem to define F(d) +µ +on a +larger space than L1(Rd). With this they are able to prove a general version of Theorem 2.5 +under slightly weaker conditions on V . However, we do not pursue this here, see Remark 2.6. +8 + +3.1 +Proof of Proposition 3.1 +Proof of Proposition 3.1. The argument is divided into several steps. +1. A priori spectral information on KTc + λV . First note that, due to Theorem 1.3 and +Definition 1.4, the critical temperature Tc is determined by the lowest eigenvalue of KT +λV +being 0 exactly for T = Tc. +2. +Birman-Schwinger principle. +Next, we employ the Birman-Schwinger principle, +which says that the compact Birman-Schwinger operator B(d) +T += λV 1/2K−1 +T |V |1/2 (denot- +ing V (x)1/2 = sgn(V (x))|V (x)|1/2) has −1 as its lowest eigenvalue exactly for T = Tc, see +[FHNS07; HS08b]. +Using the notation for the Fourier transform restricted to the rescaled Fermi sphere in- +troduced above, we now decompose the Birman-Schwinger operator as +B(d) +T += λm(d) +µ (T)V 1/2(F(d) +µ )†F(d) +µ |V |1/2 + λV 1/2M(d) +T |V |1/2, +where M(d) +T +is defined through the integral kernel +M(d) +T (x, y) = +1 +(2π)d +�� +|p|<√2µ +1 +KT(p) +� +eip·(x−y) − ei√µp/|p|·(x−y)� +dp + +� +|p|>√2µ +1 +KT +eip·(x−y) dp +� +. +(3.3) +We claim that V 1/2M(d) +T |V |1/2 is uniformly bounded. +Lemma 3.4. Let µ > 0. Let V satisfy Assumption 2.3. Then we have for all T ≥ 0 +���V 1/2M(d) +T |V |1/2��� +HS ≤ C , +where C > 0 denotes some positive constant and ∥ · ∥HS is the Hilbert-Schmidt norm. +Armed with this bound, we have that for sufficiently small λ that 1 + λV 1/2M(d) +T |V |1/2 is +invertible, and hence +1 + B(d) +T += (1 + λV 1/2M(d) +T |V |1/2) +� +1 + +λm(d) +µ (T) +1 + λV 1/2M(d) +T |V |1/2V 1/2(F(d) +µ )†F(d) +µ |V |1/2 +� +. +Thus, the fact that B(d) +T +has lowest eigenvalue −1 at T = Tc is equivalent to +λm(d) +µ (T)F(d) +µ |V |1/2 +1 +1 + λV 1/2M(d) +T |V |1/2V 1/2(F(d) +µ )† +(3.4) +having lowest eigenvalue −1, again at T = Tc, as it is isospectral to the rightmost operator on +the right-hand-side above. (Recall that for bounded operators A, B, the operators AB and +BA have the same spectrum apart from possibly at 0. However, in our case, both operators +are compact on an infinite dimensional space and hence 0 is in both spectra.) +We now prove Lemma 3.4. +9 + +Proof of Lemma 3.4. We want to bound the integral kernel (3.3) of M(d) +T +uniformly in T. +Hence, we will bound KT ≥ |p2 − µ|. The computation is slightly different in d = 1 and +d = 2, so we do them separately. +d = 1. The second integral in (3.3) is bounded by +2 +� +|p|>√2µ +1 +|p2 − µ| dp = 2 arcoth +√ +2 +√µ +. +For the first integral, we use that |eix − eiy| ≤ min{|x − y|, 2}, |p2 − µ| ≥ √µ||p| − √µ|, and +increase the domain of integration to obtain the bound +2 +√µ +� 2√µ +0 +min +� +||p − √µ||x − y|, 2 +� +|p − √µ| +dp = +8 +√µ +� +1 + ln +� +max +�|x − y|√µ +2 +, 1 +��� +≤ +8 +√µ(1 + ln(1 + √µ max{|x|, |y|}). +We conclude that |M(1) +T (x, y)| ≲ +1 +√µ(1 + ln(1 + √µ max{|x|, |y|})). Hence, +���V 1/2M(1) +T |V |1/2��� +2 +HS ≲ 1 +µ +� +∥V ∥2 +L1(R) + ∥V ∥L1(R) +� +R +|V (x)|(1 + ln(1 + √µ|x|))2 dx +� +. +d = 2. We first compute the angular integral. Note that +� +S1 eipx dω(p) = 2πJ0(|x|), where J0 is +the zeroth order Bessel function. For the second integral in (3.3) we may bound |p2−µ| ≥ cp2. +Up to some finite factor, the second integral is hence bounded by +� ∞ +√2µ +1 +p|J0(p|x − y|)| dp ≤ C +� ∞ +√2µ +1 +p1+λ|x − y|−λ dp ≤ Cλ|x − y|−λ, +for any 0 < λ ≤ 1/2 since |J0(x)| ≤ C and √xJ0(x) ≤ C, see e.g. [BSMM12, (9.55f), (9.57a)]. +For the first integral we get the bound +� √2µ +0 +p +|p2 − µ| |J0(p|x − y|) − J0(√µ|x − y|)| dp. +Here we use that J0 is Lipschitz, since its derivative J−1 is bounded (see e.g. [BSMM12, +(9.55a), (9.55f)]), so that +|J0(x) − J0(y)| ≤ C|x − y|1/3(|J0(x)| + |J0(y)|)2/3 ≤ C|x − y|1/3 � +x−1/3 + y−1/3� +. +That is +|J0(p|x − y|) − J0(√µ|x − y|)| ≤ C |p − √µ|1/3 +p1/3 + √µ1/3. +This shows that the first integral is bounded. We conclude that |M(2) +T (x, y)| ≲ 1 + +1 +|x−y|λ for +any 0 < λ ≤ 1/2. Then, by the Hardy–Littlewood–Sobolev inequality [LL01, Theorem 4.3] +we have that +���V 1/2M(2) +T |V |1/2��� +2 +HS = +�� +|V (x)||M(2) +T (x, y)||V (y)| dx dy ≲ ∥V ∥2 +L1(R2) + ∥V ∥2 +Lp(R2) +for any 1 < p ≤ 4/3. +10 + +3. First order. Evaluating (3.4) at T = Tc and expanding the geometric series to first order +we get +−1 = λm(d) +µ (Tc) inf spec +� +F(d) +µ |V |1/2 +1 +1 + λV 1/2M(d) +Tc |V |1/2V 1/2(F(d) +µ )† +� += λ m(d) +µ (Tc) inf spec V(d) +µ (1 + O(λ)) = λ m(d) +µ (Tc) e(d) +µ (1 + O(λ)) +where we used V(d) +µ += F(d) +µ V (F(d) +µ )†. Since by assumption e(d) +µ +< 0, this shows that m(d) +µ (Tc) → +∞ as λ → 0. +4. A priori bounds on Tc. By (3.1), the divergence of m(d) +µ +as λ → 0 in particular shows +that Tc/µ → 0 in the limit λ → 0. +5. Calculation of the integral m(d) +µ (Tc). This step is very similar to [HS08b, Lemma 1] +and [HRS22, Lemma 3.5], where the asymptotics have been computed for slightly different +definitions of m(d) +µ +in three and one spatial dimension, respectively. +Integrating over the +angular variable and substituting s = +���|p|2 +µ − 1 +���, we get +m(d) +µ (Tc) = µd/2−1 +� 1 +0 +tanh +� +s +2(Tc/µ) +� (1 + s)d/2−1 + (1 − s)d/2−1 +2s +ds. +According to [HS08b, Lemma 1], +lim +Tc↓0 + + +� 1 +0 +tanh +� +s +2(Tc/µ) +� +s +ds − ln µ +Tc + + = γ − ln π +2 . +By monotone convergence, it follows that +m(d) +µ (Tc) = µd/2−1 +� +ln µ +Tc ++ γ − ln π +2 + +� 1 +0 +(1 − s)d/2−1 + (1 + s)d/2−1 − 2 +2s +ds + o(1) +� +. +The remaining integral equals ln cd and we have thus proven the second item in Proposi- +tion 3.1. +Combining this with the third step, one immediately sees that the critical temperature +vanishes exponentially fast, Tc ∼ e1/λeµ, as λ → 0, recalling that e(d) +µ +< 0 by assumption. +6. Second order. Now, to show the universality, we need to compute the next order correc- +tion. To do so, we expand the geometric series in (3.4) and employ first order perturbation +theory, yielding that +m(d) +µ (Tc) = +−1 +λ +� +u +���F(d) +µ V (F(d) +µ )† +���u +� +− λ2 +� +u +���F(d) +µ V M(d) +Tc V (F(d) +µ )† +���u +� ++ O(λ3) +, +(3.5) +where u is the (normalized) ground state (eigenstate of lowest eigenvalue) of F(d) +µ V (F(d) +µ )†. (In +case of a degenerate ground state, u is the ground state minimizing the second order term.) +This second order term in the denominator of (3.5) is close to W(d) +µ . More precisely, it +holds that +lim +λ→0 +� +u +���F(d) +µ V M(d) +Tc V (F(d) +µ )†���u +� += +� +u +��W(d) +µ +��u +� +, +(3.6) +11 + +which easily follows from dominated convergence, noting that +1 +KT increases to +1 +|p2−µ| as T → 0. +We then conclude that +lim +λ→0 +� +m(d) +µ (Tc) + +π +2b(d) +µ (λ) +� += 0 , +since +� +u +���λV(d) +µ +− λ2W(d) +µ +���u +� += inf spec(λV(d) +µ −λ2W(d) +µ ) + O(λ3) = π +2b(d) +µ (λ) + O(λ3), again by +first-order perturbation theory. This concludes the proof of Proposition 3.1. +We conclude this subsection with several remarks, comparing our proof with those of similar +results from the literature. +Remark 3.5 (Structure here vs. in earlier papers on Tc). We compare the structure of our +proof to that of the different limits in three dimensions [HS08a; HS08b; Hen22]: +• Weak coupling: The structure of the proof we gave here is quite similar to that of +[HS08b], only they do Steps 5 and 6 in the opposite order. Also the leading term for Tc +was shown already in [FHNS07], where a computation somewhat similar to Steps 1–4 +is given. +• High denisty: For µ → ∞, the structure of the proof in [Hen22] is slightly different +compared to the one given here. This is basically due to the facts that (i) the necessary +a priori bound Tc = o(µ) already requires the Birman-Schwinger decomposition and +(ii) the second order requires strengthened assumptions compared to the first order. +To conclude, the order of steps in [Hen22] can be thought of as: 1, 5, 4 (establishing +Tc = O(µ)), 2, 3, 4 (establishing Tc = o(µ)), 2 (again), 6. Here the final step is much +more involved than in the other limits considered. +• Low density: As above, for the proof of the low density limit in [HS08a] the struc- +ture is slightly different. One first needs the a priori bound Tc = o(µ) on the criti- +cal temperature before one uses the Birman-Schwinger principle and decomposes the +Birman-Schwinger operator.1 Also, the decomposition of the Birman-Schwinger opera- +tor is again different. For the full decomposition and analysis of the Birman-Schwinger +operator one needs also the first-order analysis, that is Step 2, which is done in two +parts. The order of the steps in [HS08a] can then mostly be though of as: 1, 4, 5, 2, 3, +2 (again), 6. +3.2 +Proof of Proposition 3.2 +Proof of Proposition 3.2. The structure of the proof is parallel to that of Proposition 3.1 for +the critical temperature. +1. A priori spectral information on E∆ + λV . First, it is proven in [HS08b, Lemma 2] +that F0 has a unique minimizer α which has strictly positive Fourier transform. Using radial- +ity of V , it immediately follows that this minimizer is rotationally symmetric (since otherwise +1Strictly speaking, in [HS08a], it is only proven that Tc = O(µ) (which is sufficient for applying the +Birman-Schwinger principle), while the full Tc = o(µ) itself requires the Birman-Schwinger decomposition +(see [Lau20, Remark 4.12] for details). +12 + +rotating α would give a different minimizer) and hence also ∆ = −2λ ˆV ⋆ ˆα is rotation in- +variant. It directly follows from [HS08b, (43) and Lemma 3] that that E∆ + λV has lowest +eigenvalue 0, and that the minimizer α is the corresponding eigenfunction. +2. Birman-Schwinger principle. This implies, by means of the Birman-Schwinger princi- +ple, that the Birman-Schwinger operator B(d) +∆ = λV 1/2E−1 +∆ |V |1/2 has −1 as its lowest eigen- +value. As in the proof of Proposition 3.1, we decompose it as +B(d) +∆ = λm(d) +µ (∆)V 1/2(F(d) +µ )†F(d) +µ |V |1/2 + λV 1/2M(d) +∆ |V |1/2 +and prove the second summand to be uniformly bounded. +Lemma 3.6. Let µ > 0. Let V satisfy Assumption 2.3. Then, uniformly in small λ, we have +���V 1/2M(d) +∆ |V |1/2��� +HS ≤ C . +With this one may similarly factor +1 + B(d) +∆ = (1 + λV 1/2M(d) +∆ |V |1/2) +� +1 + +λm(d) +µ (∆) +1 + λV 1/2M(d) +∆ |V |1/2V 1/2(F(d) +µ )†F(d) +µ |V |1/2 +� +(3.7) +and conclude that +T (d) +∆ := λm(d) +µ (∆)F(d) +µ |V |1/2 +1 +1 + λV 1/2M(d) +∆ |V |1/2V 1/2(F(d) +µ )† +(3.8) +has lowest eigenvalue −1. +Proof of Lemma 3.6. Note that M∆ has kernel +M∆(x, y) = +1 +(2π)d +�� +|p|<√2µ +1 +E∆(p) +� +eip·(x−y) − ei√µp/|p|·(x−y)� +dp + +� +|p|>√2µ +1 +E∆(p)eip·(x−y) dp +� +. +We may bound this exactly as in the proof of Lemma 3.4 using that E∆(p) ≥ |p2 − µ|. +3. First order. Expanding the geometric series in (3.8) to first order, we see that +−1 = λm(d) +µ (∆) inf spec +� +F(d) +µ |V |1/2 +1 +1 + λV 1/2M(d) +∆ |V |1/2V 1/2(F(d) +µ )† +� += λm(d) +µ (∆) inf spec V(d) +µ (1 + O(λ)) = λe(d) +µ m(d) +µ (∆)(1 + O(λ)). +Hence, in particular, m(d) +µ (∆) ∼ − +1 +λe(d) +µ +→ ∞ as λ → 0. +4. A priori bounds on ∆. We now prepare for the computation of the integral m(d) +µ (∆) +in terms of ∆(√µ). This requires two types of bounds on ∆: One bound estimating the +gap function ∆(p) at general momentum p ∈ Rd in terms of ∆(√µ) (see (3.9)), and one +bound controlling the difference |∆(p) − ∆(q)| in some kind of H¨older-continuity estimate +(see (3.10)). +13 + +Lemma 3.7. Suppose that V is as in Assumption 2.1. Then for λ small enough +∆(p) = f(λ) +�� +Sd−1 +ˆV (p − √µq) dω(q) + ληλ(p) +� +, +where f is some function of λ and ∥ηλ∥L∞(Rd) is bounded uniformly in λ. +Proof. Recall that α is the eigenfunction of E∆ + λV with lowest eigenvalue 0. Then, by the +Birman-Schwinger principle, φ = V 1/2α satisfies +B∆φ = λV 1/2 1 +E∆ +|V |1/2V 1/2α = −φ. +With the decomposition Equation (3.7) then φ is an eigenfunction of +λm(d) +µ (∆) +1 + λV 1/2M(d) +∆ |V |1/2V 1/2(F(d) +µ )†F(d) +µ |V |1/2 +of eigenvalue −1. Thus, F(d) +µ |V |1/2φ is an eigenfunction of T (d) +∆ +of (lowest) eigenvalue −1. +Now u = |Sd−1|−1/2 is the unique eigenfunction corresponding to the lowest eigenvalue of V(d) +µ +by radiality of V and the assumption ˆV ≤ 0 (see e.g. [FHNS07]). Hence, for λ small enough, +u is the unique eigenfunction of T (d) +∆ +of smallest eigenvalue. Thus, +φ = f(λ) +1 +1 + λV 1/2M(d) +∆ |V |1/2V 1/2(F(d) +µ )†u = f(λ) +� +V 1/2(F(d) +µ )†u + λξλ +� +for some number f(λ). The function ξλ satisfies ∥ξλ∥L2(Rd) ≤ C by Lemma 3.6. Noting that +∆ = −2 � +|V |1/2φ and bounding +��� � +|V |1/2ξλ +��� +L∞ ≤ ∥V ∥1/2 +L1 ∥ξλ∥L2 we get the desired. +Evaluating the formula in Lemma 3.7 at p = √µ we get |f(λ)| ≤ C∆(√µ) for λ small enough. +This in turn implies that +∆(p) ≤ C∆(√µ) . +(3.9) +For the H¨older-continuity, we have by rotation invariance +���� +� +ˆV (p − √µr) − ˆV (q − √µr) dω(r) +���� = +���� +� +ˆV (|p|e1 − √µr) − ˆV (|q|e1 − √µr) dω(r) +���� += +���� +1 +(2π)d/2 +� +Rd dx +� +V (x) +� +ei|p|x1 − ei|q|x1� � +Sd−1 e−i√µx·r dω(r) +����� +≤ Cεµ−ε/2||p| − |q||ε +� +dx +� +|V (x)|(√µ|x|)ε +���� +� +Sd−1 e−i√µx·r dω(r) +���� +� +, +for any 0 < ε ≤ 1. For d = 2 we have V ∈ L1(R2) and +���� +� +Sd−1 e−i√µxr dω(r) +���� = |J0(√µ|x|)| ≤ (√µ|x|)−1/2. +14 + +For d = 1 we have |x|εV ∈ L1(R) for some ε > 0 and +���� +� +Sd−1 e−i√µx·r dω(r) +���� = 2| cos(√µ|x|)| ≤ 2. +We conclude that with ε = 1/2 for d = 2 and small enough ε > 0 for d = 1 +|∆(p) − ∆(q)| ≤ C|f(λ)| +� +µ−ε/2||p| − |q||ε + λ +� +≤ C|∆(√µ)| +� +µ−ε/2||p| − |q||ε + λ +� +. +(3.10) +Additionally, since m(d) +µ (∆) → ∞ we have that ∆(p) → 0 at least for some p ∈ Rd by (3.2). +Then it follows from Lemma 3.7 that f(λ) → 0, i.e. that ∆(p) → 0 for all p. +5. Calculation of the integral m(d) +µ (∆). Armed with the apriori bounds (3.9) and (3.10), +we can now compute the integral m(d) +µ (∆). Carrying out the angular integration and substi- +tuting s = +��� |p|2−µ +µ +��� we have +m(d) +µ (∆) = µd/2−1 +2 +�� 1 +0 +� +(1 − s)d/2−1 − 1 +� +s2 + x−(s)2 + (1 + s)d/2−1 − 1 +� +s2 + x+(s)2 +� +ds ++ +� 1 +0 +� +1 +� +s2 + x−(s)2 + +1 +� +s2 + x+(s)2 +� +ds +� +, +where x±(s) = ∆(√µ√1±s) +µ +. By dominated convergence, using that x±(s) → 0, the first integral +is easily seen to converge to +� 1 +0 +�(1 − s)d/2−1 − 1 +s ++ (1 + s)d/2−1 − 1 +s +� +ds = 2 ln cd +for λ → 0. For the second integral, we will now show that +� 1 +0 +� +1 +� +s2 + x±(s)2 − +1 +� +s2 + x±(0)2 +� +ds → 0 . +In fact, the integrand is bounded by +����� +1 +� +s2 + x±(s)2 − +1 +� +s2 + x±(0)2 +����� += +|x±(0)2 − x±(s)2| +� +s2 + x±(s)2� +s2 + x±(0)2( +� +s2 + x±(s)2 + +� +s2 + x±(0)2) +≤ +Cx±(0)(sε + λ) +� +s2 + x±(s)2� +s2 + x±(0)2, +using the H¨older continuity from (3.10). By continuity of ˆV there exists some s0 (independent +of λ) such that for s < s0 we have x±(s) ≥ cx±(0). We now split the integration into +� s0 +0 +and +� 1 +s0. For the first we have +� s0 +0 +����� +1 +� +s2 + x±(s)2 − +1 +� +s2 + x±(0)2 +����� ds ≤ C +� s0 +0 +x±(0) +s2 + x±(0)2(sε + λ) ds = O(x±(0)ε + λ) . +15 + +For the second we have +� 1 +s0 +����� +1 +� +s2 + x±(s)2 − +1 +� +s2 + x±(0)2 +����� ds ≤ C +� 1 +s0 +x±(0)sε + λ +s2 +ds = O(x±(0)) . +Collecting all the estimates, we have thus shown that m(d) +µ (∆) equals +µd/2−1 +� +ln cd + +� 1 +0 +1 +� +s2 + ∆(√µ)2/µ2 ds + o(1) +� += µd/2−1 +� +ln cd + ln +� +µ + +� +µ2 + ∆(√µ)2 +|∆(√µ)| +� ++ o(1) +� += µd/2−1 ln +� +2µcd +|∆(√µ)| + o(1) +� +. +This proves the third inequality in Proposition 3.2. +Combining this with the third step, one immediately sees that the gap function evaluated +on the Fermi sphere vanishes exponentially fast, ∆(√µ) ∼ e1/λeµ, as λ → 0, recalling that +e(d) +µ +< 0 by assumption. +6. Second order. To obtain the next order, we recall that T (d) +∆ +has lowest eigenvalue −1 +(see (3.8)), and hence, by first-order perturbation theory, +m(d) +µ (∆) = +−1 +λ +� +u +���F(d) +µ V (F(d) +µ )† +���u +� +− λ2 +� +u +���F(d) +µ V M(d) +∆ V (F(d) +µ )† +���u +� ++ O(λ3) +, +(3.11) +where u(p) = |Sd−1|−1/2 is the constant function on the sphere. Recall that u is the unique +ground state of V(d) +µ . +In the second order term we have that +lim +λ→0 +� +u +���F(d) +µ V M(d) +∆ V (F(d) +µ )†���u +� += +� +u +��W(d) +µ +��u +� +, +which follows from a simple dominated convergence argument as for Tc, noting that ∆(p) → 0 +pointwise. +By again employing first–order perturbation theory, similarly to the last step in the proof +of Proposition 3.1, we conclude the second equality in Proposition 3.2. +7. Comparing ∆(√µ) to Ξ. To prove the first equality in Proposition 3.2 we separately +prove upper and lower bounds. The upper bound is immediate from +Ξ = inf +p∈Rd E∆(p) = inf +p∈Rd +� +|p2 − µ| + ∆(p)2 ≤ ∆(√µ) . +Hence, for the lower bound, take p ∈ Rd with +� +|p2 − µ| ≤ Ξ ≤ ∆(√µ). Then by (3.10) +∆(p) ≥ ∆(√µ) −|∆(p) −∆(√µ)| ≥ ∆(√µ) −C∆(√µ) (||p| − √µ|ε + λ) ≥ ∆(√µ)(1 + o(1)). +In combination with the upper bound, we have thus shown that Ξ = ∆(√µ)(1 + o(1)) as +desired. This concludes the proof of Proposition 3.2. +16 + +We conclude this subsection with several remarks, comparing our proof with those of similar +results from the literature. +Remark 3.8 (Structure here vs. in earlier papers on Ξ). We now compare the proof above +to the proofs of the three different limits in 3 dimensions [HS08b; HL22; Lau21]: +• Weak coupling: The structure of our proof here is very similar to that of [HS08b]. +Essentially, only the technical details in Lemma 3.6 and the calculation of m(d) +µ (∆) in +Step 5 are different. +• High density: For the high-density limit in [HL22], we needed some additional a +priori bounds on ∆ before we could employ the Birman-Schwinger argument. Apart +from that, in [HL22] the comparison of ∆(√µ) and Ξ are done right after these a priori +bounds. Additionally, since one starts with finding a priori bounds on ∆, one does not +need the first-order analysis in Step 3. One may think of the structure in [HL22] as +being ordered in the above steps as follows: 4, 7, 1, 2, 4 (again), 5, 6. +• Low density: For the low-density limit in [Lau21] the structure is quite different. +Again, one first needs some a priori bounds on ∆ before one can use the Birman- +Schwinger argument. One then improves these bounds on ∆ using the Birman-Schwinger +argument, which in turn can be used to get better bounds on the error term in the +decomposition of the Birman–Schwinger operator. In this sense, the Steps 2–4 are too +interwoven to be meaningfully separated. Also, Step 5 is done in two parts. +3.3 +Proof of Proposition 2.4 +Note that W(d) +µ += F(d) +µ V M(d) +0 V (F(d) +µ )†, where M(d) +0 +is defined in (3.3). +By Lemma 3.4, +V 1/2M(d) +0 V 1/2 is Hilbert-Schmidt. The integral kernel of W(d) +µ +is bounded by +|W(d) +µ (p, q)| ≤ +1 +(2π)d +� +R2d |V (x)||M(d) +0 (x, y)||V (y)| dx dy ≤ +1 +(2π)d∥V ∥1∥V 1/2M(d) +0 V 1/2∥HS. +(3.12) +It follows that ∥W(d) +µ ∥HS ≤ |Sd−1| +(2π)d ∥V ∥1∥V 1/2M(d) +0 V 1/2∥HS. +Acknowledgments. We thank Robert Seiringer for comments on the manuscript. J.H. grate- +fully acknowledges partial financial support by the ERC Advanced Grant ‘RMTBeyond’ +No. 101020331. +References +[BCS57] +J. Bardeen, L. N. Cooper, and J. R. Schrieffer. “Theory of superconductivity”. +Phys. Rev. 108.5 (1957), pp. 1175–1204. doi: 10.1103/PhysRev.108.1175. +[BSMM12] +I. N. Bronˇstejn, K. A. Semendjaev, G. Musiol, and H. M¨uhlig, eds. Taschenbuch +der Mathematik. 8., vollst. ¨uberarb. Aufl. Frankfurt am Main: Deutsch, 2012. +17 + +[Cao+18] +Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. +Jarillo-Herrero. “Unconventional superconductivity in magic-angle graphene su- +perlattices”. 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Simon. “The bound state of weakly coupled Schr¨odinger operators in one +and two dimensions”. Annals of Physics 97.2 (1976), pp. 279–288. doi: 10.10 +16/0003-4916(76)90038-5. +19 + diff --git a/a9E5T4oBgHgl3EQfew9G/content/tmp_files/load_file.txt b/a9E5T4oBgHgl3EQfew9G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce497422c4b959658fe7bbc9c497f464586cf5ac --- /dev/null +++ b/a9E5T4oBgHgl3EQfew9G/content/tmp_files/load_file.txt @@ -0,0 +1,846 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf,len=845 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='05621v1 [math-ph] 13 Jan 2023 Universality in low-dimensional BCS theory Joscha Henheik∗ Asbjørn Bækgaard Lauritsen† Barbara Roos‡ Institute of Science and Technology Austria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Am Campus 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3400 Klosterneuburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Austria 16 January 2023 Abstract It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ and the critical temperature Tc is (approximately) given by a univer- sal constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' independent of the microscopic details of the fermionic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density, and high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The goal of this short note is to extend the universal behavior to lower dimensions d = 1, 2 and give an exemplary proof in the weak coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Keywords: BCS theory, energy gap, critical temperature, BCS universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Mathematics subject classification: 81Q10, 46N50, 82D55 1 Introduction The Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [BCS57] is governed by the BCS gap equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For translation invariant systems without external fields the BCS gap equation is ∆(p) = − 1 (2π)d/2 � Rd ˆV (p − q) ∆(q) E∆(p) tanh �E∆(p) 2T � dq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) with dispersion relation E∆(p) = � (p2 − µ)2 + |∆(p)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Here, T ≥ 0 denotes the temperature and µ > 0 the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We consider dimensions d ∈ { 1, 2, 3 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The Fourier transform of the potential V ∈ L1(Rd) ∩ LpV (Rd) (with a d-dependent pV ≥ 1 to be specified below), modeling their effective interaction, is denoted by ˆV (p) = (2π)−d/2 � Rd V (x)e−ip·xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' According to BCS theory, a system is in a superconducting state, if there exists a non- zero solution ∆ to the gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The question of existence of such a non-trivial solution ∆ hinges, in particular, on the temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' It turns out, there exists a critical temperature Tc ≥ 0 such that for T < Tc there exists a non-trivial solution, and for T ≥ Tc it ∗joscha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='henheik@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='at †alaurits@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='at ‡barbara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='roos@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='at 1 does not [HHSS08, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This critical temperature is one of the key (physically measurable) quantities of the theory and its asymptotic behavior, in three spatial dimensions, has been studied in three physically rather different limiting regimes: In a weak-coupling limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' replacing V → λV and taking λ → 0) [FHNS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08b], in a low-density limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' µ → 0) [HS08a], and in a high-density limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' µ → ∞) [Hen22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' As already indicated above, at zero temperature, the function E∆ may be interpreted as the dispersion relation of a certain ‘approximate’ Hamiltonian of the quantum many-body system, see [HHSS08, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In particular Ξ := inf p∈Rd E∆(p) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2) has the interpretation of an energy gap associated with the approximate BCS Hamiltonian and as such represents a second key quantity of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Analogously to the critical temperature, the asymptotic behavior of this energy gap, again in three spatial dimensions, has been studied in the same three different limiting regimes: In a weak coupling limit [HS08b], in a low density limit [Lau21], and in a high density limit [HL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In this paper, we focus on a remarkable feature of BCS theory, which is well known in the physics literature [BCS57;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' LTB19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' NS85]: The ratio of the energy gap Ξ and criti- cal temperature Tc tends to a universal constant, independent of the microscopic details of the interaction between the fermions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' More precisely, in three spatial dimension, it holds that Ξ Tc ≈ π eγ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='76 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) where γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='577 is the Euler-Mascheroni constant, in each of the three physically very different limits mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This result follows as a limiting equality by combining asymptotic formulas for the critical temperature Tc (see [FHNS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hen22]) and the energy gap Ξ (see [HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HL22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lau21]) in the three different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Although these scenarios (weak coupling, low density, and high density) are physically rather different, they all have in common that ‘superconductivity is weak’ and one can hence derive an asymptotic formula for Tc and Ξ as they depart from being zero (in the extreme cases λ = 0, µ = 0, µ = ∞, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' However, all the asymptotic expressions are not perturbative, as they depend exponentially on the natural dimensionless small parameter in the respective limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We refer to the above mentioned original works for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The goal of this note is to prove the same universal behavior (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3), which has already been established in three spatial dimension, also in dimensions d = 1, 2 in the weak coupling limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' replacing V → λV and taking λ → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This situation serves as a showcase for the methods involved in the proofs of the various limits in three dimensions (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Apart from the mathematical curiosity in d = 1, 2, there have been recent studies in lower-dimensional superconductors in the physics literature, out of which we mention one-dimensional superconducting nanowires [NTH12] and two-dimensional ‘magic angle’ graphene [Cao+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In the remainder of this introduction, we briefly recall the mathematical formulation of BCS theory, which has been developed mostly by Hainzl and Seiringer, but also other co- authors [FHNS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HHSS08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Apart from the universality discussed here, also many other properties of BCS theory have been shown using this formulation: Most prominently, 2 Ginzburg-Landau theory, as an effective theory describing superconductors close to the crit- ical temperature, has been derived from BCS theory [DHM22a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' DHM22b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' FHSS12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' FL16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' More recently, it has been shown that the effect of boundary superconductivity occurs in the BCS model [HRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We refer to [HS16] for a more comprehensive review of developments in the mathematical formulation of BCS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The universal behavior in the weak coupling limit for lower dimensions d = 1, 2 is presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Finally, in Section 3, we provide the proofs of the statements from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 Mathematical formulation of BCS theory We will now briefly recall the mathematical formulation [HHSS08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS16] of BCS theory [BCS57], which is an effective theory developed for describing superconductivity of a fermionic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In the following, we consider these fermions in Rd, d = 1, 2, at temperature T ≥ 0 and chemical potential µ ∈ R, interacting via a two-body potential V , for which we assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We have that V is real-valued, reflection symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' V (x) = V (−x) for all x ∈ Rd, and it satisfies V ∈ LpV (Rd), where pV = 1 if d = 1, pV ∈ (1, ∞) if d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Moreover, we neglect external fields, in which case the system is translation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The central object in the mathematical formulation of the theory is the BCS functional, which can naturally be viewed as a function of BCS states Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' These states are given by a pair of functions (γ, α) and can be conveniently represented as a 2 × 2 matrix valued Fourier multiplier on L2(Rd) ⊕ L2(Rd) of the form ˆΓ(p) = �ˆγ(p) ˆα(p) ˆα(p) 1 − ˆγ(p) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4) for all p ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4), ˆγ(p) denotes the Fourier transform of the one particle density matrix and ˆα(p) is the Fourier transform of the Cooper pair wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We require reflection symmetry of ˆα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' ˆα(−p) = ˆα(p), as well as 0 ≤ ˆΓ(p) ≤ 1 as a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The BCS free energy functional takes the form FT[Γ] := � Rd(p2 − µ)ˆγ(p)dp − TS[Γ] + � Rd V (x)|α(x)|2dx , Γ ∈ D, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) D := � ˆΓ(p) = �ˆγ(p) ˆα(p) ˆα(p) 1 − ˆγ(p) � : 0 ≤ ˆΓ ≤ 1 , ˆγ ∈ L1(Rd, (1 + p2)dp) , α ∈ H1 sym(Rd) � , where the entropy density is defined as S[Γ] = − � Rd TrC2 � ˆΓ(p) log ˆΓ(p) � dp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The minimization problem associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In fact, the following re- sult has only been proven for d = 3 and V ∈ L3/2(R3), but its extension to d = 1, 2 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 ([HHSS08], see also [HS16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 on V , the BCS free energy is bounded below on D and attains its minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The BCS gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) arises as the Euler–Lagrange equations of this functional [HHSS08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Namely by defining ∆ = −2 � V α, the Euler–Lagrange equation for α takes the form of the BCS gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Additionally, one has the following linear criterion for the BCS gap equation to have non-trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Again, so far, a proof has only been given in spatial dimension d = 3 and for V ∈ L3/2(R3), but its extension to d = 1, 2 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 ([HHSS08, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 and let µ ∈ R as well as T ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, writing FT[Γ] ≡ FT(γ, α), the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The minimizer of FT is not attained with α = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' inf (γ,α)∈D FT(γ, α) < inf (γ,0)∈D FT(γ, 0), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' There exists a pair (γ, α) ∈ D with α ̸= 0 such that ∆ = −2 � V α satisfies the BCS gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The linear operator KT + V , where KT (p) = p2−µ tanh((p2−µ)/(2T)) has at least one negative eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The third item immediately leads to the following definition of the critical temperature Tc for the existence of non-trivial solutions of the BCS gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4 (Critical temperature, see [FHNS07, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For V satisfying Assump- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1, we define the critical temperature Tc ≥ 0 as Tc := inf{T > 0 : KT + V ≥ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6) By KT(p) ≥ 2T and the asymptotic behavior KT(p) ∼ p2 for |p| → ∞, Sobolev’s inequality [LL01, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3] implies that the critical temperature is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The other object we study is the energy gap Ξ defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The energy gap depends on the solution ∆ of the gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' A priori, ∆ may not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' However, for potentials with non-positive Fourier transform, this possibility can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 (see [HS08b, (21)-(22) and Lemma 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 (and additionally V ∈ L1(R2) in case that d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Moreover, we assume that ˆV ≤ 0 and ˆV (0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, there exists a unique minimizer Γ of F0 (up to a constant phase in α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' One can choose the phase such that α has strictly positive Fourier transform ˆα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In particular, we conclude that ∆ is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Moreover, by means of the gap equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1), ∆ is continuous and thus Ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 4 2 Main Results As explained in the introduction, our main result in this short note is the extension of the universality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) from d = 3 to lower spatial dimensions d = 1, 2 in the limit of weak coupling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=', replacing V → λV and taking λ → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We assume the following properties for the interaction potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let d ∈ { 1, 2 } and assume that V satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 as well as ˆV ≤ 0, ˆV (0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Moreover, for d = 1 we assume that (1 + | · |ε)V ∈ L1(R1) for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Finally, in case that d = 2, we suppose that V ∈ L1(R2) is radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5, this means that, in particular, the minimizer of F0 is unique (up to a phase) and the associated energy gap at zero temperature (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2) is strictly positive, Ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We are now ready to state our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 (BCS Universality in one and two dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V be as in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then the critical temperature Tc(λ) (defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6)) and the energy gap Ξ(λ) (defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2)) are strictly positive for all λ > 0 and it holds that lim λ→0 Ξ(λ) Tc(λ) = π eγ , where γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='577 is the Euler-Mascheroni constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To prove the universality, we separately establish asymptotic formulas for Tc (see Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) and Ξ (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7), valid to second order, and compare them by taking their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The asymptotic formula for Tc is valid under weaker conditions on V than Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1, because we do not need uniqueness of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To obtain the asymptotic formulas, we first introduce two self-adjoint operators V(d) µ and W(d) µ mapping L2(Sd−1) → L2(Sd−1) and as such measuring the strength of the interaction ˆV on the (rescaled) Fermi surface (see [HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hen22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HL22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To assure that V(d) µ and W(d) µ will be well-defined and compact, we assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Additionally, assume that for d = 1, � 1 + (ln(1 + | · |))2� V ∈ L1(R1) and for d = 2, V ∈ L1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' First, in order to capture the strength to leading order, we define V(d) µ via (V(d) µ u)(p) = 1 (2π)d/2 � Sd−1 ˆV (√µ(p − q))u(q) dω(q) , where dω is the Lebesgue measure on Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Since V ∈ L1(Rd), we have that ˆV is a bounded continuous function and hence V(d) µ is a Hilbert-Schmidt operator (in fact, trace class with trace being equal to (2π)−d|Sd−1| � Rd V (x)dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Therefore, its lowest eigenvalue e(d) µ := inf spec V(d) µ satisfies e(d) µ ≤ 0 and it is strictly negative if e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' � V < 0 as in Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 5 Second, in order to capture the strength of ˆV to next to leading order, we define the operator W(d) µ via its quadratic form � u ��W(d) µ ��u � = µd/2−1 �� |p|< √ 2 1 |p2 − 1| � |ψ(√µp)|2 − |ψ(√µp/|p|)|2� dp + � |p|> √ 2 1 |p2 − 1||ψ(√µp)|2 dp � , where ψ(p) = 1 (2π)d/2 � Sd−1 ˆV (p−√µq)u(q) dω(q) and u ∈ L2(Sd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The proof of the following proposition shall be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let d ∈ { 1, 2 } and let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The operator W(d) µ is well-defined and Hilbert-Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Next, we define the self-adjoint Hilbert-Schmidt operator B(d) µ (λ) := π 2 � λV(d) µ − λ2W(d) µ � on L2(Sd−1) and its ground state energy b(d) µ (λ) := inf spec � B(d) µ (λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) Note that if e(d) µ < 0, then also b(d) µ (λ) < 0 for small enough λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' After these preparatory definitions, we are ready to state the separate asymptotic formulas for the critical temperature and the energy gap in one and two dimensions, which immediately imply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 (Critical Temperature for d = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 and additionally e(d) µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then the critical temperature Tc, given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4, is strictly positive and satisfies lim λ→0 � ln � µ Tc(λ) � + π 2 µd/2−1 b(d) µ (λ) � = −γ − ln �2cd π � , where γ denotes the Euler-Mascheroni constant and c1 = 4 1+ √ 2 and c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Here, the Assumptions on V are weaker than Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1, since ˆV (0) < 0 implies that e(d) µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We thus have the asymptotic behavior Tc(λ) = 2cd eγ π � 1 + o(1) � µ eπ/(2µd/2−1b(d) µ (λ)) in the limit of small λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 is essentially a special case of [HS10, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We give the proof here for two main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (i) There is still some work required to translate the statement of [HS10, Theorem 2] into a form in which it is comparable to that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7 (in order to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The main difficulty is that the operator W(d) µ in [HS10] is only defined via a limit, [HS10, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 6 (ii) The goal of this paper is to give an exemplary proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 in order to compare it to the proofs of the similar statements in the literature concerning the asymptotic behavior of the critical temperature in various limits [HS08a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hen22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 is complemented by the following asymptotics for the energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7 (Energy Gap for d = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 and let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then there exists a unique radially symmetric minimizer (up to a constant phase) of the BCS functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) at temperature T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The associated energy gap Ξ, given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2), is strictly positive and satisfies lim λ→0 � ln �µ Ξ � + π 2 µd/2−1 b(d) µ (λ) � = − ln(2cd) , where b(d) µ is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) and c1 = 4 1+ √ 2 and c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In other words, we have the asymptotic behavior Ξ(λ) = 2cd � 1 + o(1) � µ eπ/(2µd/2−1b(d) µ (λ)) in the limit of small λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Now, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 follows immediately from Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8 (Other limits in dimensions d = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Similarly to the presented results, one could also consider the limits of low and high density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' µ → 0 and µ → ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We expect that also here the universality Ξ Tc ≈ π eγ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Indeed, one would expect that the proofs of BCS universality in dimension d = 3 should carry over to one and two dimensions with some minor technical modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Note that, even for the (technically less demanding) case of a weak coupling limit, which we consider here, there are still some technical details that are different in dimensions d = 1, 2 compared to dimension d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, it is not a trivial matter to generalize the arguments of [HS08a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hen22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HL22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lau21] to one and two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Moreover, for the case of low density, there is even an issue of what exactly low density means in dimensions one and two: In three spatial dimensions [HS08a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lau21], the asymptotic formulas for Tc and Ξ were obtained for potentials V with negative scattering length but not creating bound states for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This latter condition ensures that µ → 0 actually corresponds to the limit of low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' However, in spatial dimensions one and two, attractive potentials, no matter how weak, always give rise to bound states of −∇2 + V , see [Sim76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Thus for µ = 0 the particle density is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We will not deal with the low- and high-density limits here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The rest of the paper is devoted to proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3 Proofs The overall structure of our proofs is as follows: The principal idea is to derive two different formulas for each of the two integrals m(d) µ (T) := 1 |Sd−1| � |p|<√2µ 1 KT(p) dp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1) 7 and m(d) µ (∆) := 1 |Sd−1| � |p|<√2µ 1 E∆(p) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2) The first set of formulas is derived by studying the Birman-Schwinger operators B(d) T := λV 1/2K−1 T |V |1/2 and B(d) ∆ := λV 1/2E−1 ∆ |V |1/2 , associated to the Schr¨odinger type operators KT +λV and E∆ +λV , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In particu- lar, spectral properties of these unbounded Schr¨odinger type operators naturally translate to their associated Birman-Schwinger operators, which are compact and as such much simpler to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The second set of formulas is obtained by just calculating the integrals m(d) µ directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Indeed, for the critical temperature we obtain the following asymptotics, which, by com- bining them, immediately prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 and additionally e(d) µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, the critical temperature Tc is positive and, as λ → 0, we have that m(d) µ (Tc) = − π 2b(d) µ (λ) + o(1) , m(d) µ (Tc) = µd/2−1 � ln � µ Tc � + γ + ln �2cd π � + o(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the energy gap we obtain the following asymptotics, which, again by combining them, immediately prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 and let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then (by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) we have a strictly positive radially symmetric gap function ∆ and associated energy gap Ξ, which, as λ → 0, satisfy the asymptotics Ξ = ∆(√µ) � 1 + o(1) � m(d) µ (∆) = − π 2b(d) µ (λ) + o(1) m(d) µ (∆) = µd/2−1 � ln � µ ∆(√µ) � + ln(2cd) + o(1) � With a slight abuse of notation, using radiality of ∆, we wrote ∆(√µ) instead of ∆(√µˆp) for some ˆp ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In the remainder of this paper, where we give the proofs of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2, we shall frequently use the notation F(d) µ : L1(Rd) → L2(Sd−1) for the (scaled) Fourier transform restricted to the (rescaled) Fermi sphere, � F(d) µ ψ � (p) := 1 (2π)d/2 � Rd ψ(x)e−i√µp·x dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Note that for an L1-function, pointwise values of its Fourier transform are well-defined by the Riemann–Lebesgue lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (In particular the restriction to a co–dimension 1 manifold of a sphere is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=') Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In [CM21], Cuenin and Merz use the Tomas-Stein theorem to define F(d) µ on a larger space than L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' With this they are able to prove a general version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 under slightly weaker conditions on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' However, we do not pursue this here, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The argument is divided into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' A priori spectral information on KTc + λV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' First note that, due to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4, the critical temperature Tc is determined by the lowest eigenvalue of KT +λV being 0 exactly for T = Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Birman-Schwinger principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Next, we employ the Birman-Schwinger principle, which says that the compact Birman-Schwinger operator B(d) T = λV 1/2K−1 T |V |1/2 (denot- ing V (x)1/2 = sgn(V (x))|V (x)|1/2) has −1 as its lowest eigenvalue exactly for T = Tc, see [FHNS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Using the notation for the Fourier transform restricted to the rescaled Fermi sphere in- troduced above, we now decompose the Birman-Schwinger operator as B(d) T = λm(d) µ (T)V 1/2(F(d) µ )†F(d) µ |V |1/2 + λV 1/2M(d) T |V |1/2, where M(d) T is defined through the integral kernel M(d) T (x, y) = 1 (2π)d �� |p|<√2µ 1 KT(p) � eip·(x−y) − ei√µp/|p|·(x−y)� dp + � |p|>√2µ 1 KT eip·(x−y) dp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) We claim that V 1/2M(d) T |V |1/2 is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then we have for all T ≥ 0 ���V 1/2M(d) T |V |1/2��� HS ≤ C , where C > 0 denotes some positive constant and ∥ · ∥HS is the Hilbert-Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Armed with this bound, we have that for sufficiently small λ that 1 + λV 1/2M(d) T |V |1/2 is invertible, and hence 1 + B(d) T = (1 + λV 1/2M(d) T |V |1/2) � 1 + λm(d) µ (T) 1 + λV 1/2M(d) T |V |1/2V 1/2(F(d) µ )†F(d) µ |V |1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Thus, the fact that B(d) T has lowest eigenvalue −1 at T = Tc is equivalent to λm(d) µ (T)F(d) µ |V |1/2 1 1 + λV 1/2M(d) T |V |1/2V 1/2(F(d) µ )† (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4) having lowest eigenvalue −1, again at T = Tc, as it is isospectral to the rightmost operator on the right-hand-side above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (Recall that for bounded operators A, B, the operators AB and BA have the same spectrum apart from possibly at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' However, in our case, both operators are compact on an infinite dimensional space and hence 0 is in both spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=') We now prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 9 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We want to bound the integral kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) of M(d) T uniformly in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, we will bound KT ≥ |p2 − µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The computation is slightly different in d = 1 and d = 2, so we do them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) is bounded by 2 � |p|>√2µ 1 |p2 − µ| dp = 2 arcoth √ 2 √µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the first integral, we use that |eix − eiy| ≤ min{|x − y|, 2}, |p2 − µ| ≥ √µ||p| − √µ|, and increase the domain of integration to obtain the bound 2 √µ � 2√µ 0 min � ||p − √µ||x − y|, 2 � |p − √µ| dp = 8 √µ � 1 + ln � max �|x − y|√µ 2 , 1 ��� ≤ 8 √µ(1 + ln(1 + √µ max{|x|, |y|}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We conclude that |M(1) T (x, y)| ≲ 1 √µ(1 + ln(1 + √µ max{|x|, |y|})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, ���V 1/2M(1) T |V |1/2��� 2 HS ≲ 1 µ � ∥V ∥2 L1(R) + ∥V ∥L1(R) � R |V (x)|(1 + ln(1 + √µ|x|))2 dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We first compute the angular integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Note that � S1 eipx dω(p) = 2πJ0(|x|), where J0 is the zeroth order Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3) we may bound |p2−µ| ≥ cp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Up to some finite factor, the second integral is hence bounded by � ∞ √2µ 1 p|J0(p|x − y|)| dp ≤ C � ∞ √2µ 1 p1+λ|x − y|−λ dp ≤ Cλ|x − y|−λ, for any 0 < λ ≤ 1/2 since |J0(x)| ≤ C and √xJ0(x) ≤ C, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' [BSMM12, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='55f), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='57a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the first integral we get the bound � √2µ 0 p |p2 − µ| |J0(p|x − y|) − J0(√µ|x − y|)| dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Here we use that J0 is Lipschitz, since its derivative J−1 is bounded (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' [BSMM12, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='55a), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='55f)]), so that |J0(x) − J0(y)| ≤ C|x − y|1/3(|J0(x)| + |J0(y)|)2/3 ≤ C|x − y|1/3 � x−1/3 + y−1/3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' That is |J0(p|x − y|) − J0(√µ|x − y|)| ≤ C |p − √µ|1/3 p1/3 + √µ1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This shows that the first integral is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We conclude that |M(2) T (x, y)| ≲ 1 + 1 |x−y|λ for any 0 < λ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, by the Hardy–Littlewood–Sobolev inequality [LL01, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3] we have that ���V 1/2M(2) T |V |1/2��� 2 HS = �� |V (x)||M(2) T (x, y)||V (y)| dx dy ≲ ∥V ∥2 L1(R2) + ∥V ∥2 Lp(R2) for any 1 < p ≤ 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' First order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Evaluating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4) at T = Tc and expanding the geometric series to first order we get −1 = λm(d) µ (Tc) inf spec � F(d) µ |V |1/2 1 1 + λV 1/2M(d) Tc |V |1/2V 1/2(F(d) µ )† � = λ m(d) µ (Tc) inf spec V(d) µ (1 + O(λ)) = λ m(d) µ (Tc) e(d) µ (1 + O(λ)) where we used V(d) µ = F(d) µ V (F(d) µ )†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Since by assumption e(d) µ < 0, this shows that m(d) µ (Tc) → ∞ as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' A priori bounds on Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1), the divergence of m(d) µ as λ → 0 in particular shows that Tc/µ → 0 in the limit λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Calculation of the integral m(d) µ (Tc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This step is very similar to [HS08b, Lemma 1] and [HRS22, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5], where the asymptotics have been computed for slightly different definitions of m(d) µ in three and one spatial dimension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Integrating over the angular variable and substituting s = ���|p|2 µ − 1 ���, we get m(d) µ (Tc) = µd/2−1 � 1 0 tanh � s 2(Tc/µ) � (1 + s)d/2−1 + (1 − s)d/2−1 2s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' According to [HS08b, Lemma 1], lim Tc↓0 \uf8eb \uf8ed � 1 0 tanh � s 2(Tc/µ) � s ds − ln µ Tc \uf8f6 \uf8f8 = γ − ln π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By monotone convergence, it follows that m(d) µ (Tc) = µd/2−1 � ln µ Tc + γ − ln π 2 + � 1 0 (1 − s)d/2−1 + (1 + s)d/2−1 − 2 2s ds + o(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The remaining integral equals ln cd and we have thus proven the second item in Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Combining this with the third step, one immediately sees that the critical temperature vanishes exponentially fast, Tc ∼ e1/λeµ, as λ → 0, recalling that e(d) µ < 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Now, to show the universality, we need to compute the next order correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To do so, we expand the geometric series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4) and employ first order perturbation theory, yielding that m(d) µ (Tc) = −1 λ � u ���F(d) µ V (F(d) µ )† ���u � − λ2 � u ���F(d) µ V M(d) Tc V (F(d) µ )† ���u � + O(λ3) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) where u is the (normalized) ground state (eigenstate of lowest eigenvalue) of F(d) µ V (F(d) µ )†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (In case of a degenerate ground state, u is the ground state minimizing the second order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=') This second order term in the denominator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5) is close to W(d) µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' More precisely, it holds that lim λ→0 � u ���F(d) µ V M(d) Tc V (F(d) µ )†���u � = � u ��W(d) µ ��u � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6) 11 which easily follows from dominated convergence, noting that 1 KT increases to 1 |p2−µ| as T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We then conclude that lim λ→0 � m(d) µ (Tc) + π 2b(d) µ (λ) � = 0 , since � u ���λV(d) µ − λ2W(d) µ ���u � = inf spec(λV(d) µ −λ2W(d) µ ) + O(λ3) = π 2b(d) µ (λ) + O(λ3), again by first-order perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This concludes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We conclude this subsection with several remarks, comparing our proof with those of similar results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='5 (Structure here vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' in earlier papers on Tc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We compare the structure of our proof to that of the different limits in three dimensions [HS08a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hen22]: Weak coupling: The structure of the proof we gave here is quite similar to that of [HS08b], only they do Steps 5 and 6 in the opposite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Also the leading term for Tc was shown already in [FHNS07], where a computation somewhat similar to Steps 1–4 is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' High denisty: For µ → ∞, the structure of the proof in [Hen22] is slightly different compared to the one given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This is basically due to the facts that (i) the necessary a priori bound Tc = o(µ) already requires the Birman-Schwinger decomposition and (ii) the second order requires strengthened assumptions compared to the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To conclude, the order of steps in [Hen22] can be thought of as: 1, 5, 4 (establishing Tc = O(µ)), 2, 3, 4 (establishing Tc = o(µ)), 2 (again), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Here the final step is much more involved than in the other limits considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Low density: As above, for the proof of the low density limit in [HS08a] the struc- ture is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' One first needs the a priori bound Tc = o(µ) on the criti- cal temperature before one uses the Birman-Schwinger principle and decomposes the Birman-Schwinger operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 Also, the decomposition of the Birman-Schwinger opera- tor is again different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the full decomposition and analysis of the Birman-Schwinger operator one needs also the first-order analysis, that is Step 2, which is done in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The order of the steps in [HS08a] can then mostly be though of as: 1, 4, 5, 2, 3, 2 (again), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The structure of the proof is parallel to that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1 for the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' A priori spectral information on E∆ + λV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' First, it is proven in [HS08b, Lemma 2] that F0 has a unique minimizer α which has strictly positive Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Using radial- ity of V , it immediately follows that this minimizer is rotationally symmetric (since otherwise 1Strictly speaking, in [HS08a], it is only proven that Tc = O(µ) (which is sufficient for applying the Birman-Schwinger principle), while the full Tc = o(µ) itself requires the Birman-Schwinger decomposition (see [Lau20, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='12] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 12 rotating α would give a different minimizer) and hence also ∆ = −2λ ˆV ⋆ ˆα is rotation in- variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' It directly follows from [HS08b, (43) and Lemma 3] that that E∆ + λV has lowest eigenvalue 0, and that the minimizer α is the corresponding eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Birman-Schwinger principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This implies, by means of the Birman-Schwinger princi- ple, that the Birman-Schwinger operator B(d) ∆ = λV 1/2E−1 ∆ |V |1/2 has −1 as its lowest eigen- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' As in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1, we decompose it as B(d) ∆ = λm(d) µ (∆)V 1/2(F(d) µ )†F(d) µ |V |1/2 + λV 1/2M(d) ∆ |V |1/2 and prove the second summand to be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Let V satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, uniformly in small λ, we have ���V 1/2M(d) ∆ |V |1/2��� HS ≤ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' With this one may similarly factor 1 + B(d) ∆ = (1 + λV 1/2M(d) ∆ |V |1/2) � 1 + λm(d) µ (∆) 1 + λV 1/2M(d) ∆ |V |1/2V 1/2(F(d) µ )†F(d) µ |V |1/2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7) and conclude that T (d) ∆ := λm(d) µ (∆)F(d) µ |V |1/2 1 1 + λV 1/2M(d) ∆ |V |1/2V 1/2(F(d) µ )† (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8) has lowest eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Note that M∆ has kernel M∆(x, y) = 1 (2π)d �� |p|<√2µ 1 E∆(p) � eip·(x−y) − ei√µp/|p|·(x−y)� dp + � |p|>√2µ 1 E∆(p)eip·(x−y) dp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We may bound this exactly as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4 using that E∆(p) ≥ |p2 − µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' First order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Expanding the geometric series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8) to first order, we see that −1 = λm(d) µ (∆) inf spec � F(d) µ |V |1/2 1 1 + λV 1/2M(d) ∆ |V |1/2V 1/2(F(d) µ )† � = λm(d) µ (∆) inf spec V(d) µ (1 + O(λ)) = λe(d) µ m(d) µ (∆)(1 + O(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, in particular, m(d) µ (∆) ∼ − 1 λe(d) µ → ∞ as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' A priori bounds on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We now prepare for the computation of the integral m(d) µ (∆) in terms of ∆(√µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This requires two types of bounds on ∆: One bound estimating the gap function ∆(p) at general momentum p ∈ Rd in terms of ∆(√µ) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='9)), and one bound controlling the difference |∆(p) − ∆(q)| in some kind of H¨older-continuity estimate (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 13 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Suppose that V is as in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then for λ small enough ∆(p) = f(λ) �� Sd−1 ˆV (p − √µq) dω(q) + ληλ(p) � , where f is some function of λ and ∥ηλ∥L∞(Rd) is bounded uniformly in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Recall that α is the eigenfunction of E∆ + λV with lowest eigenvalue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then, by the Birman-Schwinger principle, φ = V 1/2α satisfies B∆φ = λV 1/2 1 E∆ |V |1/2V 1/2α = −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' With the decomposition Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7) then φ is an eigenfunction of λm(d) µ (∆) 1 + λV 1/2M(d) ∆ |V |1/2V 1/2(F(d) µ )†F(d) µ |V |1/2 of eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Thus, F(d) µ |V |1/2φ is an eigenfunction of T (d) ∆ of (lowest) eigenvalue −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Now u = |Sd−1|−1/2 is the unique eigenfunction corresponding to the lowest eigenvalue of V(d) µ by radiality of V and the assumption ˆV ≤ 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' [FHNS07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, for λ small enough, u is the unique eigenfunction of T (d) ∆ of smallest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Thus, φ = f(λ) 1 1 + λV 1/2M(d) ∆ |V |1/2V 1/2(F(d) µ )†u = f(λ) � V 1/2(F(d) µ )†u + λξλ � for some number f(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The function ξλ satisfies ∥ξλ∥L2(Rd) ≤ C by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Noting that ∆ = −2 � |V |1/2φ and bounding ��� � |V |1/2ξλ ��� L∞ ≤ ∥V ∥1/2 L1 ∥ξλ∥L2 we get the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Evaluating the formula in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7 at p = √µ we get |f(λ)| ≤ C∆(√µ) for λ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This in turn implies that ∆(p) ≤ C∆(√µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='9) For the H¨older-continuity, we have by rotation invariance ���� � ˆV (p − √µr) − ˆV (q − √µr) dω(r) ���� = ���� � ˆV (|p|e1 − √µr) − ˆV (|q|e1 − √µr) dω(r) ���� = ���� 1 (2π)d/2 � Rd dx � V (x) � ei|p|x1 − ei|q|x1� � Sd−1 e−i√µx·r dω(r) ����� ≤ Cεµ−ε/2||p| − |q||ε � dx � |V (x)|(√µ|x|)ε ���� � Sd−1 e−i√µx·r dω(r) ���� � , for any 0 < ε ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For d = 2 we have V ∈ L1(R2) and ���� � Sd−1 e−i√µxr dω(r) ���� = |J0(√µ|x|)| ≤ (√µ|x|)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 14 For d = 1 we have |x|εV ∈ L1(R) for some ε > 0 and ���� � Sd−1 e−i√µx·r dω(r) ���� = 2| cos(√µ|x|)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We conclude that with ε = 1/2 for d = 2 and small enough ε > 0 for d = 1 |∆(p) − ∆(q)| ≤ C|f(λ)| � µ−ε/2||p| − |q||ε + λ � ≤ C|∆(√µ)| � µ−ε/2||p| − |q||ε + λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10) Additionally, since m(d) µ (∆) → ∞ we have that ∆(p) → 0 at least for some p ∈ Rd by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='7 that f(λ) → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' that ∆(p) → 0 for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Calculation of the integral m(d) µ (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Armed with the apriori bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10), we can now compute the integral m(d) µ (∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Carrying out the angular integration and substi- tuting s = ��� |p|2−µ µ ��� we have m(d) µ (∆) = µd/2−1 2 �� 1 0 � (1 − s)d/2−1 − 1 � s2 + x−(s)2 + (1 + s)d/2−1 − 1 � s2 + x+(s)2 � ds + � 1 0 � 1 � s2 + x−(s)2 + 1 � s2 + x+(s)2 � ds � , where x±(s) = ∆(√µ√1±s) µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By dominated convergence, using that x±(s) → 0, the first integral is easily seen to converge to � 1 0 �(1 − s)d/2−1 − 1 s + (1 + s)d/2−1 − 1 s � ds = 2 ln cd for λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the second integral, we will now show that � 1 0 � 1 � s2 + x±(s)2 − 1 � s2 + x±(0)2 � ds → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In fact, the integrand is bounded by ����� 1 � s2 + x±(s)2 − 1 � s2 + x±(0)2 ����� = |x±(0)2 − x±(s)2| � s2 + x±(s)2� s2 + x±(0)2( � s2 + x±(s)2 + � s2 + x±(0)2) ≤ Cx±(0)(sε + λ) � s2 + x±(s)2� s2 + x±(0)2, using the H¨older continuity from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By continuity of ˆV there exists some s0 (independent of λ) such that for s < s0 we have x±(s) ≥ cx±(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We now split the integration into � s0 0 and � 1 s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' For the first we have � s0 0 ����� 1 � s2 + x±(s)2 − 1 � s2 + x±(0)2 ����� ds ≤ C � s0 0 x±(0) s2 + x±(0)2(sε + λ) ds = O(x±(0)ε + λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 15 For the second we have � 1 s0 ����� 1 � s2 + x±(s)2 − 1 � s2 + x±(0)2 ����� ds ≤ C � 1 s0 x±(0)sε + λ s2 ds = O(x±(0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Collecting all the estimates, we have thus shown that m(d) µ (∆) equals µd/2−1 � ln cd + � 1 0 1 � s2 + ∆(√µ)2/µ2 ds + o(1) � = µd/2−1 � ln cd + ln � µ + � µ2 + ∆(√µ)2 |∆(√µ)| � + o(1) � = µd/2−1 ln � 2µcd |∆(√µ)| + o(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This proves the third inequality in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Combining this with the third step, one immediately sees that the gap function evaluated on the Fermi sphere vanishes exponentially fast, ∆(√µ) ∼ e1/λeµ, as λ → 0, recalling that e(d) µ < 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To obtain the next order, we recall that T (d) ∆ has lowest eigenvalue −1 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8)), and hence, by first-order perturbation theory, m(d) µ (∆) = −1 λ � u ���F(d) µ V (F(d) µ )† ���u � − λ2 � u ���F(d) µ V M(d) ∆ V (F(d) µ )† ���u � + O(λ3) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='11) where u(p) = |Sd−1|−1/2 is the constant function on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Recall that u is the unique ground state of V(d) µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In the second order term we have that lim λ→0 � u ���F(d) µ V M(d) ∆ V (F(d) µ )†���u � = � u ��W(d) µ ��u � , which follows from a simple dominated convergence argument as for Tc, noting that ∆(p) → 0 pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By again employing first–order perturbation theory, similarly to the last step in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='1, we conclude the second equality in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Comparing ∆(√µ) to Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' To prove the first equality in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 we separately prove upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The upper bound is immediate from Ξ = inf p∈Rd E∆(p) = inf p∈Rd � |p2 − µ| + ∆(p)2 ≤ ∆(√µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Hence, for the lower bound, take p ∈ Rd with � |p2 − µ| ≤ Ξ ≤ ∆(√µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10) ∆(p) ≥ ∆(√µ) −|∆(p) −∆(√µ)| ≥ ∆(√µ) −C∆(√µ) (||p| − √µ|ε + λ) ≥ ∆(√µ)(1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In combination with the upper bound, we have thus shown that Ξ = ∆(√µ)(1 + o(1)) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' This concludes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 16 We conclude this subsection with several remarks, comparing our proof with those of similar results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='8 (Structure here vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' in earlier papers on Ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We now compare the proof above to the proofs of the three different limits in 3 dimensions [HS08b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' HL22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Lau21]: Weak coupling: The structure of our proof here is very similar to that of [HS08b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Essentially, only the technical details in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='6 and the calculation of m(d) µ (∆) in Step 5 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' High density: For the high-density limit in [HL22], we needed some additional a priori bounds on ∆ before we could employ the Birman-Schwinger argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Apart from that, in [HL22] the comparison of ∆(√µ) and Ξ are done right after these a priori bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Additionally, since one starts with finding a priori bounds on ∆, one does not need the first-order analysis in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' One may think of the structure in [HL22] as being ordered in the above steps as follows: 4, 7, 1, 2, 4 (again), 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Low density: For the low-density limit in [Lau21] the structure is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Again, one first needs some a priori bounds on ∆ before one can use the Birman- Schwinger argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' One then improves these bounds on ∆ using the Birman-Schwinger argument, which in turn can be used to get better bounds on the error term in the decomposition of the Birman–Schwinger operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' In this sense, the Steps 2–4 are too interwoven to be meaningfully separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Also, Step 5 is done in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4 Note that W(d) µ = F(d) µ V M(d) 0 V (F(d) µ )†, where M(d) 0 is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='4, V 1/2M(d) 0 V 1/2 is Hilbert-Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' The integral kernel of W(d) µ is bounded by |W(d) µ (p, q)| ≤ 1 (2π)d � R2d |V (x)||M(d) 0 (x, y)||V (y)| dx dy ≤ 1 (2π)d∥V ∥1∥V 1/2M(d) 0 V 1/2∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='12) It follows that ∥W(d) µ ∥HS ≤ |Sd−1| (2π)d ∥V ∥1∥V 1/2M(d) 0 V 1/2∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' We thank Robert Seiringer for comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' grate- fully acknowledges partial financial support by the ERC 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='2 (1976), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 279–288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content='10 16/0003-4916(76)90038-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E5T4oBgHgl3EQfew9G/content/2301.05621v1.pdf'} diff --git a/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf b/adE3T4oBgHgl3EQfdAqs/content/2301.04531v1.pdf new file mode 100644 index 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In this context, we numerically analyze how phase- +field models converge to certain sharp-interface limits when the interface thickness tends to +zero ε → 0. In particular, we study the scaling of the Cahn-Hilliard mobility m(ε) = m0εα for +0 ≤ α ≤ ∞. In the presence of interfaces, it is known that the intended sharp-interface limit +is only valid for α < α < α. However, in the presence of moving contact lines we show that α +near α produces significant errors. +1 +Introduction +Interfaces and surfaces with surface energy are ubiquitous in nature, and their description can +employ different levels of detail. For multiphase systems with moving interfaces different mathe- +matical and numerical techniques are available. Explicit descriptions via front tracking methods +[23] allow high control over the moving boundary. Implicit description via level-set methods [8, 20] +or using phase fields lead to diffuse representations of interfaces [5]. The choice of method depends +on problem-specific features and requirements, e.g. feasibility of topological changes, treatment of +discontinuities at the interface, mass conservation and transport across interfaces, ease of imple- +mentation, availability of computational resources, precision and control over interface evolution. In +general for continuum models, sharp-interface and phase-field models are the most common levels +of abstraction with more or less details, respectively. +Such systems can be arbitrarily complicated when interfaces have a complex substructure, e.g., +with interfacial mass and thermodynamical effects. +This complexity increases even further by +considering nonlinear diffusion and phase separation, reactions and phase transitions and complex +viscoelastic properties as possible dissipative processes. +However, even for simple systems with +constant surface tension and fluid flow or elasticity in the pure phase, the connection between +common diffuse-interface models and their sharp limits is elusive. In phase-field models, the interface +is modeled by continuous functions ψ that are constant inside a phase, e.g. ψ = ±1, and whose +change in an interfacial region of width proportional to ε > 0 indicates the transition to another +phase in a continuous manner. The mathematical purpose of the phase field is twofold. Firstly, +∗Weierstrass +Institute, +Mohrenstrasse +39, +10117 +Berlin, +Germany +(leonie.schmeller@wias-berlin.de, +dirk.peschka@wias-berlin.de) +1 +arXiv:2301.04968v1 [math.AP] 12 Jan 2023 + +it should act as an indicator for the presence of a phase. Secondly, it defines a phase-field energy +density W ε +phase(ψ, F −T ∇ψ) that measures, among other things, the surface tension of the interface +with deformation gradient F . Without mechanics F = I the Cahn-Hilliard model +∂tψ = ∇ · (m∇µε) , +µε = δ +δψ W ε +phase , +W ε +phase = +3 +2 +√ +2 +� +ε +2|∇ψ|2 + 1 +4ε(1 − ψ2)2� +, +(1) +is commonly used to model phase-field evolution and phase separation. Its well-known that W ε +phase +converges to the perimeter of the interface connecting regions with ψ = ±1. The limit passage of +dynamic phase-field models to sharp-interface models can be analyzed by different techniques, e.g. +by matched asymptotics [16, 12, 2], Γ-convergence [17] and evolutionary convergence. +Degenerate +ψ-dependent mobilities m(ε, ψ) are highly relevant for the limit passage from the diffuse to the +sharp-interface model, which is studied in detail in [2, 16, 10]. +In many studies, only one set of interfacial width ε and mobility m is used for a specific appli- +cation leading to reasonable results [24, 9, 13, 7, 18]. Instead, we focus on a systematic study of +the appropriate choice of the Cahn-Hilliard mobility. We extend the work by Yue et al. [25] by con- +sidering fluid-structure interaction and by directly comparing sharp and diffuse-interface models. +The goal of this work is to establish a general numerical approach to the problem of sharp-interface +limits with moving contact lines, somewhat in the spirit of previous work by Liu et al. [14] and +Aland et al. [4]. We consider m(ε) = m0εα with an appropriate 0 ≤ α ≤ ∞ and determine the +scaling of m. Usually, for interface problems 1, m = εm0 yields the intended interface condition [2] +and gives an upper bound on the mobility. Further, the authors in [1, 21] show that there is also +a lower bound on the Cahn-Hilliard mobility which suggests that the intended limit is reached for +α ∈ (α, ¯α). However, since such techniques are inevitably much more complex for moving contact +lines and require considerations in higher-dimensions, we will resort to numerical techniques. +In order to explore the sharp-interface limit of phase-field models, the strategy of the paper is +as follows. In Section 2 the phase-field evolution (1) is extended to a ternary multiphase systems +with solid (s), liquid (ℓ) and air (a) phases. Similarly, we provide a sharp-interface model of the +supposedly same system. Nonlinear elasticity of compressible materials and appropriate coupling +terms for phase-field and sharp-interface model are introduced using an hyperelastic energy density +Welast(F , ψ) with deformation gradient F = ∇χ. At the moving contact line we assume an equi- +librium of surface forces. We provide an Eulerian formulation of the system of nonlinearly coupled +PDEs and also provide the corresponding Lagrangian weak formulation that uses the underlying +gradient flow structure of the model. +For the phase-field model we discuss different possible scaling limits of the mobility m in order +to have convergence to the sharp-interface model as ε → 0. We also review existing results on +scaling laws and options for choosing the Cahn-Hilliard mobility for models with interfaces. Then +we introduce the technical methodology with which we perform our numerical convergence analysis. +In Section 3 we perform the discretization of both phase-field and sharp-interface model in order +to characterize the sharp-interface limit with contact lines using suitable norms ∥ · ∥. The goal here +is to discuss the distance ∥χ0,∆ − χε,∆′∥ of numerical solutions of the sharp-interface model χ0,∆ +and of the phase-field model χε,∆′ after interpolating between different discrete function spaces for +different discretization parameters ∆, ∆′. +Most importantly, this requires a discussion of discretization errors of the two models with +respect to time-discretization, mesh refinement and the polynomial order of the finite element +spaces. Therefore, as the main tool we use the Bochner norms for L2(QT ; Rd) and L∞(QT ; Rd) +1excluding cases with three-phase contact lines +2 + +for the deformation to assess the error in the space-time cylinder QT = [0, T] × Ω as ε → 0 for +different m(ε) = m0εα. Finally, in Section 4 we perform a discussion of the numerical solutions +generated with the methodology and based on the error analysis introduced in Section 3. We focus +on comparing concrete solutions of the benchmark problem “liquid droplet on viscoelastic substrate”. +The main goal of Section 4 and this entire work is to identify exponents α ≤ αopt ≤ α that +give an optimal mobility mopt = m0εαopt in the sense of (error of) the sharp-interface limit and to +provide a systematic approach to identify such αopt. While such a statement usually depends on +the norm, our choice of norm and discussion is based on the prediction of the moving contact line. +2 +Models for diffuse and sharp interfaces in variational form +Cahn-Hilliard-Navier-Stokes systems are widely used for modeling multiphase flows in various appli- +cations as well as for incompressible viscous Newtonian fluids [7, 2]. Since the numerical treatment +of physical interface thicknesses ε on the nanoscale is often infeasible for phase-field models, it is +important to clearly understand the behavior for ε → 0, i.e. the sharp-interface limit. The Cahn- +Hilliard mobility m depends on ε and possibly on the phase field variable ψ [19]. An advantage of +ψ-dependent degenerate mobilities is the ability to guarantee that the phase field variables remain +in a fixed, application-specific interval, which in general is a absent property. +As a typical test case we use an outer cylinder Ω = [0, R] × [0, H] with R = 1 and H = 1. The +region Ωs = {(r, z) ∈ Ω : 0 < z < 1} defines the solid phase, Ωℓ = {(r, z) ∈ Ω : +� +r2 − (z − 1)2 < +rdrop, z > 1} with rdrop = 1/2 defines the liquid phase, and the remainder Ωa = Ω \ (Ωℓ ∪ Ωs) is the +air phase. The undeformed reference domains Ωs (red) and Ωℓ (blue) are shown in the left image +of Figure 1. The corresponding deformed domains ¯Ωi(t) = χ(t, Ωi) from the sharp-interface model +at time t = T are shown in the right image of Figure 1. In this image, the air phase is not shown +to improve visibility of the liquid and solid phases and their three-dimensional impression. +Figure 1: Slice of (left) initial liquid droplet (blue half sphere) on a flat elastic substrate (red cylinder). +For x = (r, z) ∈ Ω, at z = 0 (solid bottom) and z = H (air top) we have no-slip conditions χ(x) = +(χr(x), χz(x)) = (r, z) and at r = 0 and r = R we fix the normal component χr(r) = r and have natural +boundary conditions (symmetry/slip) for the tangential component χz. (Right) Deformed droplet ¯Ωℓ(t) +(blue) and deformed solid substrate ¯Ωs(t) (red) at time t = 1.15 as a solution of the sharp-interface model. +The interfaces are visualized using thick white curves and the deformation in the solid phase is visualized +using a thin white mesh. +3 + +Before we explain the diffuse-interface model and the sharp-interface model, we would like to +make a comment on simplifying modeling assumptions and our approach. +• We neglect inertia to avoid effects caused by elastic or capillary waves. +• We formulate the viscoelastic models entirely in Lagrangian coordinates. +• We assume no-slip conditions between phases and all phases have the same viscosity µ. +• We consider a multiphase system with solid, liquid and gas phases, where the liquid and gas +phases are characterized by the absence of elastic properties. +• We consider slightly compressible phases via κ +2 (det F − 1)2 in the elastic energy with κ ≫ 1. +Locking is avoided by using sufficiently high polynomial degree in the FE discretization [6]. +While we made these choices to make it easier to follow the presentation or avoid certain technical +difficulties, none of these choices should restrict the validity of our findings concerning the sharp- +interface limit in the presence of moving contact lines. +2.1 +Diffuse-interface model +First we present a ternary phase-field model, that is capable to describe the evolution of a liquid +phase and a soft elastic phase surrounded by a gas phase. Therefore, within a given fixed box +Ω ⊂ Rd for d = 2, 3 let ψ : [0, T] × Ω → R2 be a ternary phase-field and χε : [0, T] × Ω → Rd a +deformation field. We further decompose the phase-field ψ = (ψs, ψℓ) into solid and liquid phase, +so that the remaining air phase ψa is determined implicitly by ψa = −1 − ψs − ψℓ so that ψa = +1 +if ψs = ψℓ = −1. Using this convention, ψi = 1 indicates the presence of phase i and ψi = −1 the +absence of phase i for i ∈ {s, ℓ, a}. We denote the deformation gradient F = ∇χε and the Jacobian +determinant J = det(F ). For qε = (χε, ψ) ∈ Qε, the cornerstone of the phase-field model is the +free energy Fε : Qε → R +Fε(qε) = +� +Ω +W ε[qε] dx, +W ε[qε] = Welast(F , ψ) + W ε +phase(ψ, F −T ∇ψ), +(2) +for which here we choose +Welast(F , ψ) = G(ψ) +2 +tr(F T F − I) + κ +2 (J − 1)2 +(3a) +W ε +phase(ψ, F −T ∇ψ) = +� +i∈{s,ℓ,a} +γi +� +ε +2|F −T ∇ψi|2 + 1 +4ε(1 − ψ2 +i )2� +J +(3b) +with surface parameters γi that correspond to the usual surface tensions via +γsa = +3 +2 +√ +2(γs + γa), +γsℓ = +3 +2 +√ +2(γs + γℓ), +γℓa = +3 +2 +√ +2(γℓ + γa) . +(4) +The function G(ψ) in (3a) interpolates the elastic properties of the different phases by +G(ψ) = 1 +2Gℓ(1 + ψℓ) + 1 +2Gs(1 + ψs) + 1 +2Ga(1 + ψa) , +(5) +4 + +with Gi ∈ R, but we are going to set Gℓ = Ga = 0 and rescale Gs = 1. The material is nearly +incompressible, which we achieve by setting κ ≫ 1. On the boundary of the domain we use either +homogeneous Dirichlet boundary conditions χε(x) − x = 0 for the entire displacement or for the +normal component. In the latter case, homogeneous natural boundary conditions are implied for +the tangential component. +Motivated by the extended gradient structures concept introduced in [22], we use a dynamical +model for the evolution qε : [0, T] → Qε with chemical potentials η : [0, T] → U that satisfy +aε(η, φ) + b(φ, ∂tqε) = 0 , +b(η, v)s(∂tχε, vχ) = ⟨DFε(qε), v⟩Vε , +(6) +for all φ ∈ U = H1(Ω; R2) and all v = (vχ, vψ) ∈ Vε = H1(Ω; Rd) × H1(Ω; R2) and given initial +values qε(t = 0) ∈ Qε. For this model we use the bilinear forms +aε(η, φ) = +� +Ω +m(ε)F −T ∇η · F −T ∇φ dx, +(7a) +b(φ, v) = +� +Ω +φvψ dx, +(7b) +s(w, v) = +� +Ω +µ(ψ) ((∇w)F −1) : ((∇v)F −1) det F dx, +(7c) +with phase-dependent viscosity function +µ(ψ) = 1 +2µℓ(1 + ψℓ) + 1 +2µs(1 + ψs) + 1 +2µa(1 + ψa) , +(8) +with µi ∈ R. For simplicity here we choose µi = µ, such that the viscosity is constant µ = µ(ψ). +By testing (6) with v = ∂tqε and φ = η we directly obtain thermodynamic consistency +d +dtFε = ⟨DFε(qε), ∂tqε⟩V = −(s(∂tχε, ∂tχε) + aε(η, η)) ≤ 0. +(9) +Note that for clarity of the presentation, the dependence of the bilinear forms on the state variable +qε is not shown explicitly. +Remark 1. We make all the computations for d = 3 by assuming axisymmetry. Therefore, the +weak formulation in cylindrical coordinates is obtained by replacing the domain with a cylinder of +radius R and height H such that Ωr = [0, R] × [0, H]. Using w = (wr, wz) we make the following +replacements +dx = 2πdr dz, +∇w = +� +� +∂rwr +0 +∂zwr +0 +r−1wr +0 +∂rwz +0 +∂zwz +� +� , +∇η = +� +� +∂rη +0 +∂zη +� +� , +for the integration measure dx, for gradients of vector fields ∇w and for gradients of scalar fields +∇η. These replacements are performed consistently in the bilinear forms and the free energy. +2.2 +Sharp-interface model +Now we present the corresponding sharp-interface model of the multiphase model that is our candi- +date for the sharp-interface limits as ε → 0. Using the same fixed box Ω ⊂ Rd instead we consider +5 + +that each phase i ∈ {s, ℓ, a} occupies a sufficiently regular subdomain Ωi ⊂ Ω that do not overlap +Ωi ∩ Ωj = ∅ and fill the entire domain, i.e., Ω = Ωs ∪ Ωℓ ∪ Ωa. For q0 ≡ χ0 ∈ Q0, the cornerstone +of the sharp-interface model is also a free energy F0 : Q0 → R +F0(q0) = +� +i∈{s,ℓ,a} +� +Ωi +W i +elast(F ) dx + +� +ij∈{sℓ,sa,ℓa} +� +Γij +γij|cof(F ) · ν| ds , +(10) +where F = ∇χ0 as before and for which here we choose +W i +elast(F ) = Gi +2 tr(F T F − I) + κ +2 (J − 1)2, +(11) +and for the surface tensions γij from (4) associated to the (sharp) interfaces Γij = ∂Ωi ∩ ∂Ωj +for ij ∈ {sℓ, sa, ℓa}. Also motivated by a gradient structure, we use a dynamical model for the +evolution q0 : [0, T] → Q0 that satisfies +−s(∂tχ0, vχ) = ⟨DF0(q0), vχ⟩V , +(12) +for all vχ ∈ V0 = H1(Ω; Rd) and given initial values q0(t = 0) ∈ Q0. For this model we use the +same bilinear form as before, i.e. +s(w, v) = +� +i∈{s,ℓ,a} +� +Ωi +µi ((∇w)F −1) : ((∇v)F −1) det F dx , +(13) +with phase-dependent viscosities µi ∈ R, which here we set equal to µi = µ. By testing Equation (12) +with ∂tχ0 we directly obtain thermodynamic consistency +d +dtF0 = ⟨DF0(q0), ∂tχ0⟩V0 = −s(∂tχ0, ∂tχ0) ≤ 0. +(14) +Remark 2. Assuming axisymmetry, we additionally need to replace +dsx = 2πrdsr, +ν = +� +� +νr +0 +νz +� +� , +the surface measure dsx and the normal vector ν in the energy F0. +Remark 3 (Eulerian formulation). The Eulerian free energy for the phase-field model (2) with +¯qε = ( ¯F , ¯ψ) is +Fε(¯qε) = +� +Ω +¯J−1 � +Welast( ¯F , ¯ψ) + W ε +phase( ¯ψ, ∇ ¯ψ) +� +d¯x , +(15) +where ¯ψ ◦ χε = ψ and ¯F ◦ χε = F and ¯J = det ¯F . The free energy of the sharp-interface model +(10) in the Eulerian frame is +F0(¯q0) = +� +i∈{s,ℓ,a} +� +¯Ωi +¯J−1W i +elast( ¯F ) d¯x + +� +ij∈{sℓ,ℓa,sa} +� +¯Γij +γij d¯s , +(16) +where the Eulerian state ¯q0 = ( ¯F , ¯Ωi) contains ¯Ωi(t) = χ0(t, Ωi(0)) the domain currently occupied by +phase i ∈ {s, ℓ, a} at time t. The PDEs in Eulerian variables also contain a velocity ¯u : [0, T]×Ω → +Rd, which, however, the free energy does not depend on for viscous flows without kinetic energy. +6 + +2.3 +Scaling limits for the Cahn-Hilliard mobility +Varying choices/scalings of the Cahn-Hilliard mobility in ε, may approximate different sharp- +interface models. Generally speaking, a bound m(ε) ≲ O(ε) on the mobility is needed to prevent +artificial diffusion of the phase fields [11]. +A further bound from the other side is essential to +ensure that the mobility does not become too small, which would lead to a different/unintended +sharp-interface limit [21]. +A fundamental work on the passage from the phase-field model to the sharp-interface limit is [2]. +The sharp interface is analyzed using matched asymptotic techniques, and degenerate mobilities +are included in their study. Precisely, the following four cases are discussed: +m(ε, ψ) = +� +� +� +� +� +� +� +� +� +m0 +Case 1 +εm0 +Case 2 +m1 +ε (1 − ψ2)+ +Case 3 +m1(1 − ψ2)+ +Case 4 +, +where for Case 2 and 4 a sharp-interface limit corresponding to ours is reached. In the other cases, +additional terms appear in the kinematic conditions of the sharp-interface model. +Having introduced the diffuse and the associated sharp-interface model, we will now analyze the +limit passage ε → 0. In this work we want to generalize some considerations concerning viable sharp- +interface limits, for which the passage from the diffuse-interface model (2) to the sharp-interface +model (10) is valid including contact lines. The sharp-interface limit, ε → 0, of the diffuse-interface +model depends substantially on the scaling of the Cahn-Hilliard mobility m = m0εα, 0 ≤ α ≤ ∞. +Degenerate mobilities or other forms of Cahn-Hilliard mobility are also used and require appropriate +scaling properties [2, 12, 10]. +We believe the setup shown in Section 2 has several mathematical and numerical challenges that +make it a suitable benchmark to study the sharp-interface limit of phase-field models: +• For this setup, the stationary state is close to the initial data so that Lagrangian methods can +compute a regular deformation χ without needing ALE methods or remeshing. +• With the initially flat solid substrate, surface energies will quickly enforce a contact angle (kink +in the solid surface) via the Neumann triangle construction. While the resulting nonsmoothness +in F is a challenge, this scenario is typical for soft wetting applications. This could lead to +suboptimal convergence rates in space discretizations. +• If initial data do not satisfy the Neumann triangle construction, then solutions are also expected +to show nonsmooth behavior in time, leading to suboptimal convergence in time. +In the following, in order to improve the visibility of results we will show solutions of the sharp- +interface model and of the phase-field model only as cross sections in the (r, z)-plane. Figure 2 +shows in the left half of each droplet the result of the sharp-interface at fixed time t = 1.15, while +the right half shows the diffuse-interface model with ε = 2¯ε with ¯ε = 1.8775 · 10−3 at the same +time with two different mobilities. For m = εα with α = 5/2 used in the left/first figure, we see a +comparison. In the second/right droplet we choose m = 0 and a clear deviation between the diffuse +and the sharp-interface model can be seen. The reason for this deviation is that the phase-field +ψ upon deformation to ¯ψ ◦ χε = ψ can not relax to its optimal tanh-profile and therefore the +approximation of the Eulerian surface energy is not guaranteed [21]. Thus, this indicates that the +7 + +Cahn-Hilliard mobility is chosen too small, i.e. α < α. Furthermore, the upper bound to the Cahn- +Hilliard mobility must also be respected. The next step is to numerically analyze the convergence +of the models in appropriate norms. +3 +Discretization in space and time +In this section, we numerically estimate the distance of the the phase-field and sharp-interface +models to make a statement about the convergence to the sharp-interface limit. Therefore, the two +main assumptions that need to be verified are: +A1 The deformation converges ∥χ0 − χε∥ → 0 as ε → 0 for a suitable norm ∥ · ∥. +A2 The (Lagrangian) phase fields ψi remain close to Ωi in the sense +distance(Ωi,ε(t), Ωi) → 0, +as ε → 0 for time t and a suitable distance between Ωi,ε(t) = {x ∈ Ω : ψi(t, x) > 0} and the +(time-independent) sharp domains Ωi for i ∈ {s, ℓ, a}. +In order to make meaningful statements about this convergence, we need to estimate the numerical +errors in the respective distances and norms. Let us denote by χε,∆ a numerical solution of the +diffuse-interface model for fixed ε and by χ0,∆ a numerical solution of the sharp-interface model. +As above by χε and χ0 we denote the exact solutions. Then, using the triangle inequality, we have +the estimate for the sharp-interface limit +∥χ0 − χε∥ +� +�� +� +sharp-interface limit +≤ +∥χ0 − χ0,∆∥ +� +�� +� +error sharp interface e0,∆ ++ +∥χ0,∆ − χε,∆∥ +� +�� +� +estimate sharp-interface limit ++ +∥χε,∆ − χε∥ +� +�� +� +error diffuse interface eε,∆ +(17) +a) +b) +c) +Figure 2: In a,b) the left side shows the sharp interface at time t = T = 1.15 and the right side is the +diffuse interface model for mobility a) m = ε5/2 and b) m = 0 with ε = 2¯ε in both cases. The deformed +phase fields are plotted via ¯ψid = (2 + ¯ψs − ¯ψℓ)/2 taking values ¯ψid = 0, 1, 2 in the liquid (blue), air +(gray), solid phase (red), respectively. c) Deformed diffuse phase fields for a) showing (left) liquid phase +−1 ≤ ¯ψℓ ≤ 1 and (right) solid phase −1 ≤ ¯ψs ≤ 1. +8 + +0 +-1in an appropriate function space norm ∥ · ∥. The first goal here is to estimate various contributions +to the approximation errors e0,∆ = ∥χ0 − χ0,∆∥ and eε,∆ = ∥χε,∆ − χε∥ for simulation parameters +associated with the discretization such as ∆ = {τ, h, kχ, kψ} for time-step size τ, mesh size h, poly- +nomial degree for deformation kχ, and for the polynomial degree for the phase fields kψ. These we +verify by time-step bisection τ → τ/2, uniform refinement h → h/2, and by varying the polyno- +mial degree. Then we compute ∥e0,∆−∆′∥ = ∥χ0,∆ − χ0,∆′∥ for different τ and ∆ = {τ, h, kχ, kψ} +and ∆′ = {τ/2, h, kχ, kψ} to approximate the convergence of the time-discretization and for the +other cases respectively. In such a case we write for simplicity ∥e0,∆−∆′∥ ≡ ∥e0,τ−τ/2∥ and as- +sume all other discretization parameters are known an fixed. +For the space discretizations the +computation of the error might also involve a projection Ph : V0,h → Vε,h′ or Ph : V0,h → V0,h′ or +Ph : Vε,h → Vε,h′, but this resulting projection error was mostly negligible compared to the error +of the other approximations. +The error between the deformation fields is measured in L2 and L∞ Bochner norms which is a +concept that extends the classical Sobolev norms [3] to time-dependent problems via +∥χ∥L2(QT ;Rd) = +�� T +0 +∥χ(t)∥2 +L2(Ω;Rd)dt +� 1 +2 +, +∥χ∥L∞(QT ;Rd) = ess sup +t∈[0,T ] +∥χ(t)∥L∞(Ω;Rd) . +(18) +While the L2 Bocher norm measures the error over the whole area and neglects errors in small +regions, the L∞ norm is also sensitive to errors in small regions, e.g. on interfaces and at contact +lines. First we introduce separately the space and time discretizations of phase-field and sharp- +interface model. +Remark 4. Using the deformations, we can map A2 to an Eulerian version and require +distance(¯Ωi,ε(t), ¯Ωi(t)) → 0, +(19) +as ε → 0 for any time t with ¯Ωi,ε(t) = χε(t, Ωi,ε(t)) and ¯Ωi(t) = χ0(Ωi) and i ∈ {s, ℓ, a}. Assuming +convergence A1 in a sufficiently strong norm, then A2 and (19) should become equivalent. +We +ensure A2 by making m(ε) small enough as ε → 0. +3.1 +Space and time discretization +Space-discretization +Based on the weak formulation of the diffuse-interface model (6) and the +sharp-interface model (12) we employ the finite element method to derive a discretization in space. +The main idea of this structure preserving discretization is to adopt a weak formulation for the +phase fields with Qh,ε ⊂ Qε, Uh ⊂ U, Vε,h ⊂ Vε and for the sharp-interface limit with Qh,0 ⊂ Q0, +V0,h ⊂ V0 via a finite element method. +For the sharp-interface model we use computational meshes, where the elements and edges are +aligned with the phases Ωi and the interfaces Γij as shown in the first (left) panel of Figure 3. +For the convergence study we usually use finer meshes for the sharp-interface model, where we +perform 1-3 uniform refinements of the first mesh (mesh a) shown in Figure 3 and project the +vertices of Γℓa (interface between blue and gray domain) back onto the set +� +r2 + (z − 1)2 = 1/2. +This coarsest mesh has 453 vertices resulting in the dimension dim Vh,0 = 3 496, while after three +uniform refinements the mesh has 27 221 vertices resulting in the dimension dim Vh,0 = 216 786 +using P2 finite elements for the deformation. +9 + +For the diffuse-interface model the base mesh strongly depends on the values of the interfacial +thickness ε, where we consider for ¯ε = 0.001875 the values ε = n¯ε for n = 1, 2, 4, 8, 16, 32. The +corresponding meshes shown in Figure 3 have 1 310 (mesh b), 5 205 (mesh c), 19 502} (mesh d) +vertices corresponding to dim Vh,ε = 10 322, dim Vh,ε = 41 458, dim Vh,ε = 155 810 using P2 finite +elements for the deformation, respectively. On top of that, the phase-field model contains scalar +unknowns for the phase fields ψ and their chemical potentials η. Note that the mesh is refined near +the interfaces to resolve the transition layer of width ε and in a larger region near the contact line +to account for potential artificial diffusion. +Let Pk(T, Rr) be the set of Rr-valued polynomials of degree k restricted to triangles T ∈ Th and +Ω = � +T ∈Th T an admissible triangulation of the ternary system. We use spaces V k for vectorial +unknowns and V k for scalar unknowns, where we define the H1-conforming finite element spaces +V k +h = {v ∈ C1(Ω, Rd) : v|T ∈ Pk(T, Rd), T ∈ Th}, +V k +h = {v ∈ C1(Ω, R) : v|T ∈ Pk(T, R), T ∈ Th} . +For the sharp-interface model we use Vh,0 = V kχ +h +with kχ = 1, 2, 3 and enforce essential boundary +conditions when solving the corresponding nonlinear problem and for the diffuse-interface model +a) +b) +c) +d) +Figure 3: (Top) Different initial meshes from (left) to (right) for a) sharp interface, b) phase-field with ε = +16¯ε, c) phase-field with ε = 4¯ε and d) phase-field with ε = ¯ε for ¯ε = 0.001875 and (bottom) corresponding +phase indicators. For the phase field we show ψid. +10 + +we use +Vh,ε = V kχ +h × V kψ +h +× V kψ +h , +Uh = V kψ +h +× V kψ +h , +with kχ = 1, 2, 3 and kψ = 2, 3. As for the sharp-interface model, essential boundary conditions +for the deformation are enforced when solving the corresponding nonlinear problem. Nonlinear +elasticity problems with compressibility κ ≫ 1 are likely to show locking phenomena, which we deal +with by choosing kχ large enough. +Time-discretization +The time discretization is constructed as in [22] based on an incremental- +minimization-scheme for qk +ε = qε(kτ) and qk +0 = q0(kτ) with time-step size τ. Based on the previous +formal definition of the weak formulations for the diffuse-interface model in (6) and sharp-interface +model in (12) from Section 2 this leads to the following incremental nonlinear problems. +Incremental scheme for sharp-interface model: For given qk−1 +0 +≡ χk−1 +0 +∈ Vh,0 find qk +0 ≡ χk +0 ∈ +Vh,0 such that +− 1 +τ s(χk +0 − χk−1 +0 +, vχ) = ⟨DF0(χk +0), vχ⟩Vh,0 , +(20) +for all vχ ∈ Vh,0. While the left-hand side of the problem appears is linear if in s we use the +previous state qk−1 +0 +, it is usually nonlinear through the dependence of DF0(χk +0) on χk +0. +Incremental scheme for diffuse-interface model: For given qk−1 +ε += (χk−1 +ε +, ψk−1) ∈ Vh,ε and +ηk−1 ∈ Uh find qk +ε = (χk +ε, ψk) ∈ Vh,ε and ηk ∈ Uh such that +aε(ηk, φ) + 1 +τ b(φ, qk +ε − qk−1 +ε +) = 0 , +b(ηk, v) − 1 +τ s(χk +ε − χk−1 +ε +, vχ) = ⟨DFε(qk +ε ), v⟩Vh,ε , +(21) +for all v = (vχ, vψ) ∈ Vh,ε and all φ ∈ Uh. +To initialize the chemical potentials η(t = 0) in the diffuse-interface model and to overcome +any initial transient (in both models) we perform the first iteration with a small time-step size +0 < τ0 ≪ τ. The next 20 iterations are performed with a time-step size τ/10 and then we output +the solutions qk +0,∆ and qk +ε,∆ every multiples of 0.05 time units until the final time T is reached. Next +we will discuss space-time discretization errors for numerical solutions +q0,∆ +→ +∆ = {τ, h, P kχ}, +(22) +qε,∆ +→ +∆ = {τ, h, P kχ, Pkψ}, +(23) +of the sharp-interface model q0,∆ and the diffuse-interface model qε,∆ with the discretization pa- +rameters mentioned before, i.e. h indicates the space discretization, τ is the time step size, kχ and +kψ denote the polynomial degree of the deformation and the phase fields, respectively. +3.2 +Estimation of numerical errors +Parameters +The physical parameters for the sharp-interface and diffuse-interface model are listed +in Table 1 and the corresponding surface tensions for the sharp-interface model can be derived from +them using Equation (4). For integer n ≥ 0, interfacial widths for the diffuse-interface model are +11 + +ε = 2n¯ε with ¯ε = 1.8775 · 10−3. Our standard numerical parameters, i.e. the once causing the error +collected in ∆, are +τ = 0.005 , +kχ = 2 , +kψ = 2 , +m0 = 1 , +(24) +and τ0 = 0.5 · 10−7. The common mesh for the diffuse-interface model corresponds to 4¯ε which is +Case c) and for the phase-field model two uniform refinements of Case a) shown in Section 3.1. The +time evolution of the energy2 is shown in Figure 4. We will consider solutions for 0 < t < T = 1.15, +shown by the horizontal dotted line, after which the evolution can be considered stationary and +no nontrivial contribution to the sharp-interface limit tested using the Bochner norms is expected. +The energies of the two models agree well for m = ε5/2 but are systematically lower for m = ε1. +Note: We omitted the factor 2π in all surface and volume integrals, which modifies the L2 error +norms and energies correspondingly. +Figure 4: Free energies F0 and Fε (full lines) and surface energies F0,s = � +ij +� +¯Γij γijd¯s and Fε,s = +� +Ω W ε +phase dx (dashed lines) as a function of time 0 ≤ t ≤ T for sharp (red) and diffuse interfaces with +mobilities m = ε1 (yellow) and m = ε5/2 (blue). +parameter +Gs +Gℓ,a +γs +γℓ +γa +R +H +rdrop +κ +T +value +10 +0 +0.5 +2.5 +3.5 +1.0 +2.0 +1/2 +104 +1.15 +Table 1: Parameters for the sharp-interface and diffuse-interface model: Gi the shear modulus in the +elastic energy, γi the surface parameter for i ∈ {s, ℓ, a}, R is the width of the rotational domain, rdrop is +the initial radius of the droplet, h is the height of the substrate, H is the height of the domain, and T the +final time. +2The energy is computed using the sharp-interface model with P2 elements on the coarsest mesh a) from Figure 3. +12 + +3.5 +m +3.4 +5/2 +3.3 +sharp +energy +3.2 + 1.15 +3.1 +3 +2.9 +0 +0.5 +1 +T +1.5 +time tDiffuse-interface model +The numerical errors of the phase-field model measured using the two +Bochner norms are shown in Table 2. We observe clear convergence trends in space and time. As +expected, errors in the L2 norm are always smaller then those in the corresponding L∞ norm. The +significant decrease in error between eε,h−h/2 and eε,h/2−h/4 by a factor ten is unclear but gives a +comparable order as the time-discretization. For the time-discretization, we observe a reduction of +the error between eε,τ−2τ and eε,τ−τ/2 by roughly a factor of 2, which indicates the expected linear +convergence rate in τ. By far the largest errors on the order of 10−3 for the L∞ norm emerge from +the polynomial order, which indicate possible locking effects for kχ and a limiting resolution a the +interface for kψ. +norm +eε,h−h/2 +eε,h/2−h/4 +eε,τ−2τ +eε,τ−τ/2 +eε,ψP 2−ψP 3 +eε,χP 2−χP 3 +L2 +2.8 · 10−4 +2.7 · 10−5 +9.5 · 10−5 +5.3 · 10−5 +1.2 · 10−4 +2.9 · 10−4 +L∞ +4.6 · 10−3 +3.2 · 10−4 +9.5 · 10−4 +5.5 · 10−4 +2.7 · 10−3 +5.0 · 10−3 +Table 2: Numerical errors for the phase-field model with respect to the L2(QT ; R2) norm and with respect +to the L∞(QT ; R2) norm for vector-valued and scalar functions upon change of time-step size {2τ, τ, τ/2}, +uniform refinement {h, h/2, h/4}, and polynomial order {P2, P3} of the FE-space of ψ and χε. +Sharp-interface model +For the sharp-interface model we show the numerical errors in Table 3. +For the uniform refinement we see a reduction in errors with a factor between 2 and 4 going from +e0,h/2−h/4 to e0,h/4−h/8, which indicates a convergence order between 1 and 2 in space, as expected +for P2 elements and a less regular solution. The reduction of the errors e0,τ−τ/2 and e0,τ/2−τ/4 is by +a factor of two, showing linear convergence. As for the diffuse-interface model, by far the largest +error come from different polynomial orders for the displacement and are of order 10−3. This is +again indicative of possible locking effects. +Both diffuse-interface model and sharp-interface model clearly show numerical convergence, with +the largest contribution to the error coming from polynomial degree and being of the order 10−3 in +the L∞ norm. +norm +e0,h−h/2 +e0,h/2−h/4 +e0,h/4−h/8 +e0,τ−τ/2 +e0,τ/2−τ/4 +e0,P1−P2 +e0,P2−P3 +L2 +1.4 · 10−3 +2.8 · 10−4 +7.9 · 10−5 +5.3 · 10−5 +2.8 · 10−5 +9.7 · 10−3 +8.9 · 10−5 +L∞ +1.3 · 10−2 +4.9 · 10−3 +3.7 · 10−3 +5.5 · 10−4 +2.9 · 10−4 +6.8 · 10−2 +4.2 · 10−3 +Table 3: Numerical errors of the sharp-interface model with respect to the L2(QT ; R2) norm and with +respect to the L∞(QT ; R2) norm for deformations upon change of time-step size {τ, τ/2, τ/4}, uniform +refinement of the mesh a) {h, h/2, h/4, h/8} in Section 3.1, and polynomial order {P1, P2, P3} of the FE-space +for χ0. +4 +Convergence to sharp-interface model +Figure 5 gives a visual representation of the main results of this work and shows the direct compar- +ison of phase-field model and sharp-interface model at time t = T for different mobilities m = m0εα +13 + +and 0 ≤ α ≤ ∞. In the following we first give a detailed discussion of direct observations in this +representation for different mobilities and then we discuss the (experimental) error ∥χ0,∆ − χε,∆′∥ +in some detail for different norms. +a) +b) +c) +d) +e) +f) +g) +h) +Figure 5: Superposition of diffuse and sharp interfaces for ε = 2¯ε with kχ = 2, kψ = 2 and 2 uniform +refinements of the mesh a) in Figure 3. The shading shows the phase-field indicator ψid with solid (red), +liquid (blue) and air (gray) phases. Thick black lines display the phase-field deformation applied to the +initial position of the diffuse interfaces and the thin black mesh displays the deformation of the initial +diffuse domain of the elastic solid. Correspondingly, the thick white lines show the sharp interfaces ¯Γij(t) +and the thin white mesh shows the displacement applied to the (sharp) elastic solid. We showcase different +mobilities: a) m = 1 (α = 0), b) m = ε, c) m = ε3/2, d) m = ε2, e) m = ε5/2, f) m = ε3, g) m = ε4, h) +m = 0 (α = ∞). Panels are shown at time t = T, except for a) which is at t = 0.1. Right next to each +image a magnification of the contact line is shown. +Firstly, note that for mobility m = 1, i.e. α = 0, in panel a) of Figure 5 one can clearly see the +nonconvergence of the phase-field model, since already at an earlier time the phase field (shading) +and the displacement of the reference solid do not match. +This is consistent with large excess +14 + +diffusion leading to a state where F = I and ψ minimizing the phase-field energy separately, i.e. +the evolution of the phase field disconnects from the evolution of the displacement and assumption +A2 for the convergence is violated. This already happens early at time t = 0.1. +Next, for mobilities m = εα for α ≥ 1 in panels b)-e) of Figure 5 we see, except for slight +convergent deviations in b), the phase field (shading) and the deformed initial interfaces (thick +black lines) remain aligned, i.e. in those regions A2 is satisfied. However, for too small mobility, +around α ≥ 4, the sharp interface and the diffuse field interface (thick white and black lines) may +not be aligned as this additionally requires ∥χ0 − χε∥ → 0. +Further, note that even for α ≥ 1 in the cases b) α = 1, c) α = 3/2 we observe clear differences +of the diffuse interface (shading) and the deformed mesh (thick black lines) much larger then +the interfacial width ε near the contact lines visible in the magnification of the contact line in +Figure 5. What might even be counter-intuitive first is that the alignment of the sharp interface +(thick white lines) and the diffuse interface (shading) in these cases b, c) appears much better then +the alignment of the displacements (black and white thick lines). This again emphasizes that the +convergence requires both convergence of the indicators A2 and convergence of the displacements +A1. The main observation here is that while in a suitable (weak) norm one might prove convergence +to the sharp-interface model, for all practical questions and in the strong L∞ norm, the error of +the contact line position clearly lags behind the highly resolved interfacial thickness ε. +For the cases d) and e) with α = 2 and α = 5/2 we generally observe a very good agreement of +phase fields (shading) and deformed initial diffuse interfaces (thick black lines) and deformed sharp +interfaces (thick white lines) and the overall displacement (thin black and white mesh). This shows +that these values are suitable for simulations when in particular a good precision of the predicted +contact line position and small error in the displacements are desired. +For the cases f,g,h) with α = 3, 4, ∞ we still see the perfect alignment of phase-field (shad- +ing) and deformed diffuse interfaces (thick black lines) but no convergence of the displacement, +i.e. assumption A1 for convergence is violated. As explained earlier, this is due to the fact that +the diffuse-interface profile does not reach the optimal tanh-profile and the approximation of the +Eulerian surface energy deteriorates. +In our Lagrangian formulation, it is possible to choose extremely small Cahn-Hilliard mobilities +and even m = 0 without the need to revise the numerical procedures and introduce artificial stabi- +lization terms. While in an Eulerian framework such a stabilization would be needed and possible +restrictions due to CFL conditions might appear, the alignment of phase fields and displacement +could in principle be also verified using an Eulerian approach by integrating the velocity field to a +obtain a deformation map. This concludes the discussion of Figure 5 and clearly shows that α = 1 +produces significant errors near the contact line. +The behavior shown in the spatial representation of the sharp-interface limit is mirrored in the +behavior of the two norms shown in Figure 6. Here we haven chosen a representation, where we +plot different errors for different ε = 2n¯ε as a function of the mobility m = εα. The reasons is that +for any given ε one would like to know the best possible mobility to obtain a sufficiently precise +approximation of the sharp-interface model. From the preceding discussion its clear that both too +large and too small values of the mobility will lead to suboptimal results. Both norms in Figure 6 +clearly show that there are optimal values for m(ε) for each ε and this optimal mobility decreases +for smaller ε. For L∞, while for ε = 16¯ε we have mopt ∼ 10−4 we have for ε = ¯ε an optimal value +mopt ∼ 10−7. This indicates that mopt ∼ ε2 is optimal for a strong norm that takes errors near the +contact line into consideration. On the other hand, the minimum in the L2 error is rather broad +and suggests that here, if the convergence in the contact line area is not of interest, a broader range +15 + +Figure 6: Convergence error ∥χ0,∆ − χε,∆′∥ for different ε = 2n¯ε and different mobilities m(ε) = εα as a +function m(ε). The used mobilities are m = ε∞ = 0, m = ε4, m = ε3, m = ε5/2, m = ε2, m = ε3/2, m = ε1, +m = 1. The left panel shows the convergence in the L∞ norm and the right panel in the L2 Bochner norm. +of α-values is possible. Note that based on the discussion of numerical errors in Section 3.2, these +statements are limited by the numerical errors of order 10−3 in the L∞ norm. +We finalize our discussion by considering the evolution of the spatial norms as a function of +time t on the cylinder Ω in Figure 7. For initial times 0 < t < 0.2 we observe a steep increase in +the convergence error, which might be affected due to the use of initial data that do not satisfy +the Neumann triangle construction. +This should result in a transient boundary layer at t = 0 +an possibly also influence the convergence rate of the sharp-interface limit in time. However, in +our experience the relaxation of a droplet from a flat planar substrate is a more realistic scenario +compared to well-prepared initial data that already satisfy the Neumann triangle. In the left panel +of Figure 7 we observe convergence in the L∞ norm for decreasing ε = 2n¯ε. In the right panel we +observe that for fixed ε and mobility exponents m = εα with α = 2, 5/2, 3 we obtain the lowest +errors of the order 10−2, which is still larger than the discretization errors previously discussed in +Section 3.2. +5 +Conclusion +We have presented a Lagrangian variational formulation of phase-field and sharp-interface models +for a nonlinearly coupled fluid-structure interaction problem that involves nonlinear elasticity in a +solid phase and a fluid and air phase, moving capillary interfaces and a moving contact line. We have +discussed the sharp-interface limit depending on the mobility m(ε) = m0εα in the Cahn-Hilliard +model in different norms and shown that in the strong L∞ norm the moving contact lines suggests +αopt = 2. This discussion is also based on a detailed consideration of different contributions to +numerical errors. Similar observations have been made before [25, 15], but here we see them as a +clear consequence of the presence of the moving contact line. +In the future one needs to avoid possible limitation due to locking by choosing incompressible +models with suitable inf-sup stable finite elements. Additionally, by choosing degenerate state- +dependent mobilities m = m(ε, ψ) +and techniques as in [10] one should be able to increase the +16 + +0.15 +*-8 +-4 +$-23 +(QT; IR") +0.1 +0.05 +0 +10-10 +10~5 +1000.03 +-16 +0.025 ++-8 +4 +$-2m +0.02 +IRd +0.015 +0.01 +0.005 +0 +10-10 +10~5 +109Figure 7: +Convergence error ∥χ0,∆ − χε,∆′∥(t) for L∞(Ω; Rd) Sobolev norm as a function of time t. +(Left) for fixed mobility exponent m = ε2 but different ε and (right) for fixed ε = ¯ε but different mobility +exponents. +region of validity compared to m = m(ε). This discussion should be extended to models that take +into account Navier-slip and dynamic contact angles. Rigorous and formal asymptotic results in +this direction would be desirable but require a good understanding of both upper and lower bound +α, α. +Acknowledgement +Both authors thank Marita Thomas, for discussions about analysis and nonlinear elasticity and +Andreas Münch regarding aspects of asymptotics of phase-field models with degenerate mobilities. +Furthermore, we thank Helmut Abels and Harald Garcke for the discussion on sharp-interface limits +and lower and upper bounds of the mobility. +Both authors acknowledge the funding by the German Research Foundation (DFG) within the +DFG Priority Program SPP 2171 Dynamic Wetting of Flexible, Adaptive, and Switchable Sub- +strates through the projects #422786086 (LS) and #422792530 (DP). LS & DP thank the Berlin +Mathematics Research Center MATH+ for funding and support through project AA2-9. +References +[1] H. Abels. (Non-) convergence of solutions of the convective Allen–Cahn equation. 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The sharp-interface +limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids. Journal of Fluid Mechanics, +714:95–126, 2013. +[16] E. Meca, A. Münch, and B. Wagner. Sharp-interface formation during lithium intercalation +into silicon. European Journal of Applied Mathematics, 29(1):118–145, 2018. +[17] L. Modica. The gradient theory of phase transitions and the minimal interface criterion. Archive +for Rational Mechanics and Analysis, 98(2):123–142, 1987. +[18] D. Mokbel, H. Abels, and S. Aland. A phase-field model for fluid–structure interaction. Journal +of Computational Physics, 372:823–840, 2018. +18 + +[19] A. Novick-Cohen. The Cahn–Hilliard equation. Handbook of Differential Equations: Evolu- +tionary Equations, 4:201–228, 2008. +[20] S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: Algorithms +based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79(1):12–49, 1988. +[21] S. Schaubeck. Sharp interface limits for diffuse interface models. PhD thesis, 2014. +[22] L. Schmeller and D. Peschka. Gradient flows for coupling order parameters and mechanics. +WIAS Preprint, 2909:1–24, 2022. +[23] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, +and Y.-J. Jan. A front-tracking method for the computations of multiphase flow. Journal of +Computational Physics, 169(2):708–759, 2001. +[24] E. van Brummelen, T. Demont, and G. van Zwieten. An adaptive isogeometric analysis ap- +proach to elasto-capillary fluid-solid interaction. International Journal for Numerical Methods +in Engineering, 122(19):5331–5352, 2021. +[25] P. Yue, C. Zhou, and J. J. Feng. Sharp-interface limit of the Cahn–Hilliard model for moving +contact lines. Journal of Fluid Mechanics, 645:279–294, 2010. +19 + diff --git a/adE4T4oBgHgl3EQfOwzB/content/tmp_files/load_file.txt b/adE4T4oBgHgl3EQfOwzB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a15d923bc700133462fab5a2bdbc9bdb156772b5 --- /dev/null +++ b/adE4T4oBgHgl3EQfOwzB/content/tmp_files/load_file.txt @@ -0,0 +1,584 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf,len=583 +page_content='Sharp-interface limits of Cahn–Hilliard models and mechanics with moving contact lines Leonie Schmeller∗ Dirk Peschka∗ January 13, 2023 Abstract We construct gradient structures for free boundary problems with nonlinear elasticity and study the impact of moving contact lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In this context, we numerically analyze how phase- field models converge to certain sharp-interface limits when the interface thickness tends to zero ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In particular, we study the scaling of the Cahn-Hilliard mobility m(ε) = m0εα for 0 ≤ α ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the presence of interfaces, it is known that the intended sharp-interface limit is only valid for α < α < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' However, in the presence of moving contact lines we show that α near α produces significant errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 1 Introduction Interfaces and surfaces with surface energy are ubiquitous in nature, and their description can employ different levels of detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For multiphase systems with moving interfaces different mathe- matical and numerical techniques are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Explicit descriptions via front tracking methods [23] allow high control over the moving boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Implicit description via level-set methods [8, 20] or using phase fields lead to diffuse representations of interfaces [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The choice of method depends on problem-specific features and requirements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' feasibility of topological changes, treatment of discontinuities at the interface, mass conservation and transport across interfaces, ease of imple- mentation, availability of computational resources, precision and control over interface evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In general for continuum models, sharp-interface and phase-field models are the most common levels of abstraction with more or less details, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Such systems can be arbitrarily complicated when interfaces have a complex substructure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=', with interfacial mass and thermodynamical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This complexity increases even further by considering nonlinear diffusion and phase separation, reactions and phase transitions and complex viscoelastic properties as possible dissipative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' However, even for simple systems with constant surface tension and fluid flow or elasticity in the pure phase, the connection between common diffuse-interface models and their sharp limits is elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In phase-field models, the interface is modeled by continuous functions ψ that are constant inside a phase, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' ψ = ±1, and whose change in an interfacial region of width proportional to ε > 0 indicates the transition to another phase in a continuous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The mathematical purpose of the phase field is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Firstly, ∗Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany (leonie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='schmeller@wias-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='de, dirk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='peschka@wias-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='de) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='04968v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='AP] 12 Jan 2023 it should act as an indicator for the presence of a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Secondly, it defines a phase-field energy density W ε phase(ψ, F −T ∇ψ) that measures, among other things, the surface tension of the interface with deformation gradient F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Without mechanics F = I the Cahn-Hilliard model ∂tψ = ∇ · (m∇µε) , µε = δ δψ W ε phase , W ε phase = 3 2 √ 2 � ε 2|∇ψ|2 + 1 4ε(1 − ψ2)2� , (1) is commonly used to model phase-field evolution and phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Its well-known that W ε phase converges to the perimeter of the interface connecting regions with ψ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The limit passage of dynamic phase-field models to sharp-interface models can be analyzed by different techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' by matched asymptotics [16, 12, 2], Γ-convergence [17] and evolutionary convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Degenerate ψ-dependent mobilities m(ε, ψ) are highly relevant for the limit passage from the diffuse to the sharp-interface model, which is studied in detail in [2, 16, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In many studies, only one set of interfacial width ε and mobility m is used for a specific appli- cation leading to reasonable results [24, 9, 13, 7, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Instead, we focus on a systematic study of the appropriate choice of the Cahn-Hilliard mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We extend the work by Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' [25] by con- sidering fluid-structure interaction and by directly comparing sharp and diffuse-interface models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The goal of this work is to establish a general numerical approach to the problem of sharp-interface limits with moving contact lines, somewhat in the spirit of previous work by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' [14] and Aland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We consider m(ε) = m0εα with an appropriate 0 ≤ α ≤ ∞ and determine the scaling of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Usually, for interface problems 1, m = εm0 yields the intended interface condition [2] and gives an upper bound on the mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Further, the authors in [1, 21] show that there is also a lower bound on the Cahn-Hilliard mobility which suggests that the intended limit is reached for α ∈ (α, ¯α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' However, since such techniques are inevitably much more complex for moving contact lines and require considerations in higher-dimensions, we will resort to numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In order to explore the sharp-interface limit of phase-field models, the strategy of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In Section 2 the phase-field evolution (1) is extended to a ternary multiphase systems with solid (s), liquid (ℓ) and air (a) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Similarly, we provide a sharp-interface model of the supposedly same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Nonlinear elasticity of compressible materials and appropriate coupling terms for phase-field and sharp-interface model are introduced using an hyperelastic energy density Welast(F , ψ) with deformation gradient F = ∇χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' At the moving contact line we assume an equi- librium of surface forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We provide an Eulerian formulation of the system of nonlinearly coupled PDEs and also provide the corresponding Lagrangian weak formulation that uses the underlying gradient flow structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the phase-field model we discuss different possible scaling limits of the mobility m in order to have convergence to the sharp-interface model as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We also review existing results on scaling laws and options for choosing the Cahn-Hilliard mobility for models with interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Then we introduce the technical methodology with which we perform our numerical convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In Section 3 we perform the discretization of both phase-field and sharp-interface model in order to characterize the sharp-interface limit with contact lines using suitable norms ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The goal here is to discuss the distance ∥χ0,∆ − χε,∆′∥ of numerical solutions of the sharp-interface model χ0,∆ and of the phase-field model χε,∆′ after interpolating between different discrete function spaces for different discretization parameters ∆, ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Most importantly, this requires a discussion of discretization errors of the two models with respect to time-discretization, mesh refinement and the polynomial order of the finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Therefore, as the main tool we use the Bochner norms for L2(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rd) and L∞(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rd) 1excluding cases with three-phase contact lines 2 for the deformation to assess the error in the space-time cylinder QT = [0, T] × Ω as ε → 0 for different m(ε) = m0εα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Finally, in Section 4 we perform a discussion of the numerical solutions generated with the methodology and based on the error analysis introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We focus on comparing concrete solutions of the benchmark problem “liquid droplet on viscoelastic substrate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The main goal of Section 4 and this entire work is to identify exponents α ≤ αopt ≤ α that give an optimal mobility mopt = m0εαopt in the sense of (error of) the sharp-interface limit and to provide a systematic approach to identify such αopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' While such a statement usually depends on the norm, our choice of norm and discussion is based on the prediction of the moving contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 2 Models for diffuse and sharp interfaces in variational form Cahn-Hilliard-Navier-Stokes systems are widely used for modeling multiphase flows in various appli- cations as well as for incompressible viscous Newtonian fluids [7, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Since the numerical treatment of physical interface thicknesses ε on the nanoscale is often infeasible for phase-field models, it is important to clearly understand the behavior for ε → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' the sharp-interface limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The Cahn- Hilliard mobility m depends on ε and possibly on the phase field variable ψ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' An advantage of ψ-dependent degenerate mobilities is the ability to guarantee that the phase field variables remain in a fixed, application-specific interval, which in general is a absent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As a typical test case we use an outer cylinder Ω = [0, R] × [0, H] with R = 1 and H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The region Ωs = {(r, z) ∈ Ω : 0 < z < 1} defines the solid phase, Ωℓ = {(r, z) ∈ Ω : � r2 − (z − 1)2 < rdrop, z > 1} with rdrop = 1/2 defines the liquid phase, and the remainder Ωa = Ω \\ (Ωℓ ∪ Ωs) is the air phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The undeformed reference domains Ωs (red) and Ωℓ (blue) are shown in the left image of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The corresponding deformed domains ¯Ωi(t) = χ(t, Ωi) from the sharp-interface model at time t = T are shown in the right image of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In this image, the air phase is not shown to improve visibility of the liquid and solid phases and their three-dimensional impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Figure 1: Slice of (left) initial liquid droplet (blue half sphere) on a flat elastic substrate (red cylinder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For x = (r, z) ∈ Ω, at z = 0 (solid bottom) and z = H (air top) we have no-slip conditions χ(x) = (χr(x), χz(x)) = (r, z) and at r = 0 and r = R we fix the normal component χr(r) = r and have natural boundary conditions (symmetry/slip) for the tangential component χz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (Right) Deformed droplet ¯Ωℓ(t) (blue) and deformed solid substrate ¯Ωs(t) (red) at time t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15 as a solution of the sharp-interface model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The interfaces are visualized using thick white curves and the deformation in the solid phase is visualized using a thin white mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 3 Before we explain the diffuse-interface model and the sharp-interface model, we would like to make a comment on simplifying modeling assumptions and our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We neglect inertia to avoid effects caused by elastic or capillary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We formulate the viscoelastic models entirely in Lagrangian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We assume no-slip conditions between phases and all phases have the same viscosity µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We consider a multiphase system with solid, liquid and gas phases, where the liquid and gas phases are characterized by the absence of elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We consider slightly compressible phases via κ 2 (det F − 1)2 in the elastic energy with κ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Locking is avoided by using sufficiently high polynomial degree in the FE discretization [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' While we made these choices to make it easier to follow the presentation or avoid certain technical difficulties, none of these choices should restrict the validity of our findings concerning the sharp- interface limit in the presence of moving contact lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1 Diffuse-interface model First we present a ternary phase-field model, that is capable to describe the evolution of a liquid phase and a soft elastic phase surrounded by a gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Therefore, within a given fixed box Ω ⊂ Rd for d = 2, 3 let ψ : [0, T] × Ω → R2 be a ternary phase-field and χε : [0, T] × Ω → Rd a deformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We further decompose the phase-field ψ = (ψs, ψℓ) into solid and liquid phase, so that the remaining air phase ψa is determined implicitly by ψa = −1 − ψs − ψℓ so that ψa = +1 if ψs = ψℓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Using this convention, ψi = 1 indicates the presence of phase i and ψi = −1 the absence of phase i for i ∈ {s, ℓ, a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We denote the deformation gradient F = ∇χε and the Jacobian determinant J = det(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For qε = (χε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' ψ) ∈ Qε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' the cornerstone of the phase-field model is the free energy Fε : Qε → R Fε(qε) = � Ω W ε[qε] dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' W ε[qε] = Welast(F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' ψ) + W ε phase(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' F −T ∇ψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (2) for which here we choose Welast(F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' ψ) = G(ψ) 2 tr(F T F − I) + κ 2 (J − 1)2 (3a) W ε phase(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' F −T ∇ψ) = � i∈{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='a} γi � ε 2|F −T ∇ψi|2 + 1 4ε(1 − ψ2 i )2� J (3b) with surface parameters γi that correspond to the usual surface tensions via γsa = 3 2 √ 2(γs + γa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' γsℓ = 3 2 √ 2(γs + γℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' γℓa = 3 2 √ 2(γℓ + γa) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (4) The function G(ψ) in (3a) interpolates the elastic properties of the different phases by G(ψ) = 1 2Gℓ(1 + ψℓ) + 1 2Gs(1 + ψs) + 1 2Ga(1 + ψa) , (5) 4 with Gi ∈ R, but we are going to set Gℓ = Ga = 0 and rescale Gs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The material is nearly incompressible, which we achieve by setting κ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' On the boundary of the domain we use either homogeneous Dirichlet boundary conditions χε(x) − x = 0 for the entire displacement or for the normal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the latter case, homogeneous natural boundary conditions are implied for the tangential component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Motivated by the extended gradient structures concept introduced in [22], we use a dynamical model for the evolution qε : [0, T] → Qε with chemical potentials η : [0, T] → U that satisfy aε(η, φ) + b(φ, ∂tqε) = 0 , b(η, v)s(∂tχε, vχ) = ⟨DFε(qε), v⟩Vε , (6) for all φ ∈ U = H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) and all v = (vχ, vψ) ∈ Vε = H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rd) × H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) and given initial values qε(t = 0) ∈ Qε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For this model we use the bilinear forms aε(η, φ) = � Ω m(ε)F −T ∇η · F −T ∇φ dx, (7a) b(φ, v) = � Ω φvψ dx, (7b) s(w, v) = � Ω µ(ψ) ((∇w)F −1) : ((∇v)F −1) det F dx, (7c) with phase-dependent viscosity function µ(ψ) = 1 2µℓ(1 + ψℓ) + 1 2µs(1 + ψs) + 1 2µa(1 + ψa) , (8) with µi ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For simplicity here we choose µi = µ, such that the viscosity is constant µ = µ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' By testing (6) with v = ∂tqε and φ = η we directly obtain thermodynamic consistency d dtFε = ⟨DFε(qε), ∂tqε⟩V = −(s(∂tχε, ∂tχε) + aε(η, η)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (9) Note that for clarity of the presentation, the dependence of the bilinear forms on the state variable qε is not shown explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We make all the computations for d = 3 by assuming axisymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Therefore, the weak formulation in cylindrical coordinates is obtained by replacing the domain with a cylinder of radius R and height H such that Ωr = [0, R] × [0, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Using w = (wr, wz) we make the following replacements dx = 2πdr dz, ∇w = � � ∂rwr 0 ∂zwr 0 r−1wr 0 ∂rwz 0 ∂zwz � � , ∇η = � � ∂rη 0 ∂zη � � , for the integration measure dx, for gradients of vector fields ∇w and for gradients of scalar fields ∇η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' These replacements are performed consistently in the bilinear forms and the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 Sharp-interface model Now we present the corresponding sharp-interface model of the multiphase model that is our candi- date for the sharp-interface limits as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Using the same fixed box Ω ⊂ Rd instead we consider 5 that each phase i ∈ {s, ℓ, a} occupies a sufficiently regular subdomain Ωi ⊂ Ω that do not overlap Ωi ∩ Ωj = ∅ and fill the entire domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=', Ω = Ωs ∪ Ωℓ ∪ Ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For q0 ≡ χ0 ∈ Q0, the cornerstone of the sharp-interface model is also a free energy F0 : Q0 → R F0(q0) = � i∈{s,ℓ,a} � Ωi W i elast(F ) dx + � ij∈{sℓ,sa,ℓa} � Γij γij|cof(F ) · ν| ds , (10) where F = ∇χ0 as before and for which here we choose W i elast(F ) = Gi 2 tr(F T F − I) + κ 2 (J − 1)2, (11) and for the surface tensions γij from (4) associated to the (sharp) interfaces Γij = ∂Ωi ∩ ∂Ωj for ij ∈ {sℓ, sa, ℓa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Also motivated by a gradient structure, we use a dynamical model for the evolution q0 : [0, T] → Q0 that satisfies −s(∂tχ0, vχ) = ⟨DF0(q0), vχ⟩V , (12) for all vχ ∈ V0 = H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rd) and given initial values q0(t = 0) ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For this model we use the same bilinear form as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' s(w, v) = � i∈{s,ℓ,a} � Ωi µi ((∇w)F −1) : ((∇v)F −1) det F dx , (13) with phase-dependent viscosities µi ∈ R, which here we set equal to µi = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' By testing Equation (12) with ∂tχ0 we directly obtain thermodynamic consistency d dtF0 = ⟨DF0(q0), ∂tχ0⟩V0 = −s(∂tχ0, ∂tχ0) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (14) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Assuming axisymmetry, we additionally need to replace dsx = 2πrdsr, ν = � � νr 0 νz � � , the surface measure dsx and the normal vector ν in the energy F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Remark 3 (Eulerian formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The Eulerian free energy for the phase-field model (2) with ¯qε = ( ¯F , ¯ψ) is Fε(¯qε) = � Ω ¯J−1 � Welast( ¯F , ¯ψ) + W ε phase( ¯ψ, ∇ ¯ψ) � d¯x , (15) where ¯ψ ◦ χε = ψ and ¯F ◦ χε = F and ¯J = det ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The free energy of the sharp-interface model (10) in the Eulerian frame is F0(¯q0) = � i∈{s,ℓ,a} � ¯Ωi ¯J−1W i elast( ¯F ) d¯x + � ij∈{sℓ,ℓa,sa} � ¯Γij γij d¯s , (16) where the Eulerian state ¯q0 = ( ¯F , ¯Ωi) contains ¯Ωi(t) = χ0(t, Ωi(0)) the domain currently occupied by phase i ∈ {s, ℓ, a} at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The PDEs in Eulerian variables also contain a velocity ¯u : [0, T]×Ω → Rd, which, however, the free energy does not depend on for viscous flows without kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='3 Scaling limits for the Cahn-Hilliard mobility Varying choices/scalings of the Cahn-Hilliard mobility in ε, may approximate different sharp- interface models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Generally speaking, a bound m(ε) ≲ O(ε) on the mobility is needed to prevent artificial diffusion of the phase fields [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' A further bound from the other side is essential to ensure that the mobility does not become too small, which would lead to a different/unintended sharp-interface limit [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' A fundamental work on the passage from the phase-field model to the sharp-interface limit is [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The sharp interface is analyzed using matched asymptotic techniques, and degenerate mobilities are included in their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Precisely, the following four cases are discussed: m(ε, ψ) = � � � � � � � � � m0 Case 1 εm0 Case 2 m1 ε (1 − ψ2)+ Case 3 m1(1 − ψ2)+ Case 4 , where for Case 2 and 4 a sharp-interface limit corresponding to ours is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the other cases, additional terms appear in the kinematic conditions of the sharp-interface model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Having introduced the diffuse and the associated sharp-interface model, we will now analyze the limit passage ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In this work we want to generalize some considerations concerning viable sharp- interface limits, for which the passage from the diffuse-interface model (2) to the sharp-interface model (10) is valid including contact lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The sharp-interface limit, ε → 0, of the diffuse-interface model depends substantially on the scaling of the Cahn-Hilliard mobility m = m0εα, 0 ≤ α ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Degenerate mobilities or other forms of Cahn-Hilliard mobility are also used and require appropriate scaling properties [2, 12, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We believe the setup shown in Section 2 has several mathematical and numerical challenges that make it a suitable benchmark to study the sharp-interface limit of phase-field models: For this setup, the stationary state is close to the initial data so that Lagrangian methods can compute a regular deformation χ without needing ALE methods or remeshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' With the initially flat solid substrate, surface energies will quickly enforce a contact angle (kink in the solid surface) via the Neumann triangle construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' While the resulting nonsmoothness in F is a challenge, this scenario is typical for soft wetting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This could lead to suboptimal convergence rates in space discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' If initial data do not satisfy the Neumann triangle construction, then solutions are also expected to show nonsmooth behavior in time, leading to suboptimal convergence in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the following, in order to improve the visibility of results we will show solutions of the sharp- interface model and of the phase-field model only as cross sections in the (r, z)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Figure 2 shows in the left half of each droplet the result of the sharp-interface at fixed time t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15, while the right half shows the diffuse-interface model with ε = 2¯ε with ¯ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8775 · 10−3 at the same time with two different mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For m = εα with α = 5/2 used in the left/first figure, we see a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the second/right droplet we choose m = 0 and a clear deviation between the diffuse and the sharp-interface model can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The reason for this deviation is that the phase-field ψ upon deformation to ¯ψ ◦ χε = ψ can not relax to its optimal tanh-profile and therefore the approximation of the Eulerian surface energy is not guaranteed [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Thus, this indicates that the 7 Cahn-Hilliard mobility is chosen too small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' α < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Furthermore, the upper bound to the Cahn- Hilliard mobility must also be respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The next step is to numerically analyze the convergence of the models in appropriate norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 3 Discretization in space and time In this section, we numerically estimate the distance of the the phase-field and sharp-interface models to make a statement about the convergence to the sharp-interface limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Therefore, the two main assumptions that need to be verified are: A1 The deformation converges ∥χ0 − χε∥ → 0 as ε → 0 for a suitable norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' A2 The (Lagrangian) phase fields ψi remain close to Ωi in the sense distance(Ωi,ε(t), Ωi) → 0, as ε → 0 for time t and a suitable distance between Ωi,ε(t) = {x ∈ Ω : ψi(t, x) > 0} and the (time-independent) sharp domains Ωi for i ∈ {s, ℓ, a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In order to make meaningful statements about this convergence, we need to estimate the numerical errors in the respective distances and norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Let us denote by χε,∆ a numerical solution of the diffuse-interface model for fixed ε and by χ0,∆ a numerical solution of the sharp-interface model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As above by χε and χ0 we denote the exact solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Then, using the triangle inequality, we have the estimate for the sharp-interface limit ∥χ0 − χε∥ � �� � sharp-interface limit ≤ ∥χ0 − χ0,∆∥ � �� � error sharp interface e0,∆ + ∥χ0,∆ − χε,∆∥ � �� � estimate sharp-interface limit + ∥χε,∆ − χε∥ � �� � error diffuse interface eε,∆ (17) a) b) c) Figure 2: In a,b) the left side shows the sharp interface at time t = T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15 and the right side is the diffuse interface model for mobility a) m = ε5/2 and b) m = 0 with ε = 2¯ε in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The deformed phase fields are plotted via ¯ψid = (2 + ¯ψs − ¯ψℓ)/2 taking values ¯ψid = 0, 1, 2 in the liquid (blue), air (gray), solid phase (red), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' c) Deformed diffuse phase fields for a) showing (left) liquid phase −1 ≤ ¯ψℓ ≤ 1 and (right) solid phase −1 ≤ ¯ψs ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 8 0 1in an appropriate function space norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The first goal here is to estimate various contributions to the approximation errors e0,∆ = ∥χ0 − χ0,∆∥ and eε,∆ = ∥χε,∆ − χε∥ for simulation parameters associated with the discretization such as ∆ = {τ, h, kχ, kψ} for time-step size τ, mesh size h, poly- nomial degree for deformation kχ, and for the polynomial degree for the phase fields kψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' These we verify by time-step bisection τ → τ/2, uniform refinement h → h/2, and by varying the polyno- mial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Then we compute ∥e0,∆−∆′∥ = ∥χ0,∆ − χ0,∆′∥ for different τ and ∆ = {τ, h, kχ, kψ} and ∆′ = {τ/2, h, kχ, kψ} to approximate the convergence of the time-discretization and for the other cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In such a case we write for simplicity ∥e0,∆−∆′∥ ≡ ∥e0,τ−τ/2∥ and as- sume all other discretization parameters are known an fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the space discretizations the computation of the error might also involve a projection Ph : V0,h → Vε,h′ or Ph : V0,h → V0,h′ or Ph : Vε,h → Vε,h′, but this resulting projection error was mostly negligible compared to the error of the other approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The error between the deformation fields is measured in L2 and L∞ Bochner norms which is a concept that extends the classical Sobolev norms [3] to time-dependent problems via ∥χ∥L2(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='Rd) = �� T 0 ∥χ(t)∥2 L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='Rd)dt � 1 2 , ∥χ∥L∞(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='Rd) = ess sup t∈[0,T ] ∥χ(t)∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (18) While the L2 Bocher norm measures the error over the whole area and neglects errors in small regions, the L∞ norm is also sensitive to errors in small regions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' on interfaces and at contact lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' First we introduce separately the space and time discretizations of phase-field and sharp- interface model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Using the deformations, we can map A2 to an Eulerian version and require distance(¯Ωi,ε(t), ¯Ωi(t)) → 0, (19) as ε → 0 for any time t with ¯Ωi,ε(t) = χε(t, Ωi,ε(t)) and ¯Ωi(t) = χ0(Ωi) and i ∈ {s, ℓ, a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Assuming convergence A1 in a sufficiently strong norm, then A2 and (19) should become equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We ensure A2 by making m(ε) small enough as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1 Space and time discretization Space-discretization Based on the weak formulation of the diffuse-interface model (6) and the sharp-interface model (12) we employ the finite element method to derive a discretization in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The main idea of this structure preserving discretization is to adopt a weak formulation for the phase fields with Qh,ε ⊂ Qε, Uh ⊂ U, Vε,h ⊂ Vε and for the sharp-interface limit with Qh,0 ⊂ Q0, V0,h ⊂ V0 via a finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the sharp-interface model we use computational meshes, where the elements and edges are aligned with the phases Ωi and the interfaces Γij as shown in the first (left) panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the convergence study we usually use finer meshes for the sharp-interface model, where we perform 1-3 uniform refinements of the first mesh (mesh a) shown in Figure 3 and project the vertices of Γℓa (interface between blue and gray domain) back onto the set � r2 + (z − 1)2 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This coarsest mesh has 453 vertices resulting in the dimension dim Vh,0 = 3 496, while after three uniform refinements the mesh has 27 221 vertices resulting in the dimension dim Vh,0 = 216 786 using P2 finite elements for the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 9 For the diffuse-interface model the base mesh strongly depends on the values of the interfacial thickness ε, where we consider for ¯ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='001875 the values ε = n¯ε for n = 1, 2, 4, 8, 16, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The corresponding meshes shown in Figure 3 have 1 310 (mesh b), 5 205 (mesh c), 19 502} (mesh d) vertices corresponding to dim Vh,ε = 10 322, dim Vh,ε = 41 458, dim Vh,ε = 155 810 using P2 finite elements for the deformation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' On top of that, the phase-field model contains scalar unknowns for the phase fields ψ and their chemical potentials η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Note that the mesh is refined near the interfaces to resolve the transition layer of width ε and in a larger region near the contact line to account for potential artificial diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Let Pk(T, Rr) be the set of Rr-valued polynomials of degree k restricted to triangles T ∈ Th and Ω = � T ∈Th T an admissible triangulation of the ternary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We use spaces V k for vectorial unknowns and V k for scalar unknowns, where we define the H1-conforming finite element spaces V k h = {v ∈ C1(Ω, Rd) : v|T ∈ Pk(T, Rd), T ∈ Th}, V k h = {v ∈ C1(Ω, R) : v|T ∈ Pk(T, R), T ∈ Th} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the sharp-interface model we use Vh,0 = V kχ h with kχ = 1, 2, 3 and enforce essential boundary conditions when solving the corresponding nonlinear problem and for the diffuse-interface model a) b) c) d) Figure 3: (Top) Different initial meshes from (left) to (right) for a) sharp interface, b) phase-field with ε = 16¯ε, c) phase-field with ε = 4¯ε and d) phase-field with ε = ¯ε for ¯ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='001875 and (bottom) corresponding phase indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the phase field we show ψid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 10 we use Vh,ε = V kχ h × V kψ h × V kψ h , Uh = V kψ h × V kψ h , with kχ = 1, 2, 3 and kψ = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As for the sharp-interface model, essential boundary conditions for the deformation are enforced when solving the corresponding nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Nonlinear elasticity problems with compressibility κ ≫ 1 are likely to show locking phenomena, which we deal with by choosing kχ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Time-discretization The time discretization is constructed as in [22] based on an incremental- minimization-scheme for qk ε = qε(kτ) and qk 0 = q0(kτ) with time-step size τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Based on the previous formal definition of the weak formulations for the diffuse-interface model in (6) and sharp-interface model in (12) from Section 2 this leads to the following incremental nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Incremental scheme for sharp-interface model: For given qk−1 0 ≡ χk−1 0 ∈ Vh,0 find qk 0 ≡ χk 0 ∈ Vh,0 such that − 1 τ s(χk 0 − χk−1 0 , vχ) = ⟨DF0(χk 0), vχ⟩Vh,0 , (20) for all vχ ∈ Vh,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' While the left-hand side of the problem appears is linear if in s we use the previous state qk−1 0 , it is usually nonlinear through the dependence of DF0(χk 0) on χk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Incremental scheme for diffuse-interface model: For given qk−1 ε = (χk−1 ε , ψk−1) ∈ Vh,ε and ηk−1 ∈ Uh find qk ε = (χk ε, ψk) ∈ Vh,ε and ηk ∈ Uh such that aε(ηk, φ) + 1 τ b(φ, qk ε − qk−1 ε ) = 0 , b(ηk, v) − 1 τ s(χk ε − χk−1 ε , vχ) = ⟨DFε(qk ε ), v⟩Vh,ε , (21) for all v = (vχ, vψ) ∈ Vh,ε and all φ ∈ Uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' To initialize the chemical potentials η(t = 0) in the diffuse-interface model and to overcome any initial transient (in both models) we perform the first iteration with a small time-step size 0 < τ0 ≪ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The next 20 iterations are performed with a time-step size τ/10 and then we output the solutions qk 0,∆ and qk ε,∆ every multiples of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='05 time units until the final time T is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Next we will discuss space-time discretization errors for numerical solutions q0,∆ → ∆ = {τ, h, P kχ}, (22) qε,∆ → ∆ = {τ, h, P kχ, Pkψ}, (23) of the sharp-interface model q0,∆ and the diffuse-interface model qε,∆ with the discretization pa- rameters mentioned before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' h indicates the space discretization, τ is the time step size, kχ and kψ denote the polynomial degree of the deformation and the phase fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 Estimation of numerical errors Parameters The physical parameters for the sharp-interface and diffuse-interface model are listed in Table 1 and the corresponding surface tensions for the sharp-interface model can be derived from them using Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For integer n ≥ 0, interfacial widths for the diffuse-interface model are 11 ε = 2n¯ε with ¯ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8775 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Our standard numerical parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' the once causing the error collected in ∆, are τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='005 , kχ = 2 , kψ = 2 , m0 = 1 , (24) and τ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The common mesh for the diffuse-interface model corresponds to 4¯ε which is Case c) and for the phase-field model two uniform refinements of Case a) shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The time evolution of the energy2 is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We will consider solutions for 0 < t < T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15, shown by the horizontal dotted line, after which the evolution can be considered stationary and no nontrivial contribution to the sharp-interface limit tested using the Bochner norms is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The energies of the two models agree well for m = ε5/2 but are systematically lower for m = ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Note: We omitted the factor 2π in all surface and volume integrals, which modifies the L2 error norms and energies correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Figure 4: Free energies F0 and Fε (full lines) and surface energies F0,s = � ij � ¯Γij γijd¯s and Fε,s = � Ω W ε phase dx (dashed lines) as a function of time 0 ≤ t ≤ T for sharp (red) and diffuse interfaces with mobilities m = ε1 (yellow) and m = ε5/2 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' parameter Gs Gℓ,a γs γℓ γa R H rdrop κ T value 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='0 1/2 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15 Table 1: Parameters for the sharp-interface and diffuse-interface model: Gi the shear modulus in the elastic energy, γi the surface parameter for i ∈ {s, ℓ, a}, R is the width of the rotational domain, rdrop is the initial radius of the droplet, h is the height of the substrate, H is the height of the domain, and T the final time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 2The energy is computed using the sharp-interface model with P2 elements on the coarsest mesh a) from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='4 5/2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='3 sharp energy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 1 T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 time tDiffuse-interface model The numerical errors of the phase-field model measured using the two Bochner norms are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We observe clear convergence trends in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As expected, errors in the L2 norm are always smaller then those in the corresponding L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The significant decrease in error between eε,h−h/2 and eε,h/2−h/4 by a factor ten is unclear but gives a comparable order as the time-discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the time-discretization, we observe a reduction of the error between eε,τ−2τ and eε,τ−τ/2 by roughly a factor of 2, which indicates the expected linear convergence rate in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' By far the largest errors on the order of 10−3 for the L∞ norm emerge from the polynomial order, which indicate possible locking effects for kχ and a limiting resolution a the interface for kψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' norm eε,h−h/2 eε,h/2−h/4 eε,τ−2τ eε,τ−τ/2 eε,ψP 2−ψP 3 eε,χP 2−χP 3 L2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='7 · 10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 · 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='3 · 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 · 10−4 L∞ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='6 · 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 · 10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 · 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='7 · 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='0 · 10−3 Table 2: Numerical errors for the phase-field model with respect to the L2(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) norm and with respect to the L∞(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) norm for vector-valued and scalar functions upon change of time-step size {2τ, τ, τ/2}, uniform refinement {h, h/2, h/4}, and polynomial order {P2, P3} of the FE-space of ψ and χε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Sharp-interface model For the sharp-interface model we show the numerical errors in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the uniform refinement we see a reduction in errors with a factor between 2 and 4 going from e0,h/2−h/4 to e0,h/4−h/8, which indicates a convergence order between 1 and 2 in space, as expected for P2 elements and a less regular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The reduction of the errors e0,τ−τ/2 and e0,τ/2−τ/4 is by a factor of two, showing linear convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As for the diffuse-interface model, by far the largest error come from different polynomial orders for the displacement and are of order 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This is again indicative of possible locking effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Both diffuse-interface model and sharp-interface model clearly show numerical convergence, with the largest contribution to the error coming from polynomial degree and being of the order 10−3 in the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' norm e0,h−h/2 e0,h/2−h/4 e0,h/4−h/8 e0,τ−τ/2 e0,τ/2−τ/4 e0,P1−P2 e0,P2−P3 L2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='4 · 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8 · 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 · 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='3 · 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8 · 10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='7 · 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 · 10−5 L∞ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='3 · 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 · 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='7 · 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='5 · 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='9 · 10−4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='8 · 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 · 10−3 Table 3: Numerical errors of the sharp-interface model with respect to the L2(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) norm and with respect to the L∞(QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' R2) norm for deformations upon change of time-step size {τ, τ/2, τ/4}, uniform refinement of the mesh a) {h, h/2, h/4, h/8} in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1, and polynomial order {P1, P2, P3} of the FE-space for χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 4 Convergence to sharp-interface model Figure 5 gives a visual representation of the main results of this work and shows the direct compar- ison of phase-field model and sharp-interface model at time t = T for different mobilities m = m0εα 13 and 0 ≤ α ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the following we first give a detailed discussion of direct observations in this representation for different mobilities and then we discuss the (experimental) error ∥χ0,∆ − χε,∆′∥ in some detail for different norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' a) b) c) d) e) f) g) h) Figure 5: Superposition of diffuse and sharp interfaces for ε = 2¯ε with kχ = 2, kψ = 2 and 2 uniform refinements of the mesh a) in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The shading shows the phase-field indicator ψid with solid (red), liquid (blue) and air (gray) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Thick black lines display the phase-field deformation applied to the initial position of the diffuse interfaces and the thin black mesh displays the deformation of the initial diffuse domain of the elastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Correspondingly, the thick white lines show the sharp interfaces ¯Γij(t) and the thin white mesh shows the displacement applied to the (sharp) elastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We showcase different mobilities: a) m = 1 (α = 0), b) m = ε, c) m = ε3/2, d) m = ε2, e) m = ε5/2, f) m = ε3, g) m = ε4, h) m = 0 (α = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Panels are shown at time t = T, except for a) which is at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Right next to each image a magnification of the contact line is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Firstly, note that for mobility m = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' α = 0, in panel a) of Figure 5 one can clearly see the nonconvergence of the phase-field model, since already at an earlier time the phase field (shading) and the displacement of the reference solid do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This is consistent with large excess 14 diffusion leading to a state where F = I and ψ minimizing the phase-field energy separately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' the evolution of the phase field disconnects from the evolution of the displacement and assumption A2 for the convergence is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This already happens early at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Next, for mobilities m = εα for α ≥ 1 in panels b)-e) of Figure 5 we see, except for slight convergent deviations in b), the phase field (shading) and the deformed initial interfaces (thick black lines) remain aligned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' in those regions A2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' However, for too small mobility, around α ≥ 4, the sharp interface and the diffuse field interface (thick white and black lines) may not be aligned as this additionally requires ∥χ0 − χε∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Further, note that even for α ≥ 1 in the cases b) α = 1, c) α = 3/2 we observe clear differences of the diffuse interface (shading) and the deformed mesh (thick black lines) much larger then the interfacial width ε near the contact lines visible in the magnification of the contact line in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' What might even be counter-intuitive first is that the alignment of the sharp interface (thick white lines) and the diffuse interface (shading) in these cases b, c) appears much better then the alignment of the displacements (black and white thick lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This again emphasizes that the convergence requires both convergence of the indicators A2 and convergence of the displacements A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The main observation here is that while in a suitable (weak) norm one might prove convergence to the sharp-interface model, for all practical questions and in the strong L∞ norm, the error of the contact line position clearly lags behind the highly resolved interfacial thickness ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the cases d) and e) with α = 2 and α = 5/2 we generally observe a very good agreement of phase fields (shading) and deformed initial diffuse interfaces (thick black lines) and deformed sharp interfaces (thick white lines) and the overall displacement (thin black and white mesh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This shows that these values are suitable for simulations when in particular a good precision of the predicted contact line position and small error in the displacements are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For the cases f,g,h) with α = 3, 4, ∞ we still see the perfect alignment of phase-field (shad- ing) and deformed diffuse interfaces (thick black lines) but no convergence of the displacement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' assumption A1 for convergence is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' As explained earlier, this is due to the fact that the diffuse-interface profile does not reach the optimal tanh-profile and the approximation of the Eulerian surface energy deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In our Lagrangian formulation, it is possible to choose extremely small Cahn-Hilliard mobilities and even m = 0 without the need to revise the numerical procedures and introduce artificial stabi- lization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' While in an Eulerian framework such a stabilization would be needed and possible restrictions due to CFL conditions might appear, the alignment of phase fields and displacement could in principle be also verified using an Eulerian approach by integrating the velocity field to a obtain a deformation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This concludes the discussion of Figure 5 and clearly shows that α = 1 produces significant errors near the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The behavior shown in the spatial representation of the sharp-interface limit is mirrored in the behavior of the two norms shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Here we haven chosen a representation, where we plot different errors for different ε = 2n¯ε as a function of the mobility m = εα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The reasons is that for any given ε one would like to know the best possible mobility to obtain a sufficiently precise approximation of the sharp-interface model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' From the preceding discussion its clear that both too large and too small values of the mobility will lead to suboptimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Both norms in Figure 6 clearly show that there are optimal values for m(ε) for each ε and this optimal mobility decreases for smaller ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For L∞, while for ε = 16¯ε we have mopt ∼ 10−4 we have for ε = ¯ε an optimal value mopt ∼ 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This indicates that mopt ∼ ε2 is optimal for a strong norm that takes errors near the contact line into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' On the other hand, the minimum in the L2 error is rather broad and suggests that here, if the convergence in the contact line area is not of interest, a broader range 15 Figure 6: Convergence error ∥χ0,∆ − χε,∆′∥ for different ε = 2n¯ε and different mobilities m(ε) = εα as a function m(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The used mobilities are m = ε∞ = 0, m = ε4, m = ε3, m = ε5/2, m = ε2, m = ε3/2, m = ε1, m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' The left panel shows the convergence in the L∞ norm and the right panel in the L2 Bochner norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' of α-values is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Note that based on the discussion of numerical errors in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2, these statements are limited by the numerical errors of order 10−3 in the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We finalize our discussion by considering the evolution of the spatial norms as a function of time t on the cylinder Ω in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' For initial times 0 < t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2 we observe a steep increase in the convergence error, which might be affected due to the use of initial data that do not satisfy the Neumann triangle construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This should result in a transient boundary layer at t = 0 an possibly also influence the convergence rate of the sharp-interface limit in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' However, in our experience the relaxation of a droplet from a flat planar substrate is a more realistic scenario compared to well-prepared initial data that already satisfy the Neumann triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the left panel of Figure 7 we observe convergence in the L∞ norm for decreasing ε = 2n¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the right panel we observe that for fixed ε and mobility exponents m = εα with α = 2, 5/2, 3 we obtain the lowest errors of the order 10−2, which is still larger than the discretization errors previously discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' 5 Conclusion We have presented a Lagrangian variational formulation of phase-field and sharp-interface models for a nonlinearly coupled fluid-structure interaction problem that involves nonlinear elasticity in a solid phase and a fluid and air phase, moving capillary interfaces and a moving contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' We have discussed the sharp-interface limit depending on the mobility m(ε) = m0εα in the Cahn-Hilliard model in different norms and shown that in the strong L∞ norm the moving contact lines suggests αopt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This discussion is also based on a detailed consideration of different contributions to numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Similar observations have been made before [25, 15], but here we see them as a clear consequence of the presence of the moving contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' In the future one needs to avoid possible limitation due to locking by choosing incompressible models with suitable inf-sup stable finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Additionally, by choosing degenerate state- dependent mobilities m = m(ε, ψ) and techniques as in [10] one should be able to increase the 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='15 *-8 4 $-23 (QT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' IR") 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='05 0 10-10 10~5 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='03 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='025 +-8 4 $-2m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='02 IRd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content='005 0 10-10 10~5 109Figure 7: Convergence error ∥χ0,∆ − χε,∆′∥(t) for L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rd) Sobolev norm as a function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (Left) for fixed mobility exponent m = ε2 but different ε and (right) for fixed ε = ¯ε but different mobility exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' region of validity compared to m = m(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' This discussion should be extended to models that take into account Navier-slip and dynamic contact angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Rigorous and formal asymptotic results in this direction would be desirable but require a good understanding of both upper and lower bound α, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Acknowledgement Both authors thank Marita Thomas, for discussions about analysis and nonlinear elasticity and Andreas Münch regarding aspects of asymptotics of phase-field models with degenerate mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Furthermore, we thank Helmut Abels and Harald Garcke for the discussion on sharp-interface limits and lower and upper bounds of the mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Both authors acknowledge the funding by the German Research Foundation (DFG) within the DFG Priority Program SPP 2171 Dynamic Wetting of Flexible, Adaptive, and Switchable Sub- strates through the projects #422786086 (LS) and #422792530 (DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' LS & DP thank the Berlin Mathematics Research Center MATH+ for funding and support through project AA2-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Abels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' (Non-) convergence of solutions of the convective Allen–Cahn equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Partial Differential Equations and Applications, 3(1):1–11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Abels, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Garcke, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adE4T4oBgHgl3EQfOwzB/content/2301.04968v1.pdf'} +page_content=' Grün.' metadata={'source': 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sha256:c0561a44ab53e6f2f3de6a8d78e7c61e15b7d7cac53959d5af98c01cf1caabbb +size 150000 diff --git a/btFPT4oBgHgl3EQfxDUf/content/tmp_files/2301.13165v1.pdf.txt b/btFPT4oBgHgl3EQfxDUf/content/tmp_files/2301.13165v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..03181a6054e571676f1e01b4a2a67a5b69b1186a --- /dev/null +++ b/btFPT4oBgHgl3EQfxDUf/content/tmp_files/2301.13165v1.pdf.txt @@ -0,0 +1,6487 @@ +Label-free learning of elliptic partial differential equation solvers +with generalizability across boundary value problems +Xiaoxuan Zhang1, Krishna Garikipati1,2,3 ∗ +1Department of Mechanical Engineering, University of Michigan, United States +2Department of Mathematics, University of Michigan, United States +3Michigan Institute for Computational Discovery & Engineering, University of Michigan, United States +Abstract +Traditional, numerical discretization-based solvers of partial differential equations (PDEs) are fun- +damentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt +solvers have a limited generalizability across these elements of boundary value problems. This is +strongly true in the case of surrogate models that are typically trained on direct numerical simu- +lations of PDEs applied to one specific boundary value problem. In a departure from this direct +approach, the label-free machine learning of solvers is centered on a loss function that incorporates +the PDE and boundary conditions in residual form. However, their generalization across boundary +conditions is limited and they remain strongly domain-dependent. Here, we present a framework that +generalizes across domains, boundary conditions and coefficients simultaneously with learning the +PDE in weak form. Our work explores the ability of simple, convolutional neural network (CNN)- +based encoder-decoder architectures to learn to solve a PDE in greater generality than its restriction +to a particular boundary value problem. In this first communication, we consider the elliptic PDEs +of Fickien diffusion, linear and nonlinear elasticity. Importantly, the learning happens indepen- +dently of any labelled field data from either experiments or direct numreical solutions. We develop +probabilistic CNNs in the Bayesian setting using variational inference. Extensive results for these +problem classes demonstrate the framework’s ability to learn PDE solvers that generalize across +hundreds of thousands of domains, boundary conditions and coefficients, including extrapolation +beyond the learning regime. Once trained, the machine learning solvers are orders of magnitude +faster than discretization-based solvers. They therefore could have relevance to high-throughput +solutions of PDEs on varying domains, boundary conditions and coefficients, such as for inverse +modelling, optimization, design and decision-making. We place our work in the context of recent +continuous operator learning frameworks, and note extensions to transfer learning, active learning +and reinforcement learning. +Introduction +Partial differential equation (PDE) solvers play a central role in computational science and engineering. They bridge +between the mathematical physics of field theories to applications in engineering science. Popular, discretization-based +numerical methods to solve PDEs include, but are not limited to, the finite element method (FEM), finite difference +method, finite volume method and their variants, each with its own advantages and limitations. The FEM, in its many +variant forms, is notable for the natural treatment of complex domain geometries and boundary conditions. However, +when a large number of boundary value problems need to be solved, such as for inverse modelling, optimization, design +and decision-making or these discretization-based numerical solvers can prove very expensive. Scientific machine +learning (ML) techniques have proved to be natural candidates. +ML approaches to solving mathematical descriptions of physical systems can be categorized as surrogate models and +PDE solvers. The first category typically requires a vast amount of training data, either from measurement or direct +numerical simulations (DNSs), whose acquisition can pose challenges of availability and expense, (see [1, 2, 3, 4], and +many others). For example, in Ref [1] a Bayesian uncertainty quantification (UQ) approach to convolutional neural +networks (CNNs) was proposed for flows in heterogeneous media. CNNs are also used to predict the velocity and +pressure fields in aerodynamics [3] and the concentration field for single-species reaction-diffusion systems [4]. The +second category requires little or no pre-labeled data to solve PDEs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For example, high- +dimensional, free-boundary PDEs have been solved by fully connected NNs [6], and by the Deep Galerkin Method +∗Corresponding author. E-mail address: krishna@umich.edu +January 31, 2023 +arXiv:2301.13165v1 [math.NA] 30 Dec 2022 + +A PREPRINT - JANUARY 31, 2023 +[7] using NNs, which satisfy the differential operators, initial conditions (ICs), and boundary conditions (BCs). The +Physics-informed Neural Networks (PINNs) approach has been proposed to solve steady and transient systems [8]. +In PINNs, the strong form of PDEs, the ICs and BCs are incorporated in the loss [8]. PINNs have been extended to +solve numerous systems [15, 16, 17, 10, 11, 18, 19, 20]. NN-based PDE solvers constructed from the weak/variational +formulation also have been studied [21, 22, 23, 24, 25]. +To become viable alternates to discretization-based PDE solvers, ML frameworks have to extend to solutions of the +same PDE system, but with different ICs, BCs, and on different problem domains. However, this is difficult to achieve +with NN-based approaches that typically enforce only one specific set of BCs [8], parameterized BCs and domains +[26] via the loss function [8] or the NN architecture [13, 26]. Such NN-based solvers must be retrained for each +new BC and domain. In one recent approach, the notion of a genome has been introduced to learn solutions on +subdomains and use them to construct solutions on larger, and to some extent varying shape domains [27]. However, +labelled training data are needed. A related domain-decomposition approach with NNs to impose desired regularity at +subdomain boundaries has also been used [28]. +In this work, we address these challenges by a new class of physics-constrained NN solvers where the BCs are specified +as inputs. We draw from the FEM, where the weak formulation incorporates the governing PDE, the natural and +essential BCs, and the solution corresponds to the vanishing the discretized residual. Central to our framework is +transformation of the NN predicted solution to a discretized PDE residual that defines the loss function used to train +the NNs, through efficient, convolutional operator-based, and vectorized calculations. We introduce weak PDE loss +layers via kernels whose parameters are not trainable, and are independent of the NN that learns the PDE solver. Such +features offer us great flexibility to choose the NN architecture. +In our framework, the trainable NN learns, from a number of boundary value problems, a representation that rec- +ognizes domains, boundary conditions and coefficients. To the extent that this learning is imperfect, there will be +uncertainties in its predicted solutions. One common categorization of uncertainties in modelling frameworks is as +epistemic and aleatoric. The former represents model-form error that can be reduced by learning from more data or +using a better model–and is natural for any finite-capacity NN. The latter typically represents measurement errors, +is less prone to reduction [29]–and is not applicable to the proposed framework, in which boundary value problems +are exact statements. Probabilistic machine learning models have been developed for uncertainty quantification with +NN-based PDE solvers [30, 31, 10, 11]. Techniques, such as dropout [32], adversarial inference [31], Bayesian meth- +ods [11], Stochastic Weight Averaging Gaussian [33] have been used, among many others. A recent work [34] uses +conditional generative adversarial networks to map between images for forward and inverse solution of PDEs; the +authors treatment of domains as images bears similarity to our approach. In this communication, we present both de- +terministic and probabilistic ML-PDE solvers, with Bayesian NNs (BNNs) for the latter. More specifically, we adopt +variational inference in the Bayesian setting, motivated by its efficiency over Monte Carlo approaches. We focus on an +encoder-decoder architecture, which has been investigated for other physical systems [1, 2, 3]. The encoder-decoder +structure can be easily adapted to BNNs with the modularized probabilistic layers available in current ML platforms, +of which we use the TensorFlow Probability (TFP) library. In our framework, deterministic/probabilistic convolutional +NN layers learn representations for problem domains and the applied BCs (both Dirichlet and Neumann) through care- +fully designed input data structures. Thus, a single NN can be used to simultaneously solve different boundary value +problems that are governed by the same PDEs but on different domains with different BCs. Having learnt the PDE, +the ML solvers can make predictions for interpolated and, to a certain extent, extrapolated domains and BCs that they +were not exposed to during training. Similar to other NN-based PDE solvers, our learning approach is free of labelled +data on field solutions. +We note the recent development of operator learning approaches for nonlinear mappings between continuous func- +tional spaces as inputs and solution fields as outputs. Of particular interest in this direction are DeepONets [35, 36] +as well as graph kernel networks [37] and Fourier neural operators [38] as settings for PDE solvers. Our framework +differs from these approaches in its focus upon learning solvers for specific PDEs, but with generalizability across +boundary value problems spanning domains, BCs and coefficients with the added feature of UQ. Our proposed frame- +work is generalizable and applicable to both steady-state and transient problems. Here, we present it in detail and focus +on its application to elliptic PDEs of steady-state diffusion, linear and nonlinear elasticity. We defer the investigation +of transient problems to a subsequent work. We demonstrate that, with the proposed framework, a single NN can +learn a solver that can be applied to tens to hundreds of thousands of boundary value problems. For brevity we use +NN-PDE-S for the deterministic neural network-based PDE solver, and BNN-PDE-S for the Bayesian version. +2 + +A PREPRINT - JANUARY 31, 2023 +NN +solution +zero/non-zero Dirichlet BCs +non-zero Neumann BCs +NN +solution +(I) +(II) +NN Sol.+D.BC +N.BC +NN Sol.+D.BC +(i) +(ii) +(iii) +(iv) +1 0 +1 +1 +1 +0 0 +0 +0 0 +0 +0 +0 +0 0 +0 +a +b +c +d +Neural Networks +Encoder +Decoder +(fill random +numbers) +Rbulk +Rneu +Weak PDE loss layers +Rtot +Rtot +red +Image representation to FEM representation +Rbulk +Figure 1: Architecture of the NN-PDE-S. (a) An encoder-decoder NN to store the nonlinear mapping between NN +inputs, which consist of the geometry of problem domains and the applied BCs, and the solution of PDEs. Random +numbers are given to the pixels in the interior domain of the channel with Dirichlet BCs when working with the +augmented dataset with only a few unique boundary value problems. BCs are color coded with red for zero Dirichlet +BCs; green for non-zero Dirichlet BCs; blue for non-zero Neumann BCs. (b) Both NN inputs and outputs are passed +to the Weak PDE loss layers to calculate the discretized residual, where the bulk residual (I) is computed from NN +solutions with imposed Dirichlet BCs and the residual contribution from the Neumann BCs (II) are computed based on +NN inputs. The reduced discretized residual by excluding contributions from Dirichlet boundary locations is used to +form the loss of both deterministic and probabilistic NNs. (c) Illustration of the construction of finite element meshes +from pixelated image representations. (d) Details of the bulk residual calculation (I). Four filters are applied to the +NN solutions with imposed Dirichlet BCs to select different nodes (i), resulting in a multi-channel data representation +(ii), which has the structure of the local nodes of one finite element. The multi-channel data is reshaped into a +two-dimensional matrix to perform residual calculation (iii). The two-dimensional residual matrix is reshaped to a +multi-channel data structure (iv) and x reduced sum operations are performed to get the bulk residual, avoiding the +time-consuming assembly process in the traditional FEM. +Results +(Bayesian) NN-based PDE solver +In the proposed PDE solver (Fig. 1), NNs are used to represent the nonlinear mapping between BCs and the resulting +solutions of PDEs. The discretized residuals of PDEs are used to construct the losses and therefore regularize the NN’s +solutions of PDEs. We studied both deterministic NNs and BNNs, where the uncertainty of the latter is represented by +computing the statistical moments of their outputs via the predictive expectation and the predictive variance. +Deterministic NNs have fixed model parameter values, and their losses are mean squared Euclidean norms of the +discretized reduced residual vectors. For BNNs, the model parameters are drawn from a posterior distribution that is +computed from Bayes’ theorem. The loss of BNNs is formulated using variational inference [39], which consists of a +data-independent contribution and a data-dependent contribution. The latter is the log likelihood function, which has +the form of a joint distribution of the discretized residual with an added Gaussian noise. Both NNs are trained via a +mini-batch optimization process with standard stochastic optimizers. +In the proposed framework, the PDE loss layers (Fig. 1b) are independent of the NNs, which offers flexibility in +choosing the NN architecture. An encoder-decoder NN architecture is explored (Fig. 1a). The NNs, which store the +3 + +0 +1 + +2 + +3 - +4- +0 +1 +2 +m +4A PREPRINT - JANUARY 31, 2023 +nonlinear mappings between BCs and the PDE solutions, accept image-type inputs that contain physically meaningful +boundary values and markers for different regions to allow the convolutional NN layers to learn the BCs and problem +domain. This allows a well-trained NN to make predictions for new problem domains with new BCs when these are +provided as inputs. As the image-type NN outputs can be treated as FEM meshes (Fig. 1c), we evaluate the discretized +residuals of PDEs based on the imposed BCs and NN predictions by following the FEM and using standard numerical +integration schemes. This is achieved through an efficient, discrete, convolution operator-based, and vectorized im- +plementation. Calculation of the bulk residual is illustrated in Fig. 1d with detailed implementations and procedures +provided in the SI. +In this work, we define one unique BVP as imposing a certain PDE on a specific domain with specific boundary values +at specific boundary conditions. Changes in any of these elements defines a new boundary value problem. We found +that training a single NN to solve multiple boudnary value prooblems with both Dirichlet and Neumann BCs can +be very challenging. We introduced a zero-initialization step to address the slower convergence for problems under +Neumann BCs (see SI for detailed discussions). When training BNNs to solve multiple boundary value problems +simultaneously, their parameters can stagnate around some local minima, leading to poor performance. To address +this issue, we use a warm start approach by initializing the mean of a BNN with the optimized parameters from a +deterministic NN with the identical architecture. Our method works for both small and large datasets of boundary +value problems. If the number of unique boundary value problems is small, we replicate them to obtain an augmented +dataset for training. +Steady-state diffusion problem with small dataset +In this first example (Fig. 2a-b), we use the (B)NN-PDE-S on a single steady-state diffusion boundary value problem +on an octagonal domain with imposed mixed BCs (zero/non-zero Dirichlet and non-zero Neumann BCs) at two differ- +ent mesh resolutions. The DNS solution from FEM, NN solution from a deterministic NN, and the mean and std. of +BNN results from 50 Monte Carlo samplings for a mesh resolution of 32 × 32 are shown in Fig. 2a. The optimized pa- +rameters from the deterministic NN are used for the warm start of the BNNs. The deterministic NN results, the mean +± 2 std of BNN results, and the FEM solution, which is considered the grand truth, are quantitatively commpared +along the two dashed lines in Fig. 2a. These quantitative comparisons confirm the accuracy of the NN results. The +results in Fig. 2b for a mesh resolution of 64 × 64 show the same accuracy of the NN results. In the second example +(Fig. 2c-e), we use the NN-PDE-S to simultaneously solve three boundary value problems with identical BCs but +different material parameters at a mesh resolution of 32 × 32. The quantitative comparisons in Fig. 2c-e demonstrate +the ability of a single, trained NN to simultaneously solve multiple booundary value prooblems. +Linear elasticity problem with small dataset +It is challenging to solve linear elasticity mainly because the governing PDE is written in terms of the infinitesimal +strain, which is the gradient of the displacement field and has a very small magnitude ∼ 10−4. Here, we use the +(B)NN-PDE-S for the displacement on an L-shape domain, which is fixed in both directions on the bottom edge and +has a vertical displacement applied on the left edge. A mesh resolution of 32 × 32 is considered. The problem is +defined in Fig. 3a. We consider three incremental loading levels and treat each loading level as a different boundary +value problem. The BNN used to solve these boundary value problems is trained with a warm start. The deformed +geometries from both the FEM results and the deterministic NN results are compared in Fig. 3b, where the FEM +solution is illustrated by the mesh and the NN solution by the solid black dots. The corresponding comparison between +the FEM and the mean of the BNN solution is shown in 3c. The quantitative comparisons for ux along the vertical +lines and uy along the horizontal lines in Fig. 3(b,c) between the FEM results, the deterministic NN solution, and +the mean ± 2 std of BNN results are shown in Fig. 3(d). Those results confirm the accuracy of the NN-based solver. +Additionally, we also show the comparison of reaction forces for different loading levels in both the x− and y− +directions between the FEM solution and the deterministic NNs (Fig. 3f,g) and between the FEM solution and the +mean ± 2 std of BNN results (Fig. 3h,i). However, because the magnitude of the solution at the bottom of the L-shape +is low, the NNs have difficulty computing the total reaction forces. Another example using a single deterministic NN +and BNN to solve 30 linear elastic boundary value prooblems for five different domains with six sets of BCs applied +to each appears in the SI. +Nonlinear elasticity problem with small dataset +We use the (B)NN-PDE-S solver framework on a nonlinear elasticity problem on a square domain fixed in both +directions on the left edge, and with a horizontal displacement loading and vertical traction loading applied on the +right edge. We considered 30 incremental loading levels and treated each as a different boundary value problem, +4 + +A PREPRINT - JANUARY 31, 2023 +Figure 2: (B)NN-PDE-S for steady-state diffusion. Each row contains the FEM solution (labeled as DNS), solutions +from NN-PDE-S, the mean and std of BNN-PDE-S solutions (warm started from the NN-PDE-S), and a quantitative +comparison between the FEM solution, the NN-PDE-S solution, and the mean ± 2 std of BNN-PDE-S results along +the two dashed lines. Rows 1 and 2: results for the same boundary value problem with mixed BCs at different mesh +resolutions with 32 × 32 for Row 1 and 64 × 64 for Row 2. The problem setup for Rows 1 and 2 is shown in the +SI. Rows 3-5 results for boundary value problems with identical BCs but different material parameters simultaneously +solved by a single (B)NN-PDE-S at a mesh resolution of 32 × 32. +as shown in Fig. 4c. The BNNs are trained with a warm start for four cases. The training dataset for each case +has different distributions of loading levels. The testing dataset contains both interpolated and extrapolated loading +levels, as indicated by different colors in Fig. 4f-i. The results, including the FEM solution, the deterministic NN, +and the mean and std from the BNN, for the last extrapolated loading level for case (i) in both X- and Y -directions +are shown in Fig. 4b and 4c. Additionally, we show the quantitative comparison between FEM results and the mean +± 2 std of BNN results along the dashed lines (Fig. 4d and 4e). These results confirm the accuracy of the NN-based +solver and demonstrate its predictivity for unseen extrapolated BCs. The reaction forces computed from the NN full +field solution are plotted in Fig. 4f-i, which shows improved accuracy compared to the results for linear elasticity +(Fig. 3f-i). For each case, we observe that the results from interpolated BCs are generally accurate. For extrapolated +BCs, NN predictions are accurate to a certain degree, particularly for the reaction force in the X-direction. Details of +the treatment for computing the nonlinear kinematics, constitutive relations, and discretized residuals with NNs, NN +architecture and training scheme, and an additional example on solving 30 boundary value problems with the proposed +method are provided in the SI. +Steady-state diffusion problem with large dataset +Lastly, we use the BNN-PDE-S on the steady-state diffusion problem for two very large datasets with a 64 × 64 +resolution. In these datasets, the geometries of problem domains and the values of BCs are randomly generated. The +5 + +DNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.4 +0.0030 +-0.0025 +0.3 +0.0020 +0.2 +0.0015 +0.0010 +-0.1 +0.0005 +-0.0 +0.0000Mean ± 2 Std +0.7 +DNS +Sol. (det) +0.6 +Mean +value +0.5 +0.4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.6 +0.5 +0.4 +lue +val +0.3 +0.2 +DNS +Sol. (det) +0.1 +Mean +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.7 +DNS +Sol. (det) +0.6 +Mean +value +0.5 +0.4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.6 +0.5 +0.4 +lue +val +0.3 +0.2 +DNS +Sol. (det) +0.1 +Mean +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.40 +DNS +Sol. (det) +0.35 +Mean +lue +0.30 +0.25 +0.20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.5 +DNS +0.4 +Sol. (det) +Mean +value +0.3 +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.40 +DNS +Sol. (det) +0.35 +Mean +lue +0.30 +0.25 +0.20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.5 +DNS +0.4 +Sol. (det) +Mean +value +0.3 +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.40 +DNS +Sol. (det) +0.35 +Mean +lue +0.30 +0.25 +0.20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.5 +DNS +0.4 +Sol. (det) +Mean +value +0.3 +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.7 +-0.6 +0.0020 +0.5 +0.0015 +0.4 +0.3 +0.0010 +0.2 +0.0005 +-0.1 +0.0 +0.0000DNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.7 +0.0012 +0.6 +0.0010 +0.5 +0.0008 +-0.4 +0.0006 +-0.3 +-0.0004 +0.2 +0.1 +-0.0002 +0.0 +0.0000DNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.4 +0.0030 +0.3 +0.0025 +0.0020 +0.2 +0.0015 +0.0010 +-0.1 +0.0005 +-0.0 +0.0000DNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.4 +0.0030 +0.3 +0.0025 +0.0020 +0.2 +0.0015 +0.0010 +-0.1 +0.0005 +-0.0 +0.0000A PREPRINT - JANUARY 31, 2023 +a +b +c +f +g +h +i +d +Figure 3: (B)NN-PDE-S for linear elasticity. (a) setup of BCs for the L-shape problem. (b) comparison of solutions +between the FEM and the NN-PDE-S. Both solutions are plotted in the deformed configuration with the FEM solution +being illustrated by the mesh grid and the deterministic NN solution being illustrated by the solid black dot. (c) similar +solution comparison as in (b), but between the FEM solution and the mean of the BNN-PDE-S solutions among 50 +Monte Carlo samplings. (d) comparison of displacements in the X-direction along the vertical line and in the Y- +direction along the horizontal line among the FEM solution, the NN-PDE-S solution, whose parameters are used to +warm start the BNN-PDE-S, and the mean and std of solutions from a BNN-PDE-S for a solution resolution of 32×32. +(f, g) comparison of reaction forces in both X- and Y-direction between the FEM and the NN-PDE-S. (h, i) comparison +of reaction forces in both X and Y -direction between the FEM and the BNN-PDE-S. +locations of BCs are randomly selected from two non-adjacent edges. The boundary values along one edge could +be constant, a linear distribution, quadratic distribution, or sinusoidal distribution. The details of the data generation +scheme and the statistics of our datasets are discussed in the SI. +In the first case, the training dataset contains 168,000 unique boundary value problems with pentagons, where none of +the inner angles exceed 160◦. The testing dataset is newly generated with 80 boundary value problems on pentagons +with all angles < 160◦ and 80 on “extreme" pentagons (those with at least one angle ≥ 160◦). Some of the selected +results for different geometries are shown in Fig. 5a, which shows that the trained NNs could predict the solution for +unseen geometries, including near-degenerate polygons, and unseen BCs. The accuracy of the NN results is evaluated +by the volume averaged L2 error across all boundary value problems in the corresponding (training/testing) dataset, +which is calculated as +∥e∥2 = +1 +NBVP +NBVP +� +l=1 +� +� +� +� +� +� +� 1 +Kpx +Kpx +� +k=1 +� +yDNS +l,k +− yNN +l,k +�2 +� +� +� . +(1) +The L2 errors of NN results for both training and testing datasets are plotted in Fig. 5(d). We observe that boundary +value problems with only Dirichlet BCs generally perform better than those with Neumann BCs. +In the second case, the training dataset contains 192,000 unique boundary value problems with randomly generated +geometries of quadrilaterals, pentagons, and hexagons, whereas the testing dataset is newly generated with the number +of edges of spanning from four to eleven with 16 boundary value problems for each type of polygon. Some of the +selected results for different polygons appear in Fig. 5c, again demonstrating that the trained NNs predict solutions on +unseen geometries and unseen BCs. The L2 errors of NN results for both training and testing datasets are plotted in +Fig. 5(e). We observe some degradation of NN results as the number of polygon edges in the testing set significantly +exceed those in the training dataset. However, for training up to hexagons, this degradation in accuracy sets in only +for nonagons, decagons and hendecagons. +Discussion +Our framework successfully learns PDE-specific solvers, whether restricted to a single or multiple boundary problems. +It has been our focus to develop solvers that generalize across boundary value problems with different domains, +6 + +Mean ± 2 Std +0.020 +DNS +Sol. (det) +0.015 +Mean +ue +0.010 +val +0.005 +0.000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.03 +DNS +Sol. (det) +0.02 +Mean +lue +val +0.01 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 1 Std +-0.02 +-0.04 +DNS +-0.06 +Train +Inter. +0.010 +0.015 +0.020 +0.025 +0.030 +MyDeterministic +0.015 +0.010 +DNS +E +Train +Inter. +0.005 +0.000 +0.010 +0.015 +0.020 +0.025 +0.030 +MyMean ± 1 Std +0.03 +DNS +Train +Inter. +0.02 +E +0.01 +0.00 +0.010 +0.015 +0.020 +0.025 +0.030Deterministic +-0.01 +-0.02 +F +-0.03 +DNS +-0.04 +Train +Inter. +0.010 +0.015 +0.020 +0.025 +0.030 +MyyA PREPRINT - JANUARY 31, 2023 +a +b +x +y +d +e +f +g +h +i +case i +case ii +case iii +case iv +c +Figure 4: (B)NN-PDE-S for nonlinear elasticity. (a) definition of the boundary value problem for the interpolated and +extrapolated loading example. Three loading levels and the deformed shapes are illustrated. (b) Comparison between +the FEM solution and the NN-PDE-S solution with a resolution of 16 × 16 for the last extrapolated BCs for case i. +The deformed shapes are plotted with the FEM solution illustrated by the mesh and the NN solution by the solid black +dot. (c) The corresponding comparison between the FEM solution and the mean of the BNN-PDE-S solution over 50 +MC samplings. (d) Comparison of displacement in the X-direction along the horizontal white lines in (b, c) between +the solutions of the FEM, the NN-PDE-S whose parameters are used to warm start the BNN-PDE-S, and the mean and +std of solutions from the BNN-PDE-S. (e) Similar comparison as in (d), but for displacement in the Y -direction along +the vertical white lines in (b, c). (f-i) reaction forces in both X- and Y -directions for four different cases with each +containing a different number of training and testing data. +Table 1: Summary of the accuracy and speed of the (B)NN-PDE-S for the example presented in Fig. 4. Note: There +may be speedup of the FEM simulation with further code optimization. The wall times of (B)NN-PDE-S results do +not include the time to load the solver. +Solver +Hardware +Software +Wall-time +Averaged L2 error +FEM +Intel i7-8750, 2.2GHz (use single core) +mechanoChemFEM +110ms +- +deterministic NN +GeForce GTX 1050 Ti, 4GB memory +Tensorflow +0.22ms +2.45e-3 +BNN +GeForce GTX 1050 Ti, 4GB memory +Tensorflow +0.29ms +3.07e-3 +boundary conditions and coefficients, and are orders of magnitude faster than the traditional FEM, as shown in Table +1. Such features distinguish our approach from certain other NN-based PDE solvers [8, 26]. +We have used the framework to solve the diffusion problem over two fairly complex, and large datasets of boundary +value problems to demonstrate its performance. From the corresponding examples, we observe that the (B)NN-PDE-S +makes more accurate predictions for interpolated BCs. To improve its performance, one can manually introduce new +BCs to the training dataset to improve the poorly trained region or targeted prediction region; for instance, expanding +the dataset with many geometries if prediction across domains is the goal. If the same geometries are considered +for training and testing, the dataset could be augmented by filling the interior domain with random numbers. Nat- +7 + +Mean ± 1 Std +DNS +Train +-5 +Inter. +Extra. +-10 +-15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +Train +Inter. +1 +Extra. +F +-2 +-3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +Train +-5 +Inter. +Extra. +-10 +-15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +DNS +Train +Inter. +-1 +Extra. +-2 +-3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +-5 +Train +Inter. +-10 +Extra. +E +-15 +-20 +-25 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +Train +Inter. +Extra. +-3 +-4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +rMean ± 1 Std +0 +DNS +-5 +Train +Inter. +-10 +Extra. +F +-15 +-20 +-25 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +-3 +Train +Inter. +-4 +Extra. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +rMean ± 1 Std +0 +DNS +-5 +Train +Inter. +Extra. +-10 +E +-15 +-20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +1 +-2 +DNS +Train +Inter. +Extra. +-3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +Train +-5 +Inter. +Extra. +-10 +-15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +DNS +0 +Train +Inter. +1 +Extra. +-2 +-3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +DNS +Train +-5 +Inter. +Extra. +-10 +-15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 1 Std +0 +DNS +-2 +Train +Inter. +Extra. +-3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Mean ± 2 Std +DNS +0.65 +Mean +0.60 +value +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateX +y3Mean ± 2 Std +1.0 +DNS +0.9 +Mean +value +0.8 +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateA PREPRINT - JANUARY 31, 2023 +a +b +c +e +d +Figure 5: NN-PDE-S for steady-state diffusion with large dataset. (a) Selected good NN-PDE-S predictions for +new, randomly generated extreme pentagons. (b) Selected good NN-PDE-S predictions for new, randomly generated +regular pentagons. (c) Selected good NN-PDE-S predictions for new, randomly generated polygons with different +total numbers of edges. (d) L2 error of NN-PDE-S results for the pentagon example study. L2 error from all boundary +value problems, those with Neumann BCs, and others without Neumann BCs are plotted for both training and testing +datasets. (e) L2 error of NN-PDE-S results for the polygon example study. L2 error from all boundary value problems, +those with Neumann BCs, and others without Neumann BCs are plotted for both training and testing datasets. +urally, good sampling of BCs and boundary locations is important for learning. The performance could further be +improved via careful hyper-parameter tuning. The learning or cross-validation error computed over this dataset of +∼ 105 boundary value problems with randomly generated polygonal domains (order and shape), boundary conditions +and coefficients can be used to drive an active learning workflow that detects regimes (domains, boundary conditions, +boundary value functions and coefficients) in which additional boundary value problems are needed for improved +learning. +For BNNs, we applied a constant additive noise to the NN solutions, which is propagated through the PDE residual +calculation to compute the training loss function. While the additive noise often is used to account for aleatoric +uncertainty in Bayesian inference, here it accounts for model form error of the BNN, and thus corresponds to epistemic +uncertainty. The applied noise Σ1 functions as the convergence threshold used in the traditional numerical methods for +PDEs. The loss of BNNs consists of data-independent (K-L divergence) and data-dependent (negative log-likelihood) +contributions (Methods, Eqs (17-23)). During training, the data-dependent log-likelihood contribution to the loss is, in +general much larger than the data-independent K-L divergence, and decreases. As it does, the data-independent K-L +divergence weighs more, leading the mean of the BNN solutions to drift away from the ground truth. With training, +both contributions decrease and the mean of the BNN solutions gradually converges to the DNS solution. The K-L +divergence term tends to introduce more uncertainty to the model parameters especially with poorly informed priors. +The log-likelihood, which depends on Σ2 tends to reduce the uncertainty. It controls convergence of the NN solution +to the DNS solution as Σ2 itself converges. We report the evolution of Σ2 in Figs S21, S25 and SS29. +8 + +DNS +Pred. +0.800 +0.800 +0.775 +0.775 +0.750 +0.750 +0.725 +0.725 +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625DNS +Pred. +0.850 +0.850 +0.825 +0.825 +0.800 +0.800 +0.775 +0.775 +0.750 +0.750 +0.725 +0.725 +0.700 +0.700 +0.675 +0.675DNS +Pred. +0.75 +0.75 +0.70 +0.70 +0.65 +0.65 +0.60 +0.60 +0.55 +0.55DNS +Pred. +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2DNS +Pred. +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1DNS +Pred. +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3DNS +Pred. +0.750 +0.750 +0.725 +0.725 +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600DNS +Pred. +0.9 +0.9 +- +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3DNS +Pred. +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600 +0.575 +0.575 +0.550 +0.550DNS +Pred. +0.85 +0.85 +0.80 +0.80 +0.75 +0.75 +0.70 +0.70 +0.65 +0.65 +0.60 +0.60 +0.55 +0.55DNS +Pred. +0.90 +0.90 +0.85 +0.85 +0.80 +0.80 +0.75 +0.75 +0.70 +0.70 +0.65 +0.65 +0.60 +0.60DNS +Pred. +0.725 +0.725 +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600 +0.575 +0.575 +0.550 +0.5500.20 +all BCs +w Neu. BCs +0.15 +w/o Neu. BCs +error +0.10 +L2 +0.05 +0.00 +regular +extreme +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +err +S0.10 +0.05 +0.00 +4 +5 +6 +7 +8 +9 +10 +11 +train +testDNS +Pred. +Pointwise Error +0.85 +0.85 +0.02 +0.01 +0.80 +0.80 +0.00 +0.75 +0.75 +0.01 +0.70 + 0.70 +0.02 +0.65 +0.65 +0.03 +0.60 +0.60DNS +Pred. +Pointwise Error +0.70 +0.70 +0.005 +0.68 +0.68 +0.000 +0.66 +0.66 +0.005 +0.64 +0.64 +-0.010 +0.62 +0.62 +0.015 +0.60 +0.60 +-0.020 +0.58 +0.58DNS +Pred. +Pointwise Error +0.725 +0.725 +0.03 +0.700 +0.700 +0.02 +0.675 +0.675 +0.650 +0.650 +0.01 +0.625 +0.525 +0.00 +0.600 +0.600 +0.575 +0.575 +0.01 +0.550 +0.550DNS +Pred. +Pointwise Error +0.8 +0.8 +0.7 +0.7 +0.02 +0.01 +0.6 +0.6 +0.00 +-0.01 +0.5 +0.5 +-0.02 +0.4 +0.4 +-0.03 +0.04 +0.3 +0.3A PREPRINT - JANUARY 31, 2023 +We use a “warm start” by initializing the means of the BNN parameters to the optimized parameters from deterministic +NNs with an identical architecture. An alternative is to initialize the means of the prior to the optimized parameters +from deterministic NNs. +BVPs with very small variations in the solutions, such as the L-shape linear elasticity example, pose challenges since +the NNs have difficulty capturing small differences. Formally, of course, these problems have low information content. +Additionally, the linear elasticity residual is determined by the displacement gradient (infinitesimal strain) field, which +is invariant to data normalization. We found that though the residual is computed from the physical, un-normalized, +solution, learning is more effective with data normalization. It ensures that the variations of NN outputs (scaled +solution) is large ∈ [−1, 1] unlike the infinitesimal strain ∼ 10−4. Large variations in the NN outputs drive NN +parameter variations and favor training. This also applies to diffusion when the solution range is very small. In the +residual-based loss, however, the NN output is scaled back to the physical range, to prevent violation of physics. +Hyper-parameter searches are essential for optimal (B)NN-PDE-S. The PDEs differ in their optimal hyperparameters. +NNs targeted at solving a wider range of steady state diffusion boundary value problems required wider layers. The +vector elasticity problems, even with isotropic properties, have greater information content in their solution field and +in general demanded wider layers. The numbers of layers were more closely aligned, and optimal kernel sizes were +the same across the PDEs and targeted boundary value problem ranges. See Tables S1, S3, S6, S8, S10, S12. +The NN solvers presented here were trained on a single GeForce GTX 1050 Ti GPU with 4GB memory. Training +could take hours for networks designed to solve ∼ 20 boundary value problems, and days for those to solve ∼ 105 of +boundary value problems. In addition to more high-powered GPUs, as well as multi-GPU training, optimization of the +training workflow remains unexplored. The training time would be further reduced if transfer learning or multi-fidelity +learning is used by continued training of previously trained networks. Training the (B)NN-PDE-S takes more time +than training regular NNs because of the PDE constrained loss layer. The prediction time, however, is unaffected by +these loss layers, as they are not activated during prediction. +In this work, the problem domains are represented via pixels on a square background grid for simplicity. Thus, domain +boundaries are not smooth curves, but have a pixel-level resolution. This treatment is of importance as it applies +directly to solving PDEs on pixel-based, experimental images as domain data–a target future application for our work. +For smooth boundary representations, one can leverage recent work for approaches to map complex and irregular +domains onto a regular mesh [3, 4, 26]. Such geometric transformations can be taken into account in the proposed +PDE loss layers, for instance via the mapping of the physical domain from parent hyper-cubes, as is commonly done +in the FEM. While we have considered polygonal domains for their approximation of other geometries, the above +mapping could be exploited to remove this restriction, also. +Our approach is formally different from recent operator networks which are focused on learning nonlinear mappings +between input function spaces and output spaces, and therefore are mesh independent. In this regard we note that +a NN solver that has learnt a PDE on a given discretization (pixel resolution) can serve as the source network for a +target finer or coarser mesh within a transfer learning context. The most important difference between the presented +(B)NN-PDE-S and DeepONets [35, 36], graph kernel networks [37] and Fourier operator networks [38] is that these +operator network approaches need labelled field data for training–typically from DNS using the underlying PDE. By +not presenting and labelled field solution, but only the domain, BCs and coefficients to the network, our approach in +addition to being label free allows room for our claim that the network is forced to learn the PDE, which by definition +holds across boundary value problems. We note that the recent TL-DeepONet [36] uses a source Banach space from +a DeepONet trained on labelled data for specific boundary conditions. Further training the final layer of the TL- +DeepONet allows transfer learning to some new boundary value problems. We are not aware of the extensiveness +to which this transfer learning across boundary value problems has been studied by the authors. Our approach is +very appealing for high-throughput solution of PDEs ranging from inverse and other optimization problems through +design and decision-making. Its generalizability across domains and boundary conditions also presents opportunities +in topology optimization problems. Ongoing developments will extend it beyond elliptic PDEs. +Methods +General elliptic PDEs In this work, we develop (B)NN-PDE-S for steady-state diffusion, linear elasticity, and nonlin- +ear elasticity. These three physical systems are described by a general elliptic PDE on a continuum domain Ω ⊂ Rn +with Dirichlet BCs on Γϕ and the Neumann BCs on Γk: +∇ · A(ϕ) = 0 +on +Ω, +ϕ(X) = ¯ϕ(X) +on +Γϕ, +k(X) = ¯k(X) +on +Γk, +(2) +9 + +A PREPRINT - JANUARY 31, 2023 +where ϕ(X) represents the spatially-dependent unknown field and X ∈ Rn is the position vector. The boundary +of the continuum domain satisfies Γ = Γϕ � Γk and Γϕ � Γk = ∅. We use bold typeface for ϕ, A, and k in (2), +depending on the physical system, they could represent either scalar, vector, or tensor fields. For example, in the +diffusion problem, ϕ, A, and k represent the compositional order parameter (scalar), the diffusive flux (vector), and +the outflux (scalar), respectively, whereas in elasticity problems, ϕ, A, and k represent the deformation field (vector), +the stress field (second-order tensor), and the surface traction (vector), respectively. The details of each system appear +below. +The weak form of (2) states: For variations ω satisfying ∀ω ∈ V with V = +� +ω|ω = 0 on Γϕ� +, seek trial solutions +ϕ ∈ S with S = +� +ϕ|ϕ = ¯ϕ on Γϕ� +such that +� +Ω +∇ω · A(ϕ) dV − +� +Γk +¯k · ω dS = 0. +(3) +Eq. (3) is obtained by multiplying (21) with ω, integrating by parts, and then incorporating the Neumann BC in (23). +For the diffusion problem, ω is a scalar field. For elasticity problems, ω is a vector field. +Approximate, numerical solutions of (3) can be obtained using its finite-dimensional form. Finite-dimensional ap- +proximations of ω and ϕ, denoted by ωh and ϕh, are constructed with ωh ∈ V h = +� +ωh|ωh = 0 on Γϕ� +and +ϕh ∈ S h = +� +ϕh|ϕh = ¯ϕ on Γϕ� +. The finite-dimensional fields ωh, ∇ωh, and ϕh are computed as +ωh = Ndω, +∇ωh = Bdω, +and +ϕh = Ndϕ +(4) +in terms of the nodal solutions dω and dϕ, the basis functions N, and the basis function gradient operator B = ∇N. +Inserting (4) into (3) we obtain the discretized residual as an assembly over subdomains Ωe and their associated +boundary Γe as +R = +nelemA +e=1 +�� +Ωe BT A(ϕh)dV − +� +Γe,k N T ¯k dS +� +(5) +where A is the assembly operator and nelem represents the total number of subdomains. The volume and surface +integrations in (5) are evaluated numerically via Gaussian quadrature rules. In this work, the problem domain Ω is +treated as an image. Single component, connected graphs whose vertices are pixels form the subdomains Ωe. Pixel +connectivity is preserved as graph edges in the image data. Field values at each pixel of the image are treated as nodal +values. Additional discussion on constructing the subdomains based on image pixels is provided in the SI. Of interest, +but tangential, in this context is a recent work in which NN layers map between FE meshes of different resolutions +[40]. +Steady-state diffusion This problem is described by a linear elliptic PDE in the scalar composition field following (2) +∇ · H = 0 +on +Ω, +C(X) = ¯C(X) +on +ΓC, +H = ¯H(X) +on +ΓH. +(6) +In (6), C represents the composition, H is the diffusive flux defined as +H = −D∇C, +(7) +with D as the diffusivity, and H is the outward surface flux in the normal direction. The discretized residual function +(5) for steady-state diffusion is written as +R = +nelemA +e=1 +�� +Ωe BT HdV − +� +Γe,H N T ¯H dS +� +. +(8) +Diffusivity D ∈ [1.0, 100.0] has been used. +Linear elasticity This problem also is posed as a linear elliptic PDE, but in terms of a vector field, u ∈ Rn. Following +(2) we have: +∇ · σ = 0 +on +Ω, +u(X) = ¯u(X) +on +Γu, +T = ¯T (X) +on +ΓT . +(9) +10 + +A PREPRINT - JANUARY 31, 2023 +In (9), u represents the displacement field, σ is the stress tensor, and T is the surface traction. Here, σ is related to +the infinitesimal strain ε = 1 +2 +� +∇u + (∇u)T � +via the following constitutive relationship +σ = λtr(ε)1 + 2µε +(10) +where λ and µ are the Lamé constants, and 1 is the second-order identity tensor. The discretized residual function (5) +for the linear elasticity problem is written as +R = +nelemA +e=1 +�� +Ωe BT σdV − +� +Γe,T N T ¯T dS +� +. +(11) +We used λ = 14.4231 and µ = 9.61538 in both DNSs with FEM for comparison with the (B)NN-PDE-S and in the +PDE loss layers. +Nonlinear elasticity With the displacement u ∈ Rn as the vector field unknown, we write following (2): +∇ · P = 0 +on +Ω0, +u(X) = ¯u(X) +on +Γu +0 , +T = ¯T (X) +on +ΓT +0 , +(12) +for the nonlinear elasticity problem, with the subscript 0 indicating the reference configuration. In (12), P is the +first Piola-Kirchhoff stress tensor, and T is the surface traction. In the nonlinear elasticity problem, the deformation +gradient is defined as F = 1 + ∂u/∂X with 1 being the second-order identity tensor. The right Cauchy-Green +deformation tensor is written as C = F T F . The following compressible Neo-hookean hyperelastic free energy +function is considered +W = 1 +2µ(tr(C)3 − 3 − 2 ln(J)) + λ1 +2(J − 1)2, +(13) +with µ and λ as the Lamé constants and J = det(F). The Piola stress tensor P is computed as +P = ∂W +∂F = λ(J2 − J)F −T + µ(F − F −T ). +(14) +The discretized residual function (5) for the nonlinear elasticity problem is written as +R = +nelemA +e=1 +�� +Ωe BT P dV − +� +Γe,T N T ¯T dS +� +. +(15) +The same Lamé constants were used as in linear elasticity. +Deterministic loss When using mini-batch optimization to train the NN-PDE-S over a dataset D, where each data +point is a boundary value problem with information on problem domain and BCs, the batch loss Li is written in terms +of the reduced total residual Rred +tot , as illustrated in Fig. 1, as +Li = 1 +N +N +� +n=1 +� +Rred +tot (Di, Θ) +�2 , +(16) +for each mini-batch i = 1, 2, · · · , M with N indicating the number of data points (boundary value problems) in each +mini-batch. The detailed training of NN-PDE-S is discussed in the SI. +Probabilistic loss In BNN-PDE-S, each model parameter is sampled from a posterior distribution. We solve for the +posterior distribution of model parameters with variational inference instead of Markov Chain Monte Carlo (MCMC) +sampling, as the latter involves expensive iterative inference steps and is not suitable for systems with a large number +of parameters [39, 41]. In our work, the likelihood function is constructed based on the discretized PDE residuals. An +additive noise is often applied to the NN predicted solution to represent the aleatoric uncertainty [42, 43, 10, 1, 9]. +Here, we also applied an additive noise to the solution to represent epistemic uncertainty stemming from model form +error between BNN-PDE-S, whose perturbed solutions are still constrained by the PDEs, and the FEM solver that +yields the DNS solution. +The BNN model parameters Θ are stochastic and sampled from a posterior distribution P(Θ|D) instead of being +represented by single values as in deterministic NNs. The posterior distribution P(Θ|D) is computed based on Bayes’ +theorem +P(Θ|D) = P(D|Θ)P(Θ) +P(D) +, +(17) +11 + +A PREPRINT - JANUARY 31, 2023 +where D denotes the i.i.d. observations (training data) and P represents the probability density function. In (17), +P(D|Θ) is the likelihood, P(Θ) is the prior probability, and P(D) is the evidence, respectively. The likelihood is the +probability of D given Θ, which describes the probability of the observed data for given parameters Θ. A larger value +of P(D|Θ) implies that Θ is more likely to yield D. The prior must be specified to begin Bayesian inference [44]. +To compute the posterior distributions of Θ, one can use popular sampling-based methods, such as MCMC. However, +MCMC involves expensive iterative inference steps and would be difficult to use when datasets are large or models are +very complex [41, 45, 39]. An alternative is variational inference, which approximates the exact posterior distribution +P(Θ|D) with a more tractable surrogate distribution Q(Θ) by minimizing the Kullback-Leibler (KL) divergence +[45, 39, 46] +Q∗ = arg min DKL(Q(Θ)||P(Θ|D)). +(18) +Variational inference is faster than MCMC and easier to scale to large datasets. We therefore explore it in this work, +even though it is less rigorously studied than MCMC [39]. The KL divergence is computed as +DKL(Q(Θ)||P(Θ|D)) = EQ[log Q(Θ)] − EQ[log P(Θ, D)] + log P(D), +(19) +which requires computing the logarithm of the evidence, logP(D) in (17) [39]. However, computation of P(D) +would require marginalization over all realizations of Θ–an intractable task. It is also difficult to estimate P(D). +Consequently, it is challenging to directly evaluate the objective function in (18). Alternatively, we can optimize the +so-called evidence lower bound (ELBO) which is equivalent to the KL-divergence up to an added constant. Therefore, +an optimal distribution determined using the ELBO is also optimal for the KL-divergence. +ELBO(Q) = −DKL(Q(Θ)||P(Θ|D)) + log P(D) +ELBO(Q) = EQ[log P(Θ, D)] − EQ[log Q(Θ)] += EQ[log P(D|Θ)] − (EQ[log Q(Θ)] − EQ[log P(Θ)]) += EQ[log P(D|Θ)] − DKL (Q(Θ)||P(Θ)) . +(20) +So, the loss function for the BNN is written as +L = DKL (Q(Θ)||P(Θ)) − EQ[log P(D|Θ)], +(21) +which consists of a prior-dependent but data-independent part and a data-dependent part. The former is the KL- +divergence of the surrogate posterior distribution Q(Θ) and the prior P(Θ), and the latter is the negative log- +likelihood. For mini-batch optimization, the batch loss is written as +Li = 1 +M DKL (Q(Θ)||P(Θ)) − EQ[log P(Di|Θ(i))], +(22) +for each mini-batch i = 1, 2, · · · , M [47]. With (22), we have L = � +i Li. Following Ref [47], Monte Carlo (MC) +sampling is used to approximate the expectation in (22) as +Li ≈ 1 +M DKL (Q(Θ)||P(Θ)) − 1 +N +N +� +n=1 +log P(Dn +i |Θ(i)), +(23) +where N is the size of each mini-batch dataset, and Θ(i) denotes the ith batch sample drawn from the posterior distri- +bution Q(Θ). Even though only one set of parameters Θ(i) is drawn from Q(Θ) for each mini-batch, the perturbation +approach proposed by Flipout (see SI) ensures that parameters are de-correlated for each individual example Dn +i in +calculating the log-likelihood. Probabilistic dense layers and convolutional layers with the Flipout weight perturbation +technique have been implemented in the TFP Library 2 and are used to construct the BNNs in this work. +Neural network structure and loss function Using modularized implementation of probabilistic layers in the TFP +library it is easy to construct the BNN-PDE-S to have the encoder-decoder architecture shown in Fig. 1, which is +similar to the NN-PDE-S but with all weights being drawn from probability distributions. The loss of the BNNs is +given in (21). The probabilistic layers in the TFP library automatically calculate the prior-dependent KL-divergence +and add it to the total loss. +The data-dependent loss is accounted for by the likelihood. Assuming Gaussian noise ϵ ∼ N(0, Σ1I) with a zero- +mean and a pre-specified constant covariance Σ1, the NN representation f(x, Θ) is augmented by noise to yield the +output y: +y = f(x, Θ) + ϵ. +(24) +2www.tensorflow.org/probability/api_docs/python/tfp/layers +12 + +A PREPRINT - JANUARY 31, 2023 +Forward solutions are obtained by seeking to drive the residual to zero, and correspond to satisfaction of the weak +form. For BNN-PDE-S, the likelihood function is constructed from the residual value, rather than the NN predicted +solutions, thus ensuring that the framework remains label free. With the noise ϵ in (24) propagating through the +residual calculation, the likelihood function is written as +P(Rred +tot (Di, Θ(i))|0, ΣI) = +K +� +k=1 +N +� +Rred,k +tot +|0, Σ2 +� +(25) +where index k indicates the pixel number with K total pixels and Rred,k +tot +is the component of Rred +tot at pixel k. For +systems where nonlinear operations are involved in the residual calculation, the residual noise distribution is in general +non Gaussian even if the noise in the BNN outputs is assumed to be Gaussian. Under the conditions that Σ1 is small +and the nonlinear operations are smooth, there exists a neighborhood of any point in which the linear approximation is +valid (higher-order terms being negligible), we assume that Σ2, the noise distribution of the residual, is approximately +Gaussian. As it is challenging to directly calculate Σ2 via error propagation based on Σ1, we treat Σ2 as a learnable +parameter to be optimized based on the NN loss. In (25), Σ2 essentially serves as a convergence threshold for the +residual. The batch-wise loss of the residual constrained BNNs has the following form +Li ≈ 1 +M DKL (Q(Θ)||P(Θ)) − 1 +N +N +� +n=1 +K +� +k=1 +log +� +N +� +Rred,k +tot +(Dn +i , Θ(i))|0, Σ2 +�� +. +(26) +The detailed training scheme for BNN-PDE-S is discussed in the SI. +Uncertainty quantification The BNNs allow us to quantify the epistemic uncertainty from model parameters. With +the discretized residual constrained BNNs, the posterior predictive distribution P(y∗|x∗, D) of the BNN-predicted +full field solution y∗ for a specific testing data point x∗, i.e. one boundary value problem with information on problem +domain and BCs is [1, 42] +P(y∗|x∗, D) = +� +P(y∗|x∗, Θ)P(Θ|D)dΘ +≈ +� +P(y∗|x∗, Θ)Q(Θ)dΘ, +(27) +which can be numerically evaluated via MC sampling as +P(y∗|x∗, D) ≈ 1 +S +S +� +s=1 +P(y∗|x∗, Θs) +where +Θs ∼ Q(Θ), +(28) +with s indicating each sampling. To represent the uncertainty, we compute the statistical moments of y∗ via the +predictive expectation +E[y∗|x∗, D] ≈ 1 +S +S +� +s=1 +f(x∗, Θs) +(29) +and the predictive variance +Var[y∗|x∗, D] = E[(y∗ + ϵ)2] − (E[y∗ + ϵ])2 +≈ 1 +S +S +� +s=1 +� +f(x∗, Θs)f T (x∗, Θs) + Σ1I +� +− +� +1 +S +S +� +s=1 +f(x∗, Θs) +� � +1 +S +S +� +s=1 +f(x∗, Θs) +�T +. +(30) +Code Availability +Our modularized code implementation is publicly available3, which will assist the extension to other PDE systems. +Acknowledgements +We gratefully acknowledge the support of Toyota Research Institute, Award #849910: “Computational framework +for data-driven, predictive, multi-scale and multi-physics modeling of battery materials”. Computing resources were +3github.com/mechanoChem/mechanoChemML +13 + +A PREPRINT - JANUARY 31, 2023 +provided in part by the National Science Foundation, United States via grant 1531752 MRI: Acquisition of Conflux, +A Novel Platform for Data-Driven Computational Physics (Tech. Monitor: Ed Walker). This work also used the +Extreme Science and Engineering Discovery Environment (XSEDE) Comet at the San Diego Supercomputer Center +and Stampede2 at The University of Texas at Austin’s Texas Advanced Computing Center through allocation TG- +MSS160003 and TG-DMR180072. +Author Contributions +Competing Interests statement +References +[1] Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder-decoder networks for surrogate mod- +eling and uncertainty quantification. J. Comput. Phys., 366:415–447, 2018. +[2] Nick Winovich, Karthik Ramani, and Guang Lin. ConvPDE-UQ: Convolutional neural networks with quanti- +fied uncertainty for heterogeneous elliptic partial differential equations on varied domains. J. Comput. Phys., +394:263–279, 2019. +[3] Saakaar Bhatnagar, Yaser Afshar, Shaowu Pan, and Karthik Duraisamy. Prediciton of Aerodynamic Flow Fields +Using Convolutional Neural Networks. Comput. 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Weight uncertainty in neural net- +works. 32nd International Conference on Machine Learning, ICML 2015, 2:1613–1622, 2015. +16 + +Learning solvers for elliptic partial differential equations with +generalizability across boundary value problems - supplementary +information +Xiaoxuan Zhang1, Krishna Garikipati1,2,3 ∗ +1Department of Mechanical Engineering, University of Michigan, United States +2Department of Mathematics, University of Michigan, United States +3Michigan Institute for Computational Discovery & Engineering, University of Michigan, United States +S1 +Training NNs with stochastic weights - Flipout +Different methods are available for training neural networks (NNs) with stochastic weights, such as weight perturba- +tion [1, 2, 3], activation perturbation [4], reparameterization [5], and many others. In this work, we follow a specific +weight perturbation method, the so-called Flipout, proposed in Ref [3]. Compared with other weight perturbation algo- +rithms that suffer from high variance of the gradient estimates because the same perturbation is shared in a mini-batch +for all training examples, Flipout is an efficient method, which decorrelates the gradients in a mini-batch by implic- +itly sampling pseudo-independent weight perturbation for each example, and thus reduces the variance of NNs with +stochastic weights [3]. This method can be efficiently implemented in a vectorized manner with unbiased stochastic +gradients. +A brief description of Flipout is summarized here. Readers are directed to Ref. [3] for details. Flipout assumes that the +perturbations of different weights are independent, and the perturbation distribution is symmetric around zero. Under +such assumptions, the perturbation distribution is invariant to element-wise multiplication by a random sign matrix. +To minimize the loss L, the distribution of Q(Θ) can be described in terms of perturbations with W = W + ∆W, +where W and ∆W are the mean and a stochastic perturbation for NN parameters Θ, respectively. Flipout uses a base +perturbation � +∆W shared by all samples (training data points) in a mini-batch, and arrives at the perturbation for an +individual sample by multiplying � +∆W with a different rank-one sign matrix +∆Wn = � +∆W ◦ rnst +n, +(1) +where the subscript n indicates an individual example in a mini-batch, the superscript t denotes the transpose operation, +and rn and sn are entries of random vectors uniformly sampled from ±1. Using different perturbations for each sample +in a mini-batch rather than an identical perturbation for all the example in a mini-batch ensures the reduction of the +variance of the stochastic gradients in Flipout during training. For BNNs, the W and � +∆W are the mean and standard +deviation of the posterior distribution Q(Θ), which are obtained via backpropagation with stochastic optimization +algorithms. +S2 +Efficient implementation of the residual calculation +In this section, we describe the implementation details of the weak PDE loss layers. We heavily utilize the convolu- +tional operation, and the vector/matrix/tensor operations to achieve numerical efficiency. Readers are directed to our +source code for additional details2. As shown in Fig. ??b, the weak PDE loss layers take both NN inputs (BCs infor- +mation) and outputs (NN predicted solution) as their inputs. The data structure to represent the BCs is discussed in +detail in Section S3.1. Schematics of the major implementation steps of the bulk residual calculation and the residual +calculation of Neumann BCs in the weak PDE loss layers are shown in Fig. S1, respectively. +We choose a steady state diffusion problem with a scalar unknown at each node for the purpose of illustration, with +Dirichlet BCs being applied on the left boundary, non-zero Neumann BCs being applied on the bottom and right +boundaries, and zero Neumann BCs on the top. To fix ideas, we consider an example such that the output of the NN +shown in Fig. S1 is a 5 × 5 matrix (an image with 5 × 5 pixels), denoted as M NN +5,5 with the value of each entry being +∗Corresponding author. E-mail address: krishna@umich.edu +January 31, 2023 +2github.com/mechanoChem/mechanoChemML +arXiv:2301.13165v1 [math.NA] 30 Dec 2022 + +A PREPRINT - JANUARY 31, 2023 +(a) nodal solution +(b) selected nodes by each kernel +(c) element like nodal solution +(e) element like nodal residual +(f) assemble residual (unfold) +(g) Neumann BCs representation +(h) selected nodes by each kernel +(i) reduced total residual +(d) vectorized representation +M5,5 +M5,5,4 +kB,1 +kB,2 +kB,3 +kB,4 +M25,4 +R25,4 +bulk +R5,5,4 +bulk +(II) +kII,1 +kII,2 +residual from Neumann BCs +(I) +kI,1 +kI,2 +R5,5,1 +bulk +I5,5 +D +I5,5 +Neu,II +I5,5 +Neu,I +Figure S1: Illustration of the implementation steps of computing the residual in the weak PDE loss layers. Paddings +of zeros are not shown in (c,d,e). +the actual concentration, M NN +5,5 is equivalent to the nodal solution on a domain, which is discretized by a 4 × 4 block +of elements, as shown in Fig. S1(a).3 The implementation procedure is summarized in the Algorithm Box 1. +S2.1 +Dirichlet BCs +The channel of NN inputs with Dirichlet BCs information is denoted as I5,5 +D , as shown in Fig. S1(a). To enforce the +Dirichlet BCs, we replace the nodal values of M NN +5,5 at the location of Dirichlet boundary with the actual values of I5,5 +D +3In the source code, M NN +5,5 is stored as M NN +5,5,1 with a third dimension of 1, which indicates the DOF per node. For elasticity +problems, the third dimension has a size of 2. Here, we drop the “1” to simplify the notations. +2 + +A PREPRINT - JANUARY 31, 2023 +to obtain a new matrix, denoted as M 5,5, as indicated by the green color in Fig. S1(a). The Dirichlet BCs are then +automatically incorporated into the residual vector during the bulk residual calculation discussed in the next Section. +S2.2 +Bulk residual +The matrix representation of the nodal solution automatically contains the element connectivity information of the +mesh. To compute the bulk residual, we first apply convolutional operations to M 5,5 with the following kernels +kB,1 = +� +1 +0 +0 +0 +� +, +kB,2 = +� +0 +1 +0 +0 +� +, +kB,3 = +� +0 +0 +1 +0 +� +, +kB,4 = +� +0 +0 +0 +1 +� +. +(2) +Each convolutional operation results in a matrix with a size of 5 × 5,4 which corresponds to the selected nodes, as +highlighted with colors in Fig. S1(b). With these four convolutional operations, we now have a matrix with a size of +5 × 5 × 4 (M 5,5,4), as shown in Fig. S1(c). We then reshape the matrix to an array 25 × 4 (M 25,4), as shown in Fig. +S1(d). Each row of M 25,4 corresponds to the local nodal solution vector inside one finite element, the subdomain Ωe +in (??), which can then be used to efficiently evaluate the residual via the matrix-vector operation. But it leaves one +blank element at each of rows 5, 10,...25. +To evaluate the residual of the steady-state diffusion problem with 2×2 Gauss quadrature points, the B-operator matrix +in (??) has a size of 4 × 2 × 4 (# of Gauss quadrature points × spatial dimensions × # of nodes), denoted as B4,2,4. +Its transpose with respect to its last two slots is denoted as BT +4,4,2. The bulk residual at each Gauss quadrature point i +is evaluated as +(R25,4 +bulk )i = ωiDM 25,4Bi,4,2Bi,2,4 +(3) +with ωi denoting the weights. The total bulk residual is computed as +R25,4 +bulk = +nquad +� +i=1 +Ri +bulk, +(4) +as shown in Fig. S1(d). R25,4 +bulk is then reshaped to R5,5,4 +bulk , and stored in the element-like form, as shown in Fig. S1(e). +Next, we use the tf.roll function to shift the element-like residual to the correct nodal position, as shown Fig. S1(f), +with +R5,5,0:1 +bulk += R5,5,0:1 +bulk +R5,5,1:2 +bulk += tf.roll(R5,5,1:2 +bulk +, [1], [2]) +R5,5,2:3 +bulk += tf.roll(R5,5,2:3 +bulk +, [1], [1]) +R5,5,3:4 +bulk += tf.roll(R5,5,3:4 +bulk +, [1, 1], [1, 2]) +(5) +where the “=” sign represents an assignment operation, and the “:” sign represents the slicing operation in Python. +Readers are directed to TensorFlow documentation for the usage of tf.roll. The assemble operation in (??) for the +bulk integration is now achieved by the tf.reduce_sum(R5,5,4 +bulk ) function without looping over all the elements to +get R5,5,1 +bulk +as done traditionally in the FEM. Readers are directed to our source code for the implementation of the +linear/nonlinear elasticity problems. +S2.3 +Neumann BCs +One channel of the inputs that contains purely Neumann BCs, denoted as I5,5 +Neu, is shown in Fig. S1(g), where the +matrix contains only non-zero entries at the non-zero Neumann boundary locations. The Neumann residual needs to +be evaluated within surface elements. Similar to computing the bulk residual, we apply convolutional operations to +I5,5 +Neu to construct surface elements. Two sets of kernels are used to construct two groups of surface elements, with +group I for computing the residual contributions on edges with a surface normal in the X-direction (zero padding is +required), and group II for edges with a surface normal in the Y-direction (zero padding is required). We use the +following two kernels +kI,1 = +� +1 +0 +0 +0 +� +, +kI,2 = +� +0 +0 +1 +0 +� +. +(6) +4The resulting matrix size is 4 × 4. Zero paddings are used to ensure the resulted matrix with a dimension of 5 × 5. Keeping the +matrix size unchanged during the convolutional operations is not necessary and might require a small amount of extra floating-point +operations, but it is less prone to errors if we handle matrices with a fixed size. +3 + +A PREPRINT - JANUARY 31, 2023 +Algorithm 1 Residual calculation for the steady-state diffusion example. +Bulk residual with applied Dirichlet BCs: Rtot +1: Apply Dirichlet BCs to NN predicted solutions M NN +5,5 by replacing the nodal values at the corresponding locations +to obtain M 5,5 (Fig. S1a). +2: Convert M 5,5 from nodal value representation to a four-node element representation M 5,5,4 by convolutional +operations with kernels kB,1, kB,2, kB,3, and kB,4 (Fig. S1b, S1c). For two-dimensional elasticity problems, NN +predicted solutions have two channels to represent both the components of the displacement vector u = ϕ. The +same four kernels are applied to both channels, resulting in the element representation M with a third dimension +of 8 instead of 4. +3: Get the vectorized representation M 25,4 with each row being the local nodal solutions for one element (Fig. S1d). +4: Compute bulk residual R25,4 for each element (Fig. S1d). Readers are directed to our source code for details of +the bulk residual calculation of linear/nonlinear elasticity. +5: Switch back to matrix representation of element-like nodal residual R5,5,4 +bulk (Fig. S1e). +6: Assemble bulk residual R5,5,1 +bulk (Fig. S1f). +Residual contributions from Neumann BCs: RNeu +1: Use kernels kI,1, kI,2 and kII,1, kII,2 to construct two groups of two-node surface elements I5,5,2 +Neu,I and I5,5,2 +Neu,II +corresponding to the two edges with Neumann BCs. For two-dimensional elasticity problems, we have four +groups of surface elements with two each for the traction vector T = k. +2: Get the vectorized representation of surface elements I25,2 +Neu,I and I25,2 +Neu,II. +3: Compute residuals R25,2 +Neu,I and R25,2 +Neu,II from Neumann BCs. +4: Switch back to matrix representation of element-like nodal residual R5,5,2 +Neu,I and R5,5,2 +Neu,II. +5: Assemble residual at Neumann BCs R5,5,1 +Neu . +Reduced total residual: Rred +tot +1: Create a mask matrix M 5,5 +bulk based on I5,5 +D +to represent the pixel locations with valid bulk residual values. The +entries of M 5,5 +bulk are zero for the components of I5,5 +D +with a value of −1, which indicates the margins between the +true problem domain and the background grid (for more details see Section S3.1). For the steady-state diffusion +examples, all entries of M 5,5 +bulk are one, since the problem domain matches the background grid. +2: Create a reverse mask matrix M 5,5 +D,rev based on I5,5 +D +to represent the pixel locations that are not at the Dirichlet +boundary. The entries of M 5,5 +D,rev are zero corresponding to the elements of I5,5 +D +with a value larger than zero. +3: Compute total residual Rtot based on (13). +4: Multiply (element-wise) Rtot with M 5,5 +D,rev and M 5,5 +bulk to get Rred +tot . +to construct surface elements I5,5,2 +Neu,I for the first group, with the selected nodal information being shown in Fig. S1(h-I), +and the following kernels +kII,1 = +� +1 +0 +0 +0 +� +, +kII,2 = +� +0 +1 +0 +0 +� +, +(7) +to construct surface elements I5,5,2 +Neu,II for the second group, with the selected nodal information being shown in Fig. +S1(h-II). +Similar to the bulk residual calculation, we form two matrices, I25,2 +Neu,I and I25,2 +Neu,II, to compute the Neumann residual. +We use two Gauss quadrature points for surface integration. The shape function N in (??) has a size of 2 × 2 (# of +Gauss quadrature points × # of nodes), and is denoted by N 2,2. We evaluate the Neumann residual at each Gauss +quadrature point i via +(R25,2 +Neu,I)i = ωiI25,2 +Neu,IN i,2N i,2 +and +(R25,2 +Neu,II)i = ωiI25,2 +Neu,IIN i,2N i,2 +(8) +with ωi denoting the weights. The total Neumann residual is computed as +R25,2 +Neu,I = +nsq +� +i=1 +Ri +Neu,I +and +R25,2 +Neu,II = +nsq +� +i=1 +Ri +Neu,II. +(9) +4 + +A PREPRINT - JANUARY 31, 2023 +where nsq is the number of surface quadrature points. Again, we use the tf.roll function to unfold the element-like +residual to the correct nodal position, similar to those shown Fig. S1(f), for group I +R5,5,0:1 +Neu,I += R5,5,0:1 +Neu,I +R5,5,1:2 +Neu,I += tf.roll(R5,5,1:2 +Neu,I , [1], [1]) +(10) +and for group II +R5,5,0:1 +Neu,II += R5,5,0:1 +Neu,II +R5,5,1:2 +Neu,II += tf.roll(R5,5,1:2 +Neu,II , [1], [2]). +(11) +The assemble operation in (??) for the surface integration is now achieved by the tf.reduce_sum(R5,5,2 +Neu,I) and +tf.reduce_sum(R5,5,2 +Neu,II) without looping over elements. We obtain the final residual contributions from the Neu- +mann BCs: +R5,5,1 +Neu += RNeu,I + RNeu,II. +(12) +The total residual Rtot in (??), as shown in Fig. ??, is computed as +R5,5,1 +tot += R5,5,1 +bulk − R5,5,1 +Neu +(13) +by applying the Neumann residual to the bulk residual. To construct the deterministic loss in (??) and the likelihood +function in (??), the reduced residual Rred +tot obtained by excluding the residual at the Dirichlet boundary location from +Rtot is used, as shown Fig. S1(i). It is worth mentioning that additional auxiliary matrix/vector/tensor operations have +been introduced, which are not included in the description, to complete this efficient residual evaluation. Readers are +invited to refer to our code for the detailed implementation. +S3 +Data representation and numerical aspects +To demonstrate the performance of our methods, we investigate different definitions of BVPs for the three considered +physical systems. We prepared both small datasets, which contain one or multiple BVPs, and large datasets, which +could contain hundreds of thousands BVPs. The NN inputs corresponding to BVPs in these datasets are synthetically +generated to train the discretized residual-constrained NNs. To compare the solution accuracy between NNs and +DNSs, we also solve these BVPs with mechanoChemFEM,5 which is a publicly available multiphysics code developed +by us based on the deal.II library [6]. In general, for small datasets, the number of unique sets of BCs is much smaller +than the number of parameters of the NNs that represent the PDE solution. We therefore augment the unique sets +of BVPs by duplicating them multiple times to form an augmented dataset during training. In the remaining part of +this section, we present details on the data structure of NN inputs, domain/boundary detection, and the NN training +procedure. +S3.1 +Data structure of NN inputs +Since the discretized residual constrained NNs do not require labels for training, the NN inputs are synthetically +generated with only information on problem domains and the applied BCs. We consider a fixed square background +grid of [0, 1] × [0, 1], with nx and ny total pixels in each dimension. For both diffusion and elasticity problems, +each input data point is a three-dimensional matrix Inx,ny,3×DOF to represent a set of BCs. The first two indices of +I indicate the pixels locations in X- and Y- directions. For steady-state diffusion problem with one scalar DOF per +node, there are three channels in the third dimension, which contain information of Dirichlet BCs, Neumann BCs +on the edge with a surface normal in the X-direction, and Neumann BCs on the edge with a surface normal in the Y- +direction, respectively. For elasticity problems, there are six channels in the third dimension with the first two channels +containing Dirichlet BCs in X- and Y- directions, the third and fourth channels containing Neumann BCs in X- and Y- +directions on the edge with a surface normal in the X-direction, and the fifth and sixth channels containing Neumann +BCs in X- and Y- directions on the edge with a surface normal in the Y-direction, respectively. Data normalization +between [−1, 1] is used to ensure that all the physically meaningful data in our study has a value greater than 0. +The structure of the input data is illustrated in Fig. S2 with the diffusion problem as an example. In our study, the +problem domain does not necessarily occupy the whole background grid, which results in the margin region as shown +in Figs S2 and S9. In small datasets, for the channel(s) containing Dirichlet BCs, the problem domain is filled with −2 +5Code available at github.com/mechanoChem/mechanoChemFEM. +6For elasticity problems, the inputs contain four channels, with the first two representing Dirichlet BCs for ¯ux and ¯uy and the +last two presenting Neumann BCs for ¯Tx and ¯Ty. +5 + +A PREPRINT - JANUARY 31, 2023 +(c) Neumann BCs (II) + with jII = jn*n2 +(a) Dirichlet BCs +(b) Neumann BCs (I) + with jI = jn*n1 +(= 0) +(= 0) +(= 0) +(= -1) +(= -1) +(= -1) +(> 0) +(> 0) +(> 0) +jn = j⋅n +jI +jII +(= -2) +(= 0) +(= 0) +(= -1) +(= -1) +(= -1) +(> 0) +(> 0) +(> 0) +jn = j⋅n +jI +jII +(= -2) +(= 0) +(= 0) +(= -1) +(= -1) +(= -1) +(> 0) +(> 0) +(> 0) +jn = j⋅n +jI +jII +Small +Large +Figure S2: Illustration of the data structure of NN inputs for a steady-state diffusion problem. The NN inputs contain +three channels. (a) the Dirichlet BCs (red), (b,c) the Neumann BCs (blue) on edges with a surface normal in the X- +and Y-direction, respectively.6 Only the boundary locations have values that are greater than 0, which is physically +meaningful. Top row: input structure for small datasets, which contain one or multiple BVPs. Bottom row: input +structure for large datasets, which could contain hundreds of thousands BVPs. In the first channel, the problem +domain (gray color) is filled with a value of −2 (top) or 0 (bottom). If the value is −2, the region is filled with random +numbers during the training process. The margin (white region) is filled with a value of −1. The residual contribution +from this region is excluded when computing Rred +tot . In the second and third channel, the problem domain is filled with +a value of 0. Similarly, the margin is filled with a value of −1. We use j = [jI, jII] and n = [n1, n2] to denote the +surface flux and surface normal, with jI and jII being the projected values in X- and Y-direction, respectively. +except the Dirichlet boundary values, which is greater than 0. The auxiliary number −2 exists in augmented datasets +and serves as an indicator to be filled with random numbers during the training process. For the margin region, which +represents the space between the background grid and the problem domain, if there is any, is filled with −1. The +auxiliary number7 −1 serves as an indicator to evaluate Rred +tot with the residual in this region being excluded. In large +datasets, for the channel(s) containing Dirichlet BCs, the problem domain is filled with 0 except the Dirichlet boundary +values, which is greater than 0. For the channel(s) containing Neumann BCs, the problem domain is filled with a value +of 0 except the Neumann boundary values. When convolutional kernels operate on the problem domain, only the +boundary makes a non-zero contribution. Similarly, the margin is filled with a value of −1 for assisting the calculation +of Rred +tot . Examples of the actual inputs for steady state diffusion are shown in Fig. S8(a,b). +S3.2 +Domain/Boundary detection +As discussed in Section S3.1, a fixed value of −1 is assigned to the margins. When calculating the residual, a mask +matrix is created for domain detection. This mask matrix is created based on the information on Dirichlet BCs from the +inputs and ensures that only the residual inside the actual problem domain is evaluated. Readers are directed to Refs +[7, 8, 9] and many others for approaches to map complex and irregular domains onto a regular mesh. Such geometric +transformations can be easily taken into account in the proposed PDE loss layers with the parent to physical domain +mapping commonly used in the FEM via basis functions. The proposed approach, using a mask matrix for domain +detection, should still be applicable to other parametric domain representations, though it is not the focus of this work. +In our study, each input data point represents a unique BVP for a specific problem domain and set of applied BCs. +To detect the Dirichlet BCs in small datasets, during the NN training, the input augmented data is first passed to a +customized Keras layer, called LayerFillRandomNumber,which fills the pixel locations with values of −2 in the +Dirichlet BCs channel with uniformly generated random numbers in the range of [0, 1] to ensure that all the data +7The auxiliary numbers −1 and −2 are arbitrary choices with no physical meaning. Users can choose different values to assign +to the margin and the problem domain for the inputs. +6 + +A PREPRINT - JANUARY 31, 2023 +Figure S3: Illustration of the deterministic NNs predicted solution at different epochs for a diffusion BVP setup with +domain id 5 and BCs id 2 (flux loading from the right edge), as shown in Fig. S9(a). Top: without zero initialization. +Bottom: with zero initialization for the first 100 epochs. +Algorithm 2 Training procedure for deterministic NNs. +1: Load NN inputs with each data point being a unique set of BCs. +2: Use a large dataset D or an augmented dataset D by by duplicating the small dataset multiple times. +3: Split D into training, validation, and testing datasets. +4: Setup the encoder-decoder deterministic NN structure, with the first layer being a customized layer to fill the +locations that have values of −2 in D with uniform random numbers between [0, 1] to ensure D is i.i.d. +5: for epoch < total_epochs do +6: +Batch train the NNs +7: +if use zero initialization and epoch < total_zero_initialization_epochs then +8: +Use dummy labels with values of 0.5, which is equivalent to an actual zero before data normalization, to form +the MSE loss to train the NN. +9: +else +10: +Use Rred +tot to form the deterministic loss to train the NN. +11: +end if +12: end for +13: Make prediction. +points are independent from each other. As the problem domain is filled with random numbers, the convolutional +kernels iteratively learn the actual Dirichlet boundary values. The data structure in the Neumann BCs channel ensures +that the kernels learn to recognize the information on the Neumann BCs, as the problem domain is filled with zeros. +For large datasets, the interior domain of the Dirichlet BCs channel is filled with zero. The convolutional kernels will +learn the actual problem domain, boundary locations, and boundary values. +S3.3 +Neural network training +For deterministic NNs, a fixed learning rate is used to batch optimize the loss function (??) and solve the PDE systems. +In our study, we found that pure Dirichlet problems are learned (converge) faster than Dirichlet-Neumann problems. +In the latter case, the solution the NNs fail to learn the solution in some instances. This observation holds for all three +PDEs considered here. This is mainly because, for Dirichlet-Neumann problems, it is the gradient of the unknown +field(s) that drives the loss instead of the field(s) itself (themselves) as in the case of the Dirichlet problem. We +demonstrate this by showing the NN predicted solution at different epochs in Fig. S3 for a diffusion BVP with domain +id 5 and BCs id 2 (see Fig. S9a) with zero concentration on the left edge and non-zero flux on the right edge. The +top row of Fig. S3 shows that even though the NN predicted concentration changes, it does so very slowly via a front +progressing from the left edge (zero Dirichlet BCs) to the right edge (flux BCs). This indicates that the solution has +not yet been learned with an accuracy that is comparable to the DNS results. +This difficulty arises mainly because the parameters of NNs are randomly initialized. As a result, the NN predicted +solutions at early training stages are random numbers close to zero. Since data normalization is used, the NN solution +of zero corresponds to an actual value of −1. Such random outputs in early stages can violate the governing equations, +potentially in catastrophic manner e.g. resulting in a deformation gradient with negative determinant in nonlinear +elasticity. In order to rectify this inconsistency, we draw from the conventional initialization of the solution vector to +zero in the FEM, and adopt the same approach for the NNs. For the first few epochs, we train NNs with dummy labels +7 + +DNS epochs +10 epochs +100 epochs +500 epochs +1000 epochs +1500 epochs +1900 epochs +0.5 +0.5 +0.5 +F0.5 +F0.5 +F0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0DNS epochs +50 epochs +100 epochs +200 epochs +300 epochs +400 epochs +500 epochs +0.5 +0.5 +0.5 +0.5 +0.5 +F0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0A PREPRINT - JANUARY 31, 2023 +Algorithm 3 Training procedure for BNNs. +1: Load NN inputs with each data point being a unique set of BCs. +2: Use a large dataset D or an augmented dataset D by by duplicating the small dataset multiple times. +3: Split D into training, validation, and testing datasets. +4: Setup the encoder-decoder probabilistic NN structure, with the first layer being a customized layer to fill the +locations that have values of −2 in D with uniform random numbers between [0, 1]. +5: if use optimal parameter initialization then +6: +Load the optimized parameters from the deterministic NNs to initialize the mean of the posterior distribution of +BNN parameters. +7: else +8: +Use random initialization for the posterior distribution of BNN parameters. +9: end if +10: for epoch < total_epochs do +11: +Batch train the NNs +12: +if use zero initialization and epoch < total_zero_initialization_epochs and (not use optimal parameter initial- +ization) then +13: +Use dummy labels with values of 0.5, which is equivalent to an actual zero before data normalization, to form +the MSE loss to train the NN. +14: +else +15: +Use Rred +tot to form the likelihood loss to train the NN. +16: +end if +17: end for +18: MC sampling for UQ. +with values of 0.5 (equivalent to an actual value of 0) without enforcing the PDE constraint. We call this as the zero +initialization process. This process helps to improve the initialization of NN parameters. After the zero initialization +procedure is completed, the PDE constraints are enabled to train the NNs to solve the PDE systems. We found this +remedy to drastically improve the learned solution as well as speed up the training, as shown in the bottom row of Fig. +S3, where the NN predicted solutions approach the DNS results at 500 epochs, much faster than the case without the +zero initialization process. The training process for deterministic NNs is summarized in the Algorithm Box 2. +For probabilistic NNs, we can use the proposed approach successfully solve a single BVP. However, when we try +to solve multiple BVPs, we notice that the BNNs converge faster to purely Dirichlet problems (boundary id 1, 3 for +the diffusion problem and boundary id 1, 4 for elasticity problems) than those with non-zero Neumann BCs. Once +the BNN parameters stagnate at sub-optimal solutions, it is very difficult to optimize them further for other BVPs +with Neumann BCs. To overcome this challenge, we first train deterministic NNs with identical architectures as the +BNNs. Once the deterministic NNs have converged to a desired tolerance, we then initialize the mean of the posterior +distribution of parameters in the BNNs with the optimized parameters from the deterministic model. We refer to this +as the optimal parameter initialization process. During the subsequent training of the BNNs we use a small learning +rate to explore the local parameter space around these optimized parameters. A similar approach has been adopted in +Ref [10], The training process for BNNs is summarized in the Algorithm Box 3. +Here an epoch corresponds to a single application of an augmented dataset or a large dataset D as inputs to predict a NN +solution by training parameters. For augmented dataset, when splitting D into training, validation, and testing groups, +each group could potentially contain all the unique BCs. The evaluation of the NN results based on such validation +and testing dataset only indicates how well the NNs solve the exposed sets of BCs. To test the predictability of the +trained NNs in Sections S4.2.2 and S4.3.2, the unseen testing sets of BCs were set aside before data augmentation. +S3.4 +Data generation for large dataset +We use the scikit-geometry package to generate the Polygons that were used for the large dataset study, where the +vertices of polygons are randomly sampled approximately along a circle with applied perturbation. The detailed code +implementation is available on GitHub. To choose the edges to impose the BCs, we first generate all the possible com- +binations of two non-adjacent edges. We then apply the numpy.random.shuffle() to the combinations and select +the first few of them from the combinations. For the boundary values, we use the numpy.random.uniform()function +to sample control points from the range [0, 1]. If the boundary value has a constant distribution, a single value is sam- +pled. If the boundary value has a linear distribution, two extremes are sampled. And interpolated values are imposed +8 + +A PREPRINT - JANUARY 31, 2023 +to each pixel on the edge. One can refer to our code on Github for details to impose BCs with a quadratic or sinusoidal +distribution. +S4 +Numerical results +In this section, we provide additional information for the examples presented in the main text along with extra examples +to support the performance of the proposed method. +S4.1 +Steady state diffusion - small dataset +S4.1.1 +Single octagon domain with Dirichlet BCs - cold start versus warm start +(a) +(b) +Figure S4: Illustration of the setup of BVPs with (a) purely Dirichlet BCs and (b) mixed BCs +(a) cold start +(b) warm start +Figure S5: Comparison of BNN results between (a) cold start and (b) warm start with purely Dirichlet BCs. The +results are very similar, except the uncertainty level in (a) is higher than (b). +In this example, we use the proposed method to solve the steady-state diffusion problem on an octagon domain with +purely Dirichlet BCs, as shown in Fig. S4(a). The BNN is trained in two cases with (i) cold start and (ii) warm start. +The results are shown in Fig. S5. One can see that the BNNs with cold start can also find the correct solution for +this specific BVP, comparable with the results from the warm start. This confirms that the loss of the BNN is correct. +However, in our study, we found that it is very challenging, if not impossible, for BNNs to directly (cold start) solve +multiple BVPs. All the other BNNs results in this work are solved based on the warm start approach. +S4.1.2 +Single octagon domain with mixed BCs +In Fig. ??(a-g), we use the proposed PDE constrained NNs to solve steady-state diffusion on an octagonal domain +with mixed BCs, resulting in a solution with spatially varying gradients along both X- and Y- directions. NN structure +information for the octagonal domain simulation are summarized in Table 1 and 2. The NN hyperparameters are +manually tuned to achieve a desired performance. +The training losses for both deterministic and probabilistic NNs are given in Fig. S7(a, b). Depends on the choice of +the initial value of Σ2, Fig. S7(b) could have a negative values. The negative value is reasonable because the total +9 + +Mean ± 2 Std +1.0 +DNS +Mean +value +0.9 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.8 +DNS +Mean +value +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +1.0 +0.9 +0.003 +0.8 +-0.002 +0.7 +-0.001 +0.6 +0.5 +0.000Mean ± 2 Std +1.00 +DNS +Mean +0.95 +value +0.90 +0.85 +0.80 +0.75 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateMean ± 2 Std +0.8 +DNS +Mean +value +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +1.0 +0.008 +0.9 +0.006 +0.8 +0.004 +0.7 +0.002 +0.6 +0.5 +0.000A PREPRINT - JANUARY 31, 2023 +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 32 +ReLU +Dense +DenseFlipout +units = 32 +ReLU +Reshape +Reshape +- +[4, 4, 4] +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 1 +kernel (5,5), padding: same, ReLU +Table 1: Details of both deterministic and probabilistic NNs for solving diffusion BVPs on the octagon domain with +an output resolution of 32 × 32. +Description +Deterministic +Probabilistic +Total parameters +16,049 +31,970 +Size of D +1 × Aug: 212 +1 × Aug: 211 +Epochs +20,000 +100 +Zero initialization epochs +100 +- +Optimizer +Nadam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 2: Training related parameters for solving steady-state diffusion on the octagonal domain. Aug: data augmenta- +tion. +loss of BNNs in (??) consists two terms. The first term in (??) is non-negative, whereas the second term could be +either positive or negative depending on values of both Rred +tot and Σ2. The evolution of Σ2 from the BNN is shown +in Fig. S7(c), which converges to a sharp value during training. The evolution of Σ2 is correlated to the sign change +of the BNN loss. The logarithm of the probability density function converges to the maximum, or the negative of it +converges the minimum, as shown in Fig. S7(b), when N (0, Σ2) best represents Rred +tot . Such behavior is expected as +the BNN is initialized with optimal parameters from the deterministic NNs and is trained with a very small learning +rate to only explore the local parameter space around the optimized parameters. +S4.1.3 +Single octagon domain with mixed BCs and different material parameters +In this example, we use the proposed method to solve steady-state diffusion problem on an octagon domain with mixed +BCs and varying diffusivity. The NN architecture is shown in Fig. S6 with heterogeneous inputs, which consists of +images and scalars. The image-type input contains the information of problem domains and BCs. The scalar(s) +represent the material parameters. +S4.1.4 +Multiple rectangular domains with different BCs +We also use the proposed PDE constrained NNs to simultaneously solve 20 steady-state diffusion BVPs with a resolu- +tion of 16×16. The architectures of both deterministic and probabilistic NNs and other training related NN parameters +are summarized in Table 3 and 4, respectively. The NN hyperparameters are manually tuned to achieve a desired per- +10 + +A PREPRINT - JANUARY 31, 2023 +NN +solution +zero/non-zero Dirichlet BCs +non-zero Neumann BCs +(I) +(II) +NN Sol.+D.BC +N.BC +a +b +Neural Networks +Encoder +Decoder +(fill random +numbers) +Rbulk +Rneu +Weak PDE loss layers +Rtot +Rtot +red +Figure S6: Illustration of the NN architectures with heterogeneous data inputs to account for different material pa- +rameters. The image-type input contains the information of problem domains and BCs. The scalar(s) represent the +material parameters. +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 64 +ReLU +Dense +DenseFlipout +units = 64 +ReLU +Reshape +Reshape +- +[4, 4, 4] +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 1 +kernel (5,5), padding: same, ReLU +Table 3: Details of both deterministic and probabilistic NNs for solving 20 steady-state diffusion BVPs. Readers are +directed to TensorFlow documentation for detailed description of the functionality of each layer. +formance. We follow the training procedures in Algorithm Boxes 2 and 3 to first train the deterministic NN with +zero initialization, followed by training the BNNs with the optimal parameter initialization process. The results are +shown in Fig. S8, which confirm the accuracy of the proposed method an d demonstrate that the proposed method can +simultaneously solve multiple BVPs. The statistical moments of the BNN predictions are evaluated based on 50 MC +samplings. +11 + +A PREPRINT - JANUARY 31, 2023 +Description +Deterministic +Probabilistic +Total parameters +33,209 +66,202 +Size of D +20 × Aug: 210 +20 × Aug: 29 +Epochs +20,000 +5,000 +Zero initialization epochs +100 +- +Optimizer +Nadam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 4: Training related parameters for solving 20 steady-state diffusion BVPs. Aug: data augmentation. +(a) deterministic loss +(b) probabilistic loss +Figure S7: NN training information for octagon problem with mixed BCs. (a) Loss from the deterministic NN. (b) +Loss from the BNN. (c) Evolution of Σ2. +S4.2 +Linear elasticity - small dataset +S4.2.1 +Multiple rectangular domains with different BCs +In this section, we use the proposed PDE constrained NNs to simultaneously solve 30 linear elasticity BVPs, as shown +in Fig. S9(a), with a resolution of 16 × 16. The deformed problem domains from DNSs for three representative BVPs +are shown in Fig. S10(a). Similar architectures of both deterministic and probabilistic NNs as summarized in Table 3 +are used, except that. the last layer has two filters, representing ux and uy, instead of one for the steady-state diffusion +problem. The other training related NN parameters are summarized in Table 5. We follow the procedures described in +Section S3.3 to train both types of NNs. The NN results of three selected BVPs, as shown in Fig. S10(a), are presented +in Fig. S11The statistical moments of the BNN predictions are evaluated based on 50 MC samplings. In Fig. S11, +BVP (i), (ii), and (iii) correspond to bc id 1 (non-zero Dirichlet loading), bc id 2 (non-zero Neumann loading), bc id +3 (mixed loading) applied to domain id 5, respectively. The comparison of solutions between DNSs, the deterministic +NN, and the BNN for these threeBVPs is shown qualitatively in Fig. S11(a,c,e,g,i,k), with quantitative comparison +of the solution distribution along the dashed lines between DNSs and the BNN given in Fig. S11(b,d,f,h,j,l). Such +Description +Deterministic +Probabilistic +Total parameters +34,010 +67,803 +Size of D +30 × Aug: 29 +30 × Aug: 29 +Epochs +20,000 +100 +Zero initialization epochs +100 +- +Optimizer +Nadam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +128 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 5: Training related parameters for solving 30 linear elasticity BVPs. Aug: data augmentation. +12 + +var(sigma2) +4×10-6 +3×10-6 +2 ×10-6 +10~6 +0 +2500 +5000 +7500 +10000 +12500150001750020000 +epoch101 +loss +val_loss +100 +10~1 +10~2 +10-3 +10-4, +10-5 +10-6 +0 +2500 +5000 +7500 +10000 +12500150001750020000 +epochA PREPRINT - JANUARY 31, 2023 +Figure S8: NN results for steady-state diffusion BVPs on rectangle domains. Comparison between the FEM solution +(DNS), the deterministic NN solution, the mean and std of BNN solutions over 50 MC samplings, and solution +distributions along the dashed lines. +a comparison shows that the proposed method has successfully solved most of the BVPs with desired accuracy. The +results from NNs in Fig. S11(i,k,l) are slightly worse than the DNSs. This happens mainly because the deformation +for linear elasticity is small. The scaled results have a narrow range of [0.5, 0.55], which is challenging for NNs to +learn to distinguish, particularly for purely non-zero traction loads. For the mathematically more complex nonlinear +elasticity BVPs, in which the deformation is large, are solved by the NNs more successfully, as shown in Fig. S13. +S4.2.2 +L-shape domain with solution interpolation +NN architectures for the L-shaped domain simulation are summarized in Table 6 and 7. We follow the procedures +described in Section S3.3 to train both types of NNs with an output resolution of 32 × 32. +S4.3 +Nonlinear elasticity - small dataset +Even with the zero-initialization process, the NN outputs at early stages of training could violate the physics, e.g. with +a negative or zero determinant of the deformation gradient J. To ensure that the residual can be evaluated and to +prevent residuals from these “bad” pixels values from contributing to the final loss, we regularize the loss by omitting +the residual contribution with J < 0.1 and J > 5.0. As the training continues towards a later stage, the NN predicted +solutions gradually fulfill the governing PDEs, and the regularization on J will cease to function. +13 + +DNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.00150 +0.8 +0.8 +0.8 +0.00125 +0.00100 +0.7 +0.7 +-0.7 +- +- +0.00075 +0.00050 +0.6 +0.6 +0.6 +0.00025 +0.5 +0.5 +0.5 +0.00000Mean ± 2 Std +DNS +0.8 +Mean +lue +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.00150 +0.650 +0.650 +0.650 +0.00125 +0.625 +0.625 +0.625 +0.00100 +0.600 +0.600 +0.600 +- +0.575 +0.575 +0.575 +F0.00075 +0.550 ++0.550 +0.550 +0.00050 +0.525 +-0.525 +0.525 +0.00025 +0.500 +0.500 +0.500 +0.00000Mean ± 2 Std +DNS +0.65 +Mean +lue +0.60 +val +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.0010 +-0.8 +0.8 +-0.8 +0.0008 +0.7 +0.7 +0.7 ++0.0006 +0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.650 +0.650 +0.650 +0.00150 +0.625 +0.625 +0.625 +0.00125 +0.600 +0.600 +0.600 +0.00100 +0.575 +0.575 +0.575 +0.00075 +0.550 ++0.550 +0.550 +F0.00050 +0.525 +0.525 +0.525 +0.00025 +. +0.500 +0.500 +0.500 +0.00000Mean ± 2 Std +DNS +0.65 +Mean +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.8 +-0.00125 +0.8 +0.8 +0.00100 + 0.7 +0.7 +0.7 +0.00075 +0.00050 +0.6 +0.6 +0.6 +0.00025 +0.5 +0.5 +0.5 +0.00000Mean ± 2 Std +DNS +0.8 +Mean +lue +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.0020 +0.65 +0.65 +0.65 +0.0015 +0.60 +0.60 +0.60 +0.0010 +0.55 +0.55 +0.55 +0.0005 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.70 +DNS +Mean +0.65 +lue +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.8 +0.8 +-0.8 +0.0015 +0.7 +0.7 +0.7 +0.0010 +0.6 +0.6 +0.6 +0.0005 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +lue +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.0015 +0.65 +0.65 +0.65 +0.60 +0.60 +0.60 +0.0010 +0.55 +0.55 +-0.0005 +0.55 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.70 +DNS +Mean +0.65 +lue +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.0010 +0.8 +0.8 +0.8 +0.0008 +0.7 +0.7 +0.7 +0.0006 +0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.0012 +0.625 +0.625 +0.625 +0.0010 +0.600 +0.600 +0.600 +0.0008 +0.575 +0.575 +- +0.575 +0.0006 +0.550 +0.550 +0.550 +0.0004 +0.525 +0.525 +0.525 +0.0002 +0.500 +0.500 +0.500 +0.0000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +value +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.8 +0.8 +0.8 +0.0008 +0.0006 +0.7 +0.7 +0.7 +-0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.650 +0.650 +0.650 +0.0015 +0.625 +0.625 +0.625 +0.600 +0.600 +0.600 +0.0010 +0.575 +0.575 +0.575 +0.550 ++0.550 +0.550 +F0.0005 +0.525 +-0.525 +0.525 +. +0.500 +0.500 +0.500 +0.0000Mean ± 2 Std +DNS +0.65 +Mean +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.8 +0.8 +0.8 +0.0008 +0.0006 +0.7 +0.7 +0.7 +0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.625 +0.625 +0.625 +0.00125 +0.600 +0.600 +0.600 +0.00100 +0.575 +0.575 +0.575 +0.00075 +0.550 +0.550 +0.550 +0.00050 +0.525 +0.525 +0.525 +0.00025 +0.500 +0.500 +0.500 +0.00000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +value +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.0010 +0.8 +0.8 +0.8 +0.0008 +0.7 +0.7 +0.7 +0.0006 +0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.00150 +0.625 +0.625 +0.625 +-0.00125 +0.600 +-0.600 +-0.600 +0.00100 +0.575 +0.575 +0.575 +0.00075 +-0.550 +0.550 +0.550 +0.00050 +-0.525 +0.525 +0.525 +0.00025 +0.500 +0.500 +0.500 +0.00000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +value +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +-0.8 +0.8 +-0.8 +0.00125 +0.00100 +0.7 +0.7 +0.7 +0.00075 +0.00050 +0.6 +0.6 +0.6 +0.00025 +0.5 +0.5 +0.5 +0.00000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.650 +0.650 +0.650 +0.0015 +0.625 +0.625 +0.625 +0.600 +0.600 +0.600 +0.0010 +0.575 +0.575 +0.575 +0.550 ++0.550 +0.550 +F0.0005 +0.525 +-0.525 +0.525 +0.500 +0.500 +0.500 +0.0000Mean ± 2 Std +DNS +0.65 +Mean +0.60 +val +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.0010 +0.8 +0.8 +0.8 +0.0008 +0.7 +0.7 +0.7 +0.0006 +0.0004 +0.6 +0.6 +0.6 +0.0002 +0.5 +0.5 +0.5 +0.0000Mean ± 2 Std +DNS +0.8 +Mean +0.7 +val +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS +Sol. (det) +Mean (BNN) +Std. (BNN) +0.00150 +0.625 +0.625 +0.625 +0.00125 +0.600 +0.600 +-0.600 +0.00100 +0.575 +0.575 +0.575 ++0.00075 +0.550 +0.550 +0.550 +0.00050 +0.525 +0.525 +0.525 +0.00025 +0.500 +0.500 +0.500 +0.00000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +value +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateA PREPRINT - JANUARY 31, 2023 +y +(a) five simulation domains +(b) six sets of BCs for linear/nonlinear elasticity +x +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +6 +Figure S9: Illustration of the setup of 30 BVPs on different domains for linear/nonlinear elasticity problems. In these +drawings, red represents a zero Dirichlet BC. Green represents a non-zero Dirichlet BC. Blue represents a non-zero +Neumann BC. No color is assigned to Zero Neumann BCs. (a) Setup of five rectangle simulation domains of different +sizes and locations on a fixed background grid with different applied BCs. (b) For linear/nonlinear elasticity, 6 sets of +BCs are assigned to each simulation domain, leading to 30 linear/nonlinear elasticity BVPs. +BVP (i) +BVP(ii) +BVP (iii) +(a) three selected BVPs on rectangle domains +Figure S10: Illustration of the deformed shape of selected linear elasticity BVPs. The wireframe and the gray region +indicate the undeformed and deformed problem domain, respectively. Three selected BVPs of out the 30 BVPs solved +in section S4.2.1. +S4.3.1 +Multiple rectangular domains with different BCs +In this section, we use the proposed PDE constrained NNs to simultaneously solve 30 nonlinear elasticity BVPs, as +show in Fig. S9(a), with a resolution of 16 × 16. The deformed problem domains from DNSs for three representative +setups are shown in Fig. S12. The architectures of both deterministic and probabilistic NNs and the training related +NN parameters used in this section are identical to those used in section S4.2.1 for solving linear elasticity BVPs. We +follow the procedures described in Section S3.3 to train both types of NNs. The NN results of three selected BVPs, +as shown in Fig. S12, are presented in Fig. S13.The statistical moments of the BNN predictions are evaluated based +on 50 MC samplings. In Fig. S13, BVP (i), (ii), and (iii) correspond to bc id 1 (non-zero Dirichlet loading), bc id 2 +(non-zero Neumann loading), bc id 3 (mixed loading) applied to domain id 5. The comparison of solutions between +DNSs, the deterministic NN, and the BNN for these three BVPs is shown qualitatively in Fig. S13(a,c,e,g,i,k), with +quantitative comparison of the solution distribution along the dashed lines between DNSs and the BNN given in Fig. +S13(b,d,f,h,j,l). We draw attention to the improved accuracy of the BNN for BVP (iii) seen in Fig. S13(i,j,k,l) as a +consequence of larger deformation (strain) in comparison to the same BVP with inifinitesimal strain in Fig. S11(i,j,k,l). +The BNN is able to better learn nonlinear than linear elasticity because of the stronger expression of physics in the +nonlinear solution. The comparison demonstrates that the proposed method has successfully solved multiple BVPs +with desired accuracy. +S4.3.2 +Rectangular domain with solution interpolation and extrapolation +In this section, we explore the interpolating and extrapolating capacity of the proposed framework for the BVP setup +shown in Fig. S12, case (i). Both the DNS and NN solution have resolutions of 16 × 16. The architectures of both +deterministic and probabilistic NNs and the training related NN parameters used in this section are identical to those +14 + +A PREPRINT - JANUARY 31, 2023 +(a) ux results for BVP (i) +(b) ux UQ for BVP (i) +(c) uy results for BVP (i) +(d) uy UQ for BVP (i) +(e) ux results for BVP (ii) +(f) ux UQ for BVP (ii) +(g) uy results for BVP (ii) +(h) uy UQ for BVP (ii) +(i) ux results for BVP (iii) +(j) ux UQ for BVP (iii) +(k) uy results for BVP (iii) +(l) uy UQ for BVP (iii) +Figure S11: Results of three selected BVPs out of the 30 linear elasticity BVPs with varying domains and different +applied BCs simultaneously solved by a single deterministic or probabilistic NN with the proposed method. BVP (i), +(ii), (iii) correspond to bc id 1, 2, and 3 for domain id 5, as shown in Fig. S9(a). (a, c, e, g, i, k) Solutions from DNS, +deterministic (det) NNs, and BNNs (Mean, Std.) for different BVPs. (b, d, f, h, j, l) Quantitative comparison of the +solution distribution between DNS and BNNs along the dashed lines. +15 + +DNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.55 +0.55 +0.55 +-0.10 +-0.54 +0.54 +-0.54 +0.05 +0.53 ++0.53 +0.53 +0.00 +0.52 +0.52 +0.52 +一 +0.05 +-0.51 +0.51 +0.51 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.55 +DNS +Mean +0.54 +value +0.53 +0.52 +0.51 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN)(Y) +Std.(BNN)(Y) +-0.10 +0.525 +0.525 +0.525 +0.520 +0.520 +0.520 +0.05 +0.515 +0.515 +-0.515 +0.00 +0.510 +-0.510 +-0.510 +0.505 +0.505 +0.505 +-0.05 +0.500 +0.500 +0.500 +0.10Mean ± 2 Std +DNS +0.520 +Mean +0.515 +value +0.510 +0.505 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.55 +0.55 +0.55 +-0.10 +-0.54 +0.54 +-0.54 +0.05 +0.53 ++0.53 +0.53 +0.00 +0.52 +0.52 +0.52 +一 +0.05 +-0.51 +0.51 +0.51 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.55 +DNS +Mean +0.54 +value +0.53 +0.52 +0.51 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN)(Y) +Std.(BNN)(Y) +-0.10 +-0.53 +0.53 +0.53 +0.05 ++0.52 ++0.52 +0.52 +0.00 +0.51 +0.51 +0.51 +-0.05 +0.50 ++0.50 ++0.50 +0.10Mean ± 2 Std +DNS +0.520 +Mean +0.515 +lue +0.510 +0.505 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN)(X) +Std. (BNN)(X) +-0.54 +-0.10 +0.54 +0.54 +0.53 +0.53 +0.53 +0.05 +0.52 +0.52 +0.52 +0.00 +-0.51 +-0.51 +0.51 +-0.05 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.520 +DNS +Mean +0.515 +lue +0.510 +0.505 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN)(Y) +Std.(BNN)(Y) +0.10 +0.54 +0.54 +-0.54 +0.05 +0.53 +0.53 +0.53 +0.00 +0.52 +0.52 +0.52 +一 +-0.05 +0.51 +0.51 +0.51 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.520 +DNS +Mean +0.515 +lue +0.510 +0.505 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateA PREPRINT - JANUARY 31, 2023 +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 32 +ReLU +Dense +DenseFlipout +units = 128 +ReLU +Reshape +Reshape +- +[4, 4, 8] +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 2 +kernel (5,5), padding: same, ReLU +Table 6: Details of both deterministic and probabilistic NNs for solving linear elasticity L-shape BVPs. +Description +Deterministic +Probabilistic +Total parameters +41,346 +82,435 +Size of D +5 × Aug: 210 +5 × Aug: 29 +Epochs +10,000 +100 +Zero initialization epochs +100 +- +Optimizer +Adam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 7: Training related parameters for solving linear elasticity on L-shaped BVPs. Aug: data augmentation. +BVP (i) +BVP(ii) +BVP (iii) +Figure S12: Illustration of the deformed shape of the three selected nonlinear elasticity BVPs. The wireframe and the +gray region indicate the undeformed and deformed problem domains, respectively. +used in section S4.2.1 for solving linear elasticity BVPs. Additional interpolated and extrapolated NN prediction +results for case (i) are given in Fig. S14 and S15, respectively. +S4.4 +Steady state diffusion - large dataset +In this section, three large training datasets (D1, D2, D3) and two small testing datasets (T1, T2) with a mesh resolution +of 64 × 64 are prepared for the steady-state diffusion problem to test the performance of the proposed method. The +script to synthetically generate the BVPs in these datasets are provided in the source code on GitHub. +16 + +A PREPRINT - JANUARY 31, 2023 +(a) ux results for BVP (i) +(b) ux UQ for BVP (i) +(c) uy results for BVP (i) +(d) uy UQ for BVP (i) +(e) ux results for BVP (ii) +(f) ux UQ for BVP (ii) +(g) uy results for BVP (ii) +(h) uy UQ for BVP (ii) +(i) ux results for BVP (iii) +(j) ux UQ for BVP (iii) +(k) uy results for BVP (iii) +(l) uy UQ for BVP (iii) +Figure S13: Results of three selected BVPs out of the 30 nonlinear elasticity BVPs with varying domains and different +applied BCs simultaneously solved by a single deterministic or probabilistic NN with the proposed method. BVP (i), +(ii), (iii) correspond to bc id 1, 2, and 3 for domain id 5, as shown in Fig. S9(a). (a, c, e, g, i, k) Solutions from DNS, +deterministic (det) NNs, and BNNs (Mean, Std.) for different BVPs. (b, d, f, h, j, l) Quantitative comparison of the +solution distribution between DNS and BNNs along the dashed lines. +17 + +DNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.75 +0.75 +0.75 +-0.10 +0.70 +0.70 +0.70 +0.05 +0.65 ++0.65 +0.65 +0.00 +0.60 +0.60 +0.60 +一 +0.05 +0.55 +0.55 +0.55 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.75 +DNS +Mean +0.70 +value +0.65 +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN)(Y) +Std.(BNN)(Y) +-0.10 +0.625 +0.625 +-0.625 +0.600 +0.600 +0.600 +0.05 +0.575 +0.575 +0.575 +0.00 +0.550 +0.550 +0.550 +0.525 +0.525 +0.525 +-0.05 +0.500 +0.500 +0.500 +-0.10Mean ± 2 Std +DNS +0.60 +Mean +0.58 +value +0.56 +0.54 +0.52 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.75 +0.75 +0.75 +-0.10 +0.70 +0.70 +0.70 +0.05 +0.65 ++0.65 +0.65 +0.00 +0.60 +0.60 +0.60 +一 +0.05 +0.55 +0.55 +0.55 +0.50 +0.50 +0.50 +0.10Mean ± 2 Std +0.75 +DNS +Mean +0.70 +value +0.65 +0.60 +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN) (Y) +Std. (BNN)(Y) +-0.10 +0.625 +0.625 +0.625 +0.600 +0.600 +0.600 +0.05 +0.575 ++0.575 +0.575 ++0.00 +0.550 +0.550 +0.550 ++0.525 +0.525 ++0.525 ++0.500 +0.500 ++0.500 +1 +0.10Mean ± 2 Std +0.600 +DNS +Mean +0.575 +value +0.550 +0.525 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN)(X) +-0.10 +0.650 +0.650 +0.650 +0.625 +0.625 +0.625 +0.05 +0.600 +0.600 +0.600 +0.575 +0.575 +0.575 +0.00 +0.550 +0.550 +0.550 +-0.05 +0.525 +0.525 +0.525 +0.500 +0.500 +0.500 +0.10Mean ± 2 Std +0.625 +DNS +Mean +0.600 +value +0.575 +0.550 +0.525 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN)(Y) +Std.(BNN)(Y) +-0.10 +I +0.625 +0.625 +0.625 +0.05 ++0.600 +0.600 +0.600 +-0.575 +-0.575 +0.575 +0.00 +0.550 +0.550 +0.550 +-0.05 +0.525 +0.525 +-0.525 +0.500 +0.500 ++0.500 +0.10Mean ± 2 Std +0.58 +DNS +Mean +0.56 +lue +0.54 +val +0.52 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateA PREPRINT - JANUARY 31, 2023 +Figure S14: Additional interpolated NN prediction results for case (i). +Figure S15: Additional extrapolated NN prediction results for case (i). +A1 +A2 +regular +extreme +boundary value distribution +constant +linear +quadratic +sinusoidal +Different geometries +Figure S16: Illustration of dataset D1/T1. D1 contains regular pentagons, whose inner angles are all smaller than 160 +degree. T1 contains both regular and irregular pentagons. The latter has one innger angle greater than 160 degree. The +boundary values could have a constant/linear/quadratic/sinusoidal distribution. +S4.4.1 +D1/T1: BVPs on regular pentagon domains +D1 contains 176K unique BVPs, which covers 1100 regular pentagons, whose inner angles are all smaller than 160 +degree, as shown in Fig. S16. The shape of the pentagons are randomly generated. We apply boundary conditions +to two non-adjacent edges, which are randomly sampled from all the possible locations. The boundary values are +randomly generated, which could have a constant/linear/quadratic/sinusoidal distribution along the edges, as shown +in Fig. S16. The testing dataset T1 contains 1000 BVPs, which covers both regular shapes and extreme shapes. The +latter has one inner angle greater than 160 degree. Same strategies to generate the BCs for D1 are used to generate +BCs for T1. The training loss is shown in Fig. S17. NN architectures and training details are summarized in Table 8 +and 9. +Selected NN predicted results are shown in Fig. ??(a,b), with the L2 error of all the training and testing results given in +Fig. ??(d). We can observe that, if the boundary locations and the extremes of boundary values in the training dataset +could be more uniformly distributed, the training and prediction are in generally good. Also, the predictions are more +accurate if extremes of the testing dataset are not near the extremes of training datasets. This suggests that if we know +a targeting range to use the NNs to make predictions, we can then design the NN training dataset to have a wider range, +so the accuracy of NN predicted solution can be further improved. In another word, make interpolated prediction. This +study also shows that a single NN could simultaneously solve hundreds of thousands BVPs with reasonable accuracy +and make prediction for unseen domains. +18 + +DNS (Y) +Sol. (det) (Y) +Mean(BNN)(Y) +Std. (BNN) (Y) +0.0020 +0.70 +0.70 +0.70 +0.65 +0.65 +0.65 +0.0015 +: +0.60 +0.60 +0.60 +0.0010 +0.55 +0.55 +0.55 +0.0005 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +lue +val +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN)(X) +Std. (BNN) (X) +0.55 +0.55 +0.55 +-0.00150 +0.54 +0.54 +0.54 +0.00125 +0.00100 +0.53 ++0.53 +0.53 +0.00075 +0.52 +0.52 +0.52 +0.00050 +0.51 +-0.51 +0.51 +0.00025 +0.50 +0.50 +0.50 +0.00000Mean ± 2 Std +0.55 +DNS +Mean +0.54 +value +0.53 +0.52 +0.51 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean(BNN)(Y) +Std. (BNN)(Y) +0.53 +0.53 +0.53 +-0.0006 +0.52 +0.52 +0.52 +-0.0004 +0.51 +-0.51 +0.51 +0.0002 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.52 +DNS +Mean +value +0.51 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN)(X) +0.56 +0.56 +0.56 +0.0025 +0.55 +0.55 +0.55 +0.0020 +0.54 +-0.54 +0.54 +0.0015 +0.53 +0.53 +0.53 +0.0010 +0.52 +0.52 +0.52 +0.51 +0.51 +0.51 +0.0005 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +DNS +0.56 +Mean +lue +0.54 +val +0.52 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det)(Y) +Mean (BNN)(Y) +Std. (BNN) (Y) +0.54 +-0.54 +-0.54 +-0.00125 +0.53 +0.53 +0.53 +0.00100 +0.52 +0.52 +0.52 +0.00075 +! +0.51 +0.51 +0.51 +0.00050 +0.00025 +0.50 +0.50 +0.50 +0.00000Mean ± 2 Std +DNS +Mean +0.52 +lue +0.51 +val +0.50 +0.49 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.62 +0.00150 +0.62 +0.62 +0.60 +0.60 +0.60 +0.00125 +0.58 +0.58 +0.58 +0.00100 +0.56 +0.56 +0.56 +F0.00075 +-0.54 +0.54 +0.54 +0.00050 +0.52 +0.52 +0.52 +0.00025 +0.50 +0.50 +0.50 +0.00000Mean ± 2 Std +0.625 +DNS +Mean +0.600 +Lue +0.575 +0.550 +0.525 +0.500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean(BNN)(Y) +Std. (BNN)(Y) +0.56 +0.56 +0.56 +0.0006 +0.54 +0.54 +0.54 +0.0004 +0.52 +0.52 +0.52 +0.0002 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +DNS +0.54 +Mean +lue +0.52 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.9 +0.9 +0.9 +0.003 +0.8 +-0.8 +0.8 +-0.002 +0.7 +0.7 ++0.7 +0.001 +0.6 +-0.6 ++0.6 +0.5 +0.5 +0.5 +0.000Mean ± 2 Std +0.9 +DNS +Mean +0.8 +lue +Val +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean (BNN) (Y +Std. (BNN) (Y) ++0.0020 +0.70 +0.70 +0.70 +0.0015 +0.65 +0.65 +0.65 +: +0.60 +0.60 +0.60 +0.0010 +0.55 +0.55 +0.55 +0.0005 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +lue +Val +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.9 +0.9 +0.9 +0.003 +0.8 +-0.8 +0.8 +-0.002 +0.7 +0.7 +0.7 +-0.001 +0.6 +0.6 +0.6 +0.5 +0.5 +0.5 +0.000Mean ± 2 Std +DNS +0.9 +Mean +0.8 +lue +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (Y) +Sol. (det) (Y) +Mean(BNN)(Y) +Std. (BNN) (Y) +0.70 +0.70 +0.70 +0.0015 +0.65 +0.65 +0.65 +: +0.0010 +0.60 +0.60 +0.60 +0.55 +0.55 +0.55 +0.0005 +0.50 +0.50 +0.50 +0.0000Mean ± 2 Std +0.65 +DNS +Mean +0.60 +lue +val +0.55 +0.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDNS (X) +Sol. (det) (X) +Mean (BNN) (X) +Std. (BNN) (X) +0.9 +0.9 ++0.9 +0.003 +0.8 +0.8 +0.8 +0.002 +0.7 +0.7 +0.7 +0.001 +0.6 +0.6 +0.6 +0.5 +0.5 +0.5 +0.000Mean ± 2 Std +DNS +0.9 +Mean +0.8 +lue +0.7 +0.6 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +coordinateDirichletBC +Neumann BC (x) +NeumannBC (y) +DNS +Pred.Mean +Pointwise Error + 0.8 +0.66 +0.72 + 0.8 +F 0.8 +0.05 +0.70 + 0.6 +0.64 +0.6 +0.6 +0.00 +0.68 + 0.4 +0.62 +0.66 + 0.4 +0.4 +0.05 +F 0.64Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred.Mean +Pointwise Error +0.10 +0.10 +0.010 +0.65 +0.65 +0.65 +0.05 +0.05 +0.005 +0.60 + 0.00 + 0.00 +0.60 +0.60 +0.000 +0.05 +0.05 +0.005 +0.55 +0.55 +0.55 +-0.010 +0.10 +-0.10Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred.Mean +Pointwise Error +0.900 +0.10 +0.10 +0.900 +0.900 +0.000 +0.875 +0.05 +0.05 +0.875 +0.875 +0.025 +0.850 + 0.00 + 0.00 +0.850 +0.850 +0.825 +0.825 +0.825 +0.050 +0.05 +0.05 +0.800 +0.800 +0.800 +0.075 +0.10 +0.10Dirichlet BC +Neumann BC (x) +NeumannBC (y) +DNS +Pred. Mean +Pointwise Error +0.10 +0.10 +0.9 + 0.9 + 0.9 +0.00 +0.05 +0.05 + 0.8 + 0.00 +0.00 +0.8 +0.8 +0.02 + 0.7 +0.05 +0.05 +0.7 +0.7 +-0.04 +0.6 +-0.10 +0.10Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred.Mean +Pointwise Error +0.9 +0.10 +0.10 +0.90 +0.90 +0.025 +0.05 +0.05 +0.85 +0.85 +0.000 +0.8 + 0.00 + 0.00 +0.80 +0.80 +0.025 + 0.7 +0.05 +0.05 +0.75 +0.75 +0.050 +0.075 +0.10 +0.10 +0.70 +0.70A PREPRINT - JANUARY 31, 2023 +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 32 +ReLU +Dense +DenseFlipout +units = 128 +ReLU +Reshape +Reshape +- +[4, 4, 8] +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 2 +kernel (5,5), padding: same, ReLU +Table 8: Details of both deterministic and probabilistic NNs for solving D1. +Description +Deterministic +Probabilistic +Total parameters +41,346 +82,435 +Size of D +5 × Aug: 210 +5 × Aug: 29 +Epochs +10,000 +100 +Zero initialization epochs +100 +- +Optimizer +Adam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 9: Training related parameters for solving D1. +Figure S17: The training loss of the residual constrained NNs to solve all BVPs in D1. +S4.4.2 +D2/T2: BVPs on quadrilateral/pentagon/hexagon +D2 contains 192K unique BVPs, which consists of 32K BVPs on quadrilateral, 64K BVPs on pentagons, and 96K +BVPs on hexagon. These BVPs covers 400 quadrilaterals, 400 pentagons, and 400 hexagons. The shape of these +polygons are randomly generated. As for D1/T1, we apply boundary conditions to two non-adjacent edges, which are +randomly sampled from all the possible locations. The boundary values are randomly generated, which could have +a constant/linear/quadratic/sinusoidal distribution along the edges. The testing dataset T2 contains 320 BVPs, which +19 + +Training +104 +Validation +103 +102 +SSOT +101 +100 +10-1 +10-2 +0 +10000 +20000 +30000 +40000 +50000 +60000 +epoch105 +Training +104 +Validation +103 +102 +101 +0 +-100 +-101 +-102 +-103 +-104, +0 +2500 +5000 +7500 +10000 +12500 +epoch10-5 +0 +2500 +5000 +7500 +10000 +12500 +epochA PREPRINT - JANUARY 31, 2023 +Figure S18: Poor results from NN predictions for T1 from the NN trained over D1. +Figure S19: Statistics of NN results for different training to solve D1 to confirm the repeatability of the proposed +method. +8 +9 +10 +11 +4 +5 +6 +7 +Figure S20: Illustration of dataset D2/T2. D2 contains BVPs that covers quadrilateral, pentagons, and hexagon. T2 +contains polygons with the total number of edges ranging from four to eleven. Similar as for D1/T1, the boundary +values could have a constant/linear/quadratic/sinusoidal distribution. +covers polygons with the total number of edges range from four to eleven. The training loss is shown in Fig. S21. NN +architectures and training details are summarized in Table 10 and 11. Selected NN predicted results are shown in Fig. +??(c), with the L2 error of all the training and testing results given in Fig. ??(e). +S4.4.3 +D3/T2: BVPs on quadrilateral/pentagon/hexagon/nonagon +D3 contains 288K unique BVPs, which consists of 32K BVPs on quadrilateral, 64K BVPs on pentagons, 96K BVPs +on hexagon and 96K BVPs on nonagon. These BVPs covers 400 quadrilaterals, 400 pentagons, 400 hexagons, and +400 nonagons. The shape of these polygons are randomly generated. As for other large datasets, we apply boundary +conditions to two non-adjacent edges, which are randomly sampled from all the possible locations. The boundary +values are randomly generated, which could have a constant/linear/quadratic/sinusoidal distribution along the edges. +The same testing dataset T2 are used for this problem. The training loss is shown in Fig. S25. NN architectures and +training details are summarized in Table 12 and 13. Selected NN predicted results are shown in Fig. S26(a), with the +L2 error of all the training and testing results given in Fig. S26(b). +20 + +Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. Mear +Pointwise Error +Pred. (actual range) +0.6Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. Mear +Pointwise Error +0.955Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred. Mean +Pointwise Error +Pred. (actual range) +0.100 +0.075 +0.075 + 0.96 +0.95 +0.96 + 0.96 +0.9 4 +0.2 +0.000 +0.92 +0.92 +0.25 +-0.050 +0.90 +.90 +0.050 +-0.150 +0.075 +0.880.20 +all BCs +w Neu. BCs +0.15 +w/o Neu. BCs +error +0.10 +L2 +0.05 +0.00 +regular +extreme +train +test0.20 +all BCs +w Neu. BCs +0.15 +w/o Neu. BCs +error +0.10 +L2 +0.05 +0.00 +regular +extreme +train +test0.20 +all BCs +w Neu. BCs +0.15 +w/o Neu. BCs +error +0.10 +L2 +0.05 +0.00 +regular +extreme +train +test0.20 +all BCs +w Neu. BCs +0.15 +w/o Neu. BCs +error +0.10 +L2 +0.05 +0.00 +regular +extreme +train +testA PREPRINT - JANUARY 31, 2023 +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 32 +ReLU +Dense +DenseFlipout +units = 128 +ReLU +Reshape +Reshape +- +[4, 4, 8] +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 2 +kernel (5,5), padding: same, ReLU +Table 10: Details of both deterministic and probabilistic NNs for solving D2. +Description +Deterministic +Probabilistic +Total parameters +41,346 +82,435 +Size of D +5 × Aug: 210 +5 × Aug: 29 +Epochs +10,000 +100 +Zero initialization epochs +100 +- +Optimizer +Adam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 11: Training related parameters for solving D2. +Deterministic +Probabilistic +Size +Layer arguments +Input +Input +- +- +LayerFillRandomNumber +LayerFillRandomNumber +- +- +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 8 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +MaxPooling2D +MaxPooling2D +- +kernel (2,2), padding: same +Flatten +Flatten +- +- +Dense +DenseFlipout +units = 32 +ReLU +Dense +DenseFlipout +units = 128 +ReLU +Reshape +Reshape +- +[4, 4, 8] +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +UpSampling2D +UpSampling2D +- +size (2,2) +Conv2D +Convolution2DFlipout +filters = 16 +kernel (5,5), padding: same, ReLU +Conv2D +Convolution2DFlipout +filters = 2 +kernel (5,5), padding: same, ReLU +Table 12: Details of both deterministic and probabilistic NNs for solving D3. +21 + +A PREPRINT - JANUARY 31, 2023 +Figure S21: The training loss of the residual constrained NNs to solve all BVPs in D2. +Figure S22: Poor results from NN predictions for T2 from the NN trained over D2. +References +[1] Alex Graves. Practical variational inference for neural networks. Advances in Neural Information Processing +Systems 24: 25th Annual Conference on Neural Information Processing Systems 2011, NIPS 2011, pages 1–9, +2011. +[2] Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural net- +works. 32nd International Conference on Machine Learning, ICML 2015, 2:1613–1622, 2015. +[3] Yeming Wen, Paul Vicol, Jimmy Ba, Dustin Tran, and Roger Grosse. Flipout: Efficient pseudo-independent +weight perturbations on mini-batches. 6th International Conference on Learning Representations, ICLR 2018 - +Conference Track Proceedings, pages 1–16, 2018. +[4] Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing +Internal Covariate Shift. arXiv preprint arXiv:1502.03167, 2015. +[5] Diederik P. Kingma and Max Welling. +Auto-encoding variational bayes. +2nd International Conference on +Learning Representations, ICLR 2014 - Conference Track Proceedings, pages 1–14, 2014. +22 + +Training +104 +Validation +103 +102 +SSOT +101 +100 +10-1 +10-2 +0 +10000 +20000 +30000 +40000 +50000 +60000 +epoch105 +Training +104 +Validation +103 +102 +101 +0 +-100 +-101 +-102 +-103 +-104, +0 +2500 +5000 +7500 +10000 +12500 +epoch10-5 +0 +2500 +5000 +7500 +10000 +12500 +epochDirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. +Pointwise Error +Pred. (actual range) +0.525 +0.55 + 0.90 +0.85 +0.520 +0.85 +0.54 +0.80 +0.10 +0. 75 +0.70 +0.505 +0.65 +0.65Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred. +Pointwise Error +Pred. (actual range) +1.00 +0.100 +1.00 +1.00 + 0.075 +0.00 +0.95 +0.95 +- 0.95 +S6°0 +0.050 +0.025 +-0.04 +0.90 +0.90 +0.90 +0.90 +-0.06 +0.85 +0.85 +0.025 +0.85 +-0.08 +0.85 +0.10 +0.80 +0.075 +0.075 +0.75Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred. +Pointwise Error +Pred. (actual range) +0.90 +90 +0.506 +0.85 +0.506 +0.505 +0.80 +0.505 +0.504 +0.75 + 0.504 +EOs'0 +0.70 +0.70 +0.70 +0.503 +0.65 +0.65 + 0.65 +0.502 +0.502 +0.501 +0.500Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. +Pointwise Error +Pred. (actual range) + 0.95 +0.5035 +0.5175 + 0.5150 +0.90 +0.90 +0.5030 +0.025 +0.5025 +0.5125 +-0.050 +0.5 + 0.85 +0.85 +0.5020 +0.5100 +-0.075 +08°0 +0.5015 +-0.5075 +-0.100 +0.75 +0.5010 +0.5050 +0.125 +0.70 +0.5005 +0.5025 +.65Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred +Pointwise Error +Pred. (actual range) +0.58 + 0.80 +-0 0 + 0.00 +-0.72 + 0.72 +- 0.78 +0.56 +-0.70 +-0 68 +0.54 +-0.08 + 0.66 +0.52 +0.72 +- 0.72 +-0.10 + 0.64 +0.64 +0.50 +0.12A PREPRINT - JANUARY 31, 2023 +Figure S23: Statistics of NN results for different training cases to solve D2 to confirm the repeatability of the proposed +method. +4 +5 +6 +7 +8 +9 +10 +11 +Figure S24: Illustration of dataset D3/T2. D3 contains BVPs that covers quadrilateral, pentagons, and hexagon. T2 +contains polygons with the total number of edges ranging from four to eleven. Similar as other large datasets, the +boundary values could have a constant/linear/quadratic/sinusoidal distribution. +Figure S25: The training loss of the residual constrained NNs to solve all BVPs in D3. +23 + +Training +104 +Validation +103 +102 +SSOT +101 +100 +10-1 +10-2 +0 +10000 +20000 +30000 +40000 +50000 +60000 +epoch105 +Training +104 +Validation +103 +102 +101 +0 +-100 +-101 +-102 +-103 +-104, +0 +2500 +5000 +7500 +10000 +12500 +epoch10-5 +0 +2500 +5000 +7500 +10000 +12500 +epoch0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +0.20 +w Neu. BCs +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu.BCs +errc ++ft+t + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +errc + 0.10 +0.05 +0.00 +4 +5 +6 +4 +5 +6 +7 +8 +9 +10 +11 +train +testA PREPRINT - JANUARY 31, 2023 +Description +Deterministic +Probabilistic +Total parameters +41,346 +82,435 +Size of D +5 × Aug: 210 +5 × Aug: 29 +Epochs +10,000 +100 +Zero initialization epochs +100 +- +Optimizer +Adam +Nadam +Learning Rate +2.5e-4 +1e-8 +Batch Size +256 +64 +Σ1 +- +1e-8 +Initial value of Σ2 +- +1e-8 +Table 13: Training related parameters for solving D3. +Figure S26: Selected good NN predictions for T2 (newly randomly generated polygons with different total number of +edges) from the NN trained over D3. +Figure S27: Poor results from NN predictions for T2 from the NN trained over D3. +[6] G Alzetta, D Arndt, Wolfgang Bangerth, V Boddu, B Brands, D Davydov, R Gassmoeller, T Heister, L Heltai, +K Kormann, M Kronbichler, M Maier, J.-P. Pelteret, B Turcksin, and D Wells. The deal.II Library, Version 9.0. +Journal of Numerical Mathematics, 2018. +[7] Saakaar Bhatnagar, Yaser Afshar, Shaowu Pan, and Karthik Duraisamy. Prediciton of Aerodynamic Flow Fields +24 + +Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. +Pointwise Error +Pred. (actual range) +0.525 +0.55 + 0.90 +0.85 +0.520 +0.85 +0.54 +0.80 +0.10 +0. 75 +0.70 +0.505 +0.65 +0.65Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred. +Pointwise Error +Pred. (actual range) +1.00 +0.100 +1.00 +1.00 + 0.075 +0.00 +0.95 +0.95 +- 0.95 +S6°0 +0.050 +0.025 +-0.04 +0.90 +0.90 +0.90 +0.90 +-0.06 +0.85 +0.85 +0.025 +0.85 +-0.08 +0.85 +0.10 +0.80 +0.075 +0.075 +0.75Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred. +Pointwise Error +Pred. (actual range) +0.90 +90 +0.506 +0.85 +0.506 +0.505 +0.80 +0.505 +0.504 +0.75 + 0.504 +EOs'0 +0.70 +0.70 +0.70 +0.503 +0.65 +0.65 + 0.65 +0.502 +0.502 +0.501 +0.500Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS + Pred. +Pointwise Error +Pred. (actual range) + 0.95 +0.5035 +0.5175 + 0.5150 +0.90 +0.90 +0.5030 +0.025 +0.5025 +0.5125 +-0.050 +0.5 + 0.85 +0.85 +0.5020 +0.5100 +-0.075 +08°0 +0.5015 +-0.5075 +-0.100 +0.75 +0.5010 +0.5050 +0.125 +0.70 +0.5005 +0.5025 +.65Dirichlet BC +Neumann BC (x) +Neumann BC (y) +DNS +Pred +Pointwise Error +Pred. (actual range) +0.58 + 0.80 +-0 0 + 0.00 +-0.72 + 0.72 +- 0.78 +0.56 +-0.70 +-0 68 +0.54 +-0.08 + 0.66 +0.52 +0.72 +- 0.72 +-0.10 + 0.64 +0.64 +0.50 +0.12DNS +Pred. +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600 +0.575 +0.575 +0.550 +0.550DNS +Pred. +0.85 +0.85 +0.80 +0.80 +0.75 +0.75 +0.70 +0.70 +0.65 +0.65 +0.60 +0.60 +0.55 +0.55DNS +Pred. +0.90 +0.90 +0.85 +0.85 +0.80 +0.80 +0.75 +0.75 +0.70 +0.70 +0.65 +0.65 +0.60 +0.60DNS +Pred. +0.725 +0.725 +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600 +0.575 +0.575 +0.550 +0.550DNS +Pred. +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1DNS +Pred. +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3DNS +Pred. +0.750 +0.750 +0.725 +0.725 +0.700 +0.700 +0.675 +0.675 +0.650 +0.650 +0.625 +0.625 +0.600 +0.600DNS +Pred. +0.9 +0.9 +- +0.8 +0.8 +0.7 +0.7 +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3A PREPRINT - JANUARY 31, 2023 +Figure S28: Statistics of NN results for different training cases to solve D3 to confirm the repeatability of the proposed +method. +Using Convolutional Neural Networks. Comput. Mech., 5:1–30, 2019. +[8] Angran Li, Ruijia Chen, Amir Barati Farimani, and Yongjie Jessica Zhang. Reaction diffusion system prediction +based on convolutional neural network. Sci. Rep., 10:1–9, 2020. +[9] Han Gao, Luning Sun, and Jian Xun Wang. PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional +Neural Networks for Solving Parametric PDEs on Irregular Domain. arXiv, pages 1–45, 2020. +[10] Nicholas Geneva and Nicholas Zabaras. Modeling the dynamics of PDE systems with physics-constrained deep +auto-regressive networks. J. Comput. Phys., 403:109056, 2020. +25 + +0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs + 0.10 +0.05 +0.00 +4 +5 +6 +9 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +err + 0.10 +0.05 +0.00 +4 +5 +6 +9 +4 +5 +6 +7 +8 +9 +10 +11 +train +test0.25 +all BCs +w Neu. BCs +0.20 +w/o Neu. BCs +err + 0.10 +0.05 +0.00 +4 +5 +6 +9 +4 +5 +6 +7 +8 +9 +10 +11 +train +test \ No newline at end of file diff --git a/btFPT4oBgHgl3EQfxDUf/content/tmp_files/load_file.txt b/btFPT4oBgHgl3EQfxDUf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe0ee20008fa376bc710f82a43751a51fcdc4790 --- /dev/null +++ b/btFPT4oBgHgl3EQfxDUf/content/tmp_files/load_file.txt @@ -0,0 +1,4415 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf,len=4414 +page_content='Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems Xiaoxuan Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Krishna Garikipati1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 ∗ 1Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States 2Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States 3Michigan Institute for Computational Discovery & Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States Abstract Traditional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' numerical discretization-based solvers of partial differential equations (PDEs) are fun- damentally agnostic to domains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' boundary conditions and coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In contrast, machine learnt solvers have a limited generalizability across these elements of boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This is strongly true in the case of surrogate models that are typically trained on direct numerical simu- lations of PDEs applied to one specific boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In a departure from this direct approach, the label-free machine learning of solvers is centered on a loss function that incorporates the PDE and boundary conditions in residual form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, their generalization across boundary conditions is limited and they remain strongly domain-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, we present a framework that generalizes across domains, boundary conditions and coefficients simultaneously with learning the PDE in weak form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our work explores the ability of simple, convolutional neural network (CNN)- based encoder-decoder architectures to learn to solve a PDE in greater generality than its restriction to a particular boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this first communication, we consider the elliptic PDEs of Fickien diffusion, linear and nonlinear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Importantly, the learning happens indepen- dently of any labelled field data from either experiments or direct numreical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We develop probabilistic CNNs in the Bayesian setting using variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Extensive results for these problem classes demonstrate the framework’s ability to learn PDE solvers that generalize across hundreds of thousands of domains, boundary conditions and coefficients, including extrapolation beyond the learning regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Once trained, the machine learning solvers are orders of magnitude faster than discretization-based solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' They therefore could have relevance to high-throughput solutions of PDEs on varying domains, boundary conditions and coefficients, such as for inverse modelling, optimization, design and decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We place our work in the context of recent continuous operator learning frameworks, and note extensions to transfer learning, active learning and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Introduction Partial differential equation (PDE) solvers play a central role in computational science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' They bridge between the mathematical physics of field theories to applications in engineering science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Popular, discretization-based numerical methods to solve PDEs include, but are not limited to, the finite element method (FEM), finite difference method, finite volume method and their variants, each with its own advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The FEM, in its many variant forms, is notable for the natural treatment of complex domain geometries and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, when a large number of boundary value problems need to be solved, such as for inverse modelling, optimization, design and decision-making or these discretization-based numerical solvers can prove very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Scientific machine learning (ML) techniques have proved to be natural candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ML approaches to solving mathematical descriptions of physical systems can be categorized as surrogate models and PDE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The first category typically requires a vast amount of training data, either from measurement or direct numerical simulations (DNSs), whose acquisition can pose challenges of availability and expense, (see [1, 2, 3, 4], and many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For example, in Ref [1] a Bayesian uncertainty quantification (UQ) approach to convolutional neural networks (CNNs) was proposed for flows in heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' CNNs are also used to predict the velocity and pressure fields in aerodynamics [3] and the concentration field for single-species reaction-diffusion systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The second category requires little or no pre-labeled data to solve PDEs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For example, high- dimensional, free-boundary PDEs have been solved by fully connected NNs [6], and by the Deep Galerkin Method ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' E-mail address: krishna@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='edu January 31, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='13165v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='NA] 30 Dec 2022 A PREPRINT - JANUARY 31, 2023 [7] using NNs, which satisfy the differential operators, initial conditions (ICs), and boundary conditions (BCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The Physics-informed Neural Networks (PINNs) approach has been proposed to solve steady and transient systems [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In PINNs, the strong form of PDEs, the ICs and BCs are incorporated in the loss [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' PINNs have been extended to solve numerous systems [15, 16, 17, 10, 11, 18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NN-based PDE solvers constructed from the weak/variational formulation also have been studied [21, 22, 23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To become viable alternates to discretization-based PDE solvers, ML frameworks have to extend to solutions of the same PDE system, but with different ICs, BCs, and on different problem domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, this is difficult to achieve with NN-based approaches that typically enforce only one specific set of BCs [8], parameterized BCs and domains [26] via the loss function [8] or the NN architecture [13, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such NN-based solvers must be retrained for each new BC and domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In one recent approach, the notion of a genome has been introduced to learn solutions on subdomains and use them to construct solutions on larger, and to some extent varying shape domains [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, labelled training data are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A related domain-decomposition approach with NNs to impose desired regularity at subdomain boundaries has also been used [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this work, we address these challenges by a new class of physics-constrained NN solvers where the BCs are specified as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We draw from the FEM, where the weak formulation incorporates the governing PDE, the natural and essential BCs, and the solution corresponds to the vanishing the discretized residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Central to our framework is transformation of the NN predicted solution to a discretized PDE residual that defines the loss function used to train the NNs, through efficient, convolutional operator-based, and vectorized calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We introduce weak PDE loss layers via kernels whose parameters are not trainable, and are independent of the NN that learns the PDE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such features offer us great flexibility to choose the NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our framework, the trainable NN learns, from a number of boundary value problems, a representation that rec- ognizes domains, boundary conditions and coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To the extent that this learning is imperfect, there will be uncertainties in its predicted solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' One common categorization of uncertainties in modelling frameworks is as epistemic and aleatoric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The former represents model-form error that can be reduced by learning from more data or using a better model–and is natural for any finite-capacity NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The latter typically represents measurement errors, is less prone to reduction [29]–and is not applicable to the proposed framework, in which boundary value problems are exact statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Probabilistic machine learning models have been developed for uncertainty quantification with NN-based PDE solvers [30, 31, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Techniques, such as dropout [32], adversarial inference [31], Bayesian meth- ods [11], Stochastic Weight Averaging Gaussian [33] have been used, among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A recent work [34] uses conditional generative adversarial networks to map between images for forward and inverse solution of PDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' the authors treatment of domains as images bears similarity to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this communication, we present both de- terministic and probabilistic ML-PDE solvers, with Bayesian NNs (BNNs) for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' More specifically, we adopt variational inference in the Bayesian setting, motivated by its efficiency over Monte Carlo approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We focus on an encoder-decoder architecture, which has been investigated for other physical systems [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The encoder-decoder structure can be easily adapted to BNNs with the modularized probabilistic layers available in current ML platforms, of which we use the TensorFlow Probability (TFP) library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our framework, deterministic/probabilistic convolutional NN layers learn representations for problem domains and the applied BCs (both Dirichlet and Neumann) through care- fully designed input data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Thus, a single NN can be used to simultaneously solve different boundary value problems that are governed by the same PDEs but on different domains with different BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Having learnt the PDE, the ML solvers can make predictions for interpolated and, to a certain extent, extrapolated domains and BCs that they were not exposed to during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar to other NN-based PDE solvers, our learning approach is free of labelled data on field solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We note the recent development of operator learning approaches for nonlinear mappings between continuous func- tional spaces as inputs and solution fields as outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Of particular interest in this direction are DeepONets [35, 36] as well as graph kernel networks [37] and Fourier neural operators [38] as settings for PDE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our framework differs from these approaches in its focus upon learning solvers for specific PDEs, but with generalizability across boundary value problems spanning domains, BCs and coefficients with the added feature of UQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our proposed frame- work is generalizable and applicable to both steady-state and transient problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, we present it in detail and focus on its application to elliptic PDEs of steady-state diffusion, linear and nonlinear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We defer the investigation of transient problems to a subsequent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We demonstrate that, with the proposed framework, a single NN can learn a solver that can be applied to tens to hundreds of thousands of boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For brevity we use NN-PDE-S for the deterministic neural network-based PDE solver, and BNN-PDE-S for the Bayesian version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2 A PREPRINT - JANUARY 31, 2023 NN solution zero/non-zero Dirichlet BCs non-zero Neumann BCs NN solution (I) (II) NN Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='BC N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='BC NN Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='BC (i) (ii) (iii) (iv) 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 a b c d Neural Networks Encoder Decoder (fill random numbers) Rbulk Rneu Weak PDE loss layers Rtot Rtot red Image representation to FEM representation Rbulk Figure 1: Architecture of the NN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) An encoder-decoder NN to store the nonlinear mapping between NN inputs, which consist of the geometry of problem domains and the applied BCs, and the solution of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Random numbers are given to the pixels in the interior domain of the channel with Dirichlet BCs when working with the augmented dataset with only a few unique boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs are color coded with red for zero Dirichlet BCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' green for non-zero Dirichlet BCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' blue for non-zero Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) Both NN inputs and outputs are passed to the Weak PDE loss layers to calculate the discretized residual, where the bulk residual (I) is computed from NN solutions with imposed Dirichlet BCs and the residual contribution from the Neumann BCs (II) are computed based on NN inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The reduced discretized residual by excluding contributions from Dirichlet boundary locations is used to form the loss of both deterministic and probabilistic NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c) Illustration of the construction of finite element meshes from pixelated image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (d) Details of the bulk residual calculation (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Four filters are applied to the NN solutions with imposed Dirichlet BCs to select different nodes (i), resulting in a multi-channel data representation (ii), which has the structure of the local nodes of one finite element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The multi-channel data is reshaped into a two-dimensional matrix to perform residual calculation (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The two-dimensional residual matrix is reshaped to a multi-channel data structure (iv) and x reduced sum operations are performed to get the bulk residual, avoiding the time-consuming assembly process in the traditional FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Results (Bayesian) NN-based PDE solver In the proposed PDE solver (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1), NNs are used to represent the nonlinear mapping between BCs and the resulting solutions of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The discretized residuals of PDEs are used to construct the losses and therefore regularize the NN’s solutions of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We studied both deterministic NNs and BNNs, where the uncertainty of the latter is represented by computing the statistical moments of their outputs via the predictive expectation and the predictive variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Deterministic NNs have fixed model parameter values, and their losses are mean squared Euclidean norms of the discretized reduced residual vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For BNNs, the model parameters are drawn from a posterior distribution that is computed from Bayes’ theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The loss of BNNs is formulated using variational inference [39], which consists of a data-independent contribution and a data-dependent contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The latter is the log likelihood function, which has the form of a joint distribution of the discretized residual with an added Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Both NNs are trained via a mini-batch optimization process with standard stochastic optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the proposed framework, the PDE loss layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1b) are independent of the NNs, which offers flexibility in choosing the NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' An encoder-decoder NN architecture is explored (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NNs, which store the 3 0 1 + 2 + 3 - 4- 0 1 2 m 4A PREPRINT - JANUARY 31, 2023 nonlinear mappings between BCs and the PDE solutions, accept image-type inputs that contain physically meaningful boundary values and markers for different regions to allow the convolutional NN layers to learn the BCs and problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This allows a well-trained NN to make predictions for new problem domains with new BCs when these are provided as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As the image-type NN outputs can be treated as FEM meshes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1c), we evaluate the discretized residuals of PDEs based on the imposed BCs and NN predictions by following the FEM and using standard numerical integration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This is achieved through an efficient, discrete, convolution operator-based, and vectorized im- plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Calculation of the bulk residual is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1d with detailed implementations and procedures provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this work, we define one unique BVP as imposing a certain PDE on a specific domain with specific boundary values at specific boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Changes in any of these elements defines a new boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We found that training a single NN to solve multiple boudnary value prooblems with both Dirichlet and Neumann BCs can be very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We introduced a zero-initialization step to address the slower convergence for problems under Neumann BCs (see SI for detailed discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' When training BNNs to solve multiple boundary value problems simultaneously, their parameters can stagnate around some local minima, leading to poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To address this issue, we use a warm start approach by initializing the mean of a BNN with the optimized parameters from a deterministic NN with the identical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our method works for both small and large datasets of boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' If the number of unique boundary value problems is small, we replicate them to obtain an augmented dataset for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Steady-state diffusion problem with small dataset In this first example (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2a-b), we use the (B)NN-PDE-S on a single steady-state diffusion boundary value problem on an octagonal domain with imposed mixed BCs (zero/non-zero Dirichlet and non-zero Neumann BCs) at two differ- ent mesh resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The DNS solution from FEM, NN solution from a deterministic NN, and the mean and std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' of BNN results from 50 Monte Carlo samplings for a mesh resolution of 32 × 32 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The optimized pa- rameters from the deterministic NN are used for the warm start of the BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deterministic NN results, the mean ± 2 std of BNN results, and the FEM solution, which is considered the grand truth, are quantitatively commpared along the two dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' These quantitative comparisons confirm the accuracy of the NN results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2b for a mesh resolution of 64 × 64 show the same accuracy of the NN results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the second example (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2c-e), we use the NN-PDE-S to simultaneously solve three boundary value problems with identical BCs but different material parameters at a mesh resolution of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The quantitative comparisons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2c-e demonstrate the ability of a single, trained NN to simultaneously solve multiple booundary value prooblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Linear elasticity problem with small dataset It is challenging to solve linear elasticity mainly because the governing PDE is written in terms of the infinitesimal strain, which is the gradient of the displacement field and has a very small magnitude ∼ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, we use the (B)NN-PDE-S for the displacement on an L-shape domain, which is fixed in both directions on the bottom edge and has a vertical displacement applied on the left edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A mesh resolution of 32 × 32 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The problem is defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We consider three incremental loading levels and treat each loading level as a different boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The BNN used to solve these boundary value problems is trained with a warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deformed geometries from both the FEM results and the deterministic NN results are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3b, where the FEM solution is illustrated by the mesh and the NN solution by the solid black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The corresponding comparison between the FEM and the mean of the BNN solution is shown in 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The quantitative comparisons for ux along the vertical lines and uy along the horizontal lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3(b,c) between the FEM results, the deterministic NN solution, and the mean ± 2 std of BNN results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Those results confirm the accuracy of the NN-based solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Additionally, we also show the comparison of reaction forces for different loading levels in both the x− and y− directions between the FEM solution and the deterministic NNs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3f,g) and between the FEM solution and the mean ± 2 std of BNN results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3h,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, because the magnitude of the solution at the bottom of the L-shape is low, the NNs have difficulty computing the total reaction forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Another example using a single deterministic NN and BNN to solve 30 linear elastic boundary value prooblems for five different domains with six sets of BCs applied to each appears in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Nonlinear elasticity problem with small dataset We use the (B)NN-PDE-S solver framework on a nonlinear elasticity problem on a square domain fixed in both directions on the left edge, and with a horizontal displacement loading and vertical traction loading applied on the right edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We considered 30 incremental loading levels and treated each as a different boundary value problem, 4 A PREPRINT - JANUARY 31, 2023 Figure 2: (B)NN-PDE-S for steady-state diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Each row contains the FEM solution (labeled as DNS), solutions from NN-PDE-S, the mean and std of BNN-PDE-S solutions (warm started from the NN-PDE-S), and a quantitative comparison between the FEM solution, the NN-PDE-S solution, and the mean ± 2 std of BNN-PDE-S results along the two dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Rows 1 and 2: results for the same boundary value problem with mixed BCs at different mesh resolutions with 32 × 32 for Row 1 and 64 × 64 for Row 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The problem setup for Rows 1 and 2 is shown in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Rows 3-5 results for boundary value problems with identical BCs but different material parameters simultaneously solved by a single (B)NN-PDE-S at a mesh resolution of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The BNNs are trained with a warm start for four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training dataset for each case has different distributions of loading levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The testing dataset contains both interpolated and extrapolated loading levels, as indicated by different colors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4f-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results, including the FEM solution, the deterministic NN, and the mean and std from the BNN, for the last extrapolated loading level for case (i) in both X- and Y -directions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4b and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Additionally, we show the quantitative comparison between FEM results and the mean ± 2 std of BNN results along the dashed lines (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4d and 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' These results confirm the accuracy of the NN-based solver and demonstrate its predictivity for unseen extrapolated BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The reaction forces computed from the NN full field solution are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4f-i, which shows improved accuracy compared to the results for linear elasticity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3f-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For each case, we observe that the results from interpolated BCs are generally accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For extrapolated BCs, NN predictions are accurate to a certain degree, particularly for the reaction force in the X-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Details of the treatment for computing the nonlinear kinematics, constitutive relations, and discretized residuals with NNs, NN architecture and training scheme, and an additional example on solving 30 boundary value problems with the proposed method are provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Steady-state diffusion problem with large dataset Lastly, we use the BNN-PDE-S on the steady-state diffusion problem for two very large datasets with a 64 × 64 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In these datasets, the geometries of problem domains and the values of BCs are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The 5 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) Mean (BNN) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0000Mean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 Mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 lue val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 Mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 lue val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='40 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='35 Mean lue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 DNS 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0000A PREPRINT - JANUARY 31, 2023 a b c f g h i d Figure 3: (B)NN-PDE-S for linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) setup of BCs for the L-shape problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) comparison of solutions between the FEM and the NN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Both solutions are plotted in the deformed configuration with the FEM solution being illustrated by the mesh grid and the deterministic NN solution being illustrated by the solid black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c) similar solution comparison as in (b), but between the FEM solution and the mean of the BNN-PDE-S solutions among 50 Monte Carlo samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (d) comparison of displacements in the X-direction along the vertical line and in the Y- direction along the horizontal line among the FEM solution, the NN-PDE-S solution, whose parameters are used to warm start the BNN-PDE-S, and the mean and std of solutions from a BNN-PDE-S for a solution resolution of 32×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (f, g) comparison of reaction forces in both X- and Y-direction between the FEM and the NN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (h, i) comparison of reaction forces in both X and Y -direction between the FEM and the BNN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' locations of BCs are randomly selected from two non-adjacent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary values along one edge could be constant, a linear distribution, quadratic distribution, or sinusoidal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The details of the data generation scheme and the statistics of our datasets are discussed in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the first case, the training dataset contains 168,000 unique boundary value problems with pentagons, where none of the inner angles exceed 160◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The testing dataset is newly generated with 80 boundary value problems on pentagons with all angles < 160◦ and 80 on “extreme" pentagons (those with at least one angle ≥ 160◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Some of the selected results for different geometries are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5a, which shows that the trained NNs could predict the solution for unseen geometries, including near-degenerate polygons, and unseen BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The accuracy of the NN results is evaluated by the volume averaged L2 error across all boundary value problems in the corresponding (training/testing) dataset, which is calculated as ∥e∥2 = 1 NBVP NBVP � l=1 � � � � � � � 1 Kpx Kpx � k=1 � yDNS l,k − yNN l,k �2 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (1) The L2 errors of NN results for both training and testing datasets are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We observe that boundary value problems with only Dirichlet BCs generally perform better than those with Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the second case, the training dataset contains 192,000 unique boundary value problems with randomly generated geometries of quadrilaterals, pentagons, and hexagons, whereas the testing dataset is newly generated with the number of edges of spanning from four to eleven with 16 boundary value problems for each type of polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Some of the selected results for different polygons appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5c, again demonstrating that the trained NNs predict solutions on unseen geometries and unseen BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The L2 errors of NN results for both training and testing datasets are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We observe some degradation of NN results as the number of polygon edges in the testing set significantly exceed those in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, for training up to hexagons, this degradation in accuracy sets in only for nonagons, decagons and hendecagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Discussion Our framework successfully learns PDE-specific solvers, whether restricted to a single or multiple boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' It has been our focus to develop solvers that generalize across boundary value problems with different domains, 6 Mean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 Mean ue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 Mean lue val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 1 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='04 DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='06 Train Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='030 MyDeterministic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 DNS E Train Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='030 MyMean ± 1 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 DNS Train Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='030Deterministic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='04 Train Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='030 MyyA PREPRINT - JANUARY 31, 2023 a b x y d e f g h i case i case ii case iii case iv c Figure 4: (B)NN-PDE-S for nonlinear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) definition of the boundary value problem for the interpolated and extrapolated loading example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Three loading levels and the deformed shapes are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) Comparison between the FEM solution and the NN-PDE-S solution with a resolution of 16 × 16 for the last extrapolated BCs for case i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deformed shapes are plotted with the FEM solution illustrated by the mesh and the NN solution by the solid black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c) The corresponding comparison between the FEM solution and the mean of the BNN-PDE-S solution over 50 MC samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (d) Comparison of displacement in the X-direction along the horizontal white lines in (b, c) between the solutions of the FEM, the NN-PDE-S whose parameters are used to warm start the BNN-PDE-S, and the mean and std of solutions from the BNN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (e) Similar comparison as in (d), but for displacement in the Y -direction along the vertical white lines in (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (f-i) reaction forces in both X- and Y -directions for four different cases with each containing a different number of training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Table 1: Summary of the accuracy and speed of the (B)NN-PDE-S for the example presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Note: There may be speedup of the FEM simulation with further code optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The wall times of (B)NN-PDE-S results do not include the time to load the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Solver Hardware Software Wall-time Averaged L2 error FEM Intel i7-8750, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2GHz (use single core) mechanoChemFEM 110ms deterministic NN GeForce GTX 1050 Ti, 4GB memory Tensorflow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='22ms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='45e-3 BNN GeForce GTX 1050 Ti, 4GB memory Tensorflow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='29ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='07e-3 boundary conditions and coefficients, and are orders of magnitude faster than the traditional FEM, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such features distinguish our approach from certain other NN-based PDE solvers [8, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We have used the framework to solve the diffusion problem over two fairly complex, and large datasets of boundary value problems to demonstrate its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' From the corresponding examples, we observe that the (B)NN-PDE-S makes more accurate predictions for interpolated BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To improve its performance, one can manually introduce new BCs to the training dataset to improve the poorly trained region or targeted prediction region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' for instance, expanding the dataset with many geometries if prediction across domains is the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' If the same geometries are considered for training and testing, the dataset could be augmented by filling the interior domain with random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Nat- 7 Mean ± 1 Std DNS Train 5 Inter.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateA PREPRINT - JANUARY 31, 2023 a b c e d Figure 5: NN-PDE-S for steady-state diffusion with large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) Selected good NN-PDE-S predictions for new, randomly generated extreme pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) Selected good NN-PDE-S predictions for new, randomly generated regular pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c) Selected good NN-PDE-S predictions for new, randomly generated polygons with different total numbers of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (d) L2 error of NN-PDE-S results for the pentagon example study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' L2 error from all boundary value problems, those with Neumann BCs, and others without Neumann BCs are plotted for both training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (e) L2 error of NN-PDE-S results for the polygon example study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' L2 error from all boundary value problems, those with Neumann BCs, and others without Neumann BCs are plotted for both training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' urally, good sampling of BCs and boundary locations is important for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The performance could further be improved via careful hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The learning or cross-validation error computed over this dataset of ∼ 105 boundary value problems with randomly generated polygonal domains (order and shape), boundary conditions and coefficients can be used to drive an active learning workflow that detects regimes (domains, boundary conditions, boundary value functions and coefficients) in which additional boundary value problems are needed for improved learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For BNNs, we applied a constant additive noise to the NN solutions, which is propagated through the PDE residual calculation to compute the training loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' While the additive noise often is used to account for aleatoric uncertainty in Bayesian inference, here it accounts for model form error of the BNN, and thus corresponds to epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The applied noise Σ1 functions as the convergence threshold used in the traditional numerical methods for PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The loss of BNNs consists of data-independent (K-L divergence) and data-dependent (negative log-likelihood) contributions (Methods, Eqs (17-23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' During training, the data-dependent log-likelihood contribution to the loss is, in general much larger than the data-independent K-L divergence, and decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As it does, the data-independent K-L divergence weighs more, leading the mean of the BNN solutions to drift away from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' With training, both contributions decrease and the mean of the BNN solutions gradually converges to the DNS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The K-L divergence term tends to introduce more uncertainty to the model parameters especially with poorly informed priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The log-likelihood, which depends on Σ2 tends to reduce the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' It controls convergence of the NN solution to the DNS solution as Σ2 itself converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We report the evolution of Σ2 in Figs S21, S25 and SS29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8 DNS 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 all BCs w Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='15 w/o Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 regular extreme train test0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='25 all BCs w Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 w/o Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs err S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 4 5 6 7 8 9 10 11 train testDNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='58DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3A PREPRINT - JANUARY 31, 2023 We use a “warm start” by initializing the means of the BNN parameters to the optimized parameters from deterministic NNs with an identical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' An alternative is to initialize the means of the prior to the optimized parameters from deterministic NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BVPs with very small variations in the solutions, such as the L-shape linear elasticity example, pose challenges since the NNs have difficulty capturing small differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Formally, of course, these problems have low information content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Additionally, the linear elasticity residual is determined by the displacement gradient (infinitesimal strain) field, which is invariant to data normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We found that though the residual is computed from the physical, un-normalized, solution, learning is more effective with data normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' It ensures that the variations of NN outputs (scaled solution) is large ∈ [−1, 1] unlike the infinitesimal strain ∼ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Large variations in the NN outputs drive NN parameter variations and favor training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This also applies to diffusion when the solution range is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the residual-based loss, however, the NN output is scaled back to the physical range, to prevent violation of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Hyper-parameter searches are essential for optimal (B)NN-PDE-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The PDEs differ in their optimal hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NNs targeted at solving a wider range of steady state diffusion boundary value problems required wider layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The vector elasticity problems, even with isotropic properties, have greater information content in their solution field and in general demanded wider layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The numbers of layers were more closely aligned, and optimal kernel sizes were the same across the PDEs and targeted boundary value problem ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' See Tables S1, S3, S6, S8, S10, S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN solvers presented here were trained on a single GeForce GTX 1050 Ti GPU with 4GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Training could take hours for networks designed to solve ∼ 20 boundary value problems, and days for those to solve ∼ 105 of boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In addition to more high-powered GPUs, as well as multi-GPU training, optimization of the training workflow remains unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training time would be further reduced if transfer learning or multi-fidelity learning is used by continued training of previously trained networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Training the (B)NN-PDE-S takes more time than training regular NNs because of the PDE constrained loss layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The prediction time, however, is unaffected by these loss layers, as they are not activated during prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this work, the problem domains are represented via pixels on a square background grid for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Thus, domain boundaries are not smooth curves, but have a pixel-level resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This treatment is of importance as it applies directly to solving PDEs on pixel-based, experimental images as domain data–a target future application for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For smooth boundary representations, one can leverage recent work for approaches to map complex and irregular domains onto a regular mesh [3, 4, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such geometric transformations can be taken into account in the proposed PDE loss layers, for instance via the mapping of the physical domain from parent hyper-cubes, as is commonly done in the FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' While we have considered polygonal domains for their approximation of other geometries, the above mapping could be exploited to remove this restriction, also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our approach is formally different from recent operator networks which are focused on learning nonlinear mappings between input function spaces and output spaces, and therefore are mesh independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this regard we note that a NN solver that has learnt a PDE on a given discretization (pixel resolution) can serve as the source network for a target finer or coarser mesh within a transfer learning context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The most important difference between the presented (B)NN-PDE-S and DeepONets [35, 36], graph kernel networks [37] and Fourier operator networks [38] is that these operator network approaches need labelled field data for training–typically from DNS using the underlying PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' By not presenting and labelled field solution, but only the domain, BCs and coefficients to the network, our approach in addition to being label free allows room for our claim that the network is forced to learn the PDE, which by definition holds across boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We note that the recent TL-DeepONet [36] uses a source Banach space from a DeepONet trained on labelled data for specific boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Further training the final layer of the TL- DeepONet allows transfer learning to some new boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We are not aware of the extensiveness to which this transfer learning across boundary value problems has been studied by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Our approach is very appealing for high-throughput solution of PDEs ranging from inverse and other optimization problems through design and decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Its generalizability across domains and boundary conditions also presents opportunities in topology optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Ongoing developments will extend it beyond elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Methods General elliptic PDEs In this work, we develop (B)NN-PDE-S for steady-state diffusion, linear elasticity, and nonlin- ear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' These three physical systems are described by a general elliptic PDE on a continuum domain Ω ⊂ Rn with Dirichlet BCs on Γϕ and the Neumann BCs on Γk: ∇ · A(ϕ) = 0 on Ω, ϕ(X) = ¯ϕ(X) on Γϕ, k(X) = ¯k(X) on Γk, (2) 9 A PREPRINT - JANUARY 31, 2023 where ϕ(X) represents the spatially-dependent unknown field and X ∈ Rn is the position vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary of the continuum domain satisfies Γ = Γϕ � Γk and Γϕ � Γk = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We use bold typeface for ϕ, A, and k in (2), depending on the physical system, they could represent either scalar, vector, or tensor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For example, in the diffusion problem, ϕ, A, and k represent the compositional order parameter (scalar), the diffusive flux (vector), and the outflux (scalar), respectively, whereas in elasticity problems, ϕ, A, and k represent the deformation field (vector), the stress field (second-order tensor), and the surface traction (vector), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The details of each system appear below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The weak form of (2) states: For variations ω satisfying ∀ω ∈ V with V = � ω|ω = 0 on Γϕ� , seek trial solutions ϕ ∈ S with S = � ϕ|ϕ = ¯ϕ on Γϕ� such that � Ω ∇ω · A(ϕ) dV − � Γk ¯k · ω dS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (3) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (3) is obtained by multiplying (21) with ω, integrating by parts, and then incorporating the Neumann BC in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the diffusion problem, ω is a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For elasticity problems, ω is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Approximate, numerical solutions of (3) can be obtained using its finite-dimensional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Finite-dimensional ap- proximations of ω and ϕ, denoted by ωh and ϕh, are constructed with ωh ∈ V h = � ωh|ωh = 0 on Γϕ� and ϕh ∈ S h = � ϕh|ϕh = ¯ϕ on Γϕ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The finite-dimensional fields ωh, ∇ωh, and ϕh are computed as ωh = Ndω, ∇ωh = Bdω, and ϕh = Ndϕ (4) in terms of the nodal solutions dω and dϕ, the basis functions N, and the basis function gradient operator B = ∇N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Inserting (4) into (3) we obtain the discretized residual as an assembly over subdomains Ωe and their associated boundary Γe as R = nelemA e=1 �� Ωe BT A(ϕh)dV − � Γe,k N T ¯k dS � (5) where A is the assembly operator and nelem represents the total number of subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The volume and surface integrations in (5) are evaluated numerically via Gaussian quadrature rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this work, the problem domain Ω is treated as an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Single component, connected graphs whose vertices are pixels form the subdomains Ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pixel connectivity is preserved as graph edges in the image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Field values at each pixel of the image are treated as nodal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Additional discussion on constructing the subdomains based on image pixels is provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Of interest, but tangential, in this context is a recent work in which NN layers map between FE meshes of different resolutions [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Steady-state diffusion This problem is described by a linear elliptic PDE in the scalar composition field following (2) ∇ · H = 0 on Ω, C(X) = ¯C(X) on ΓC, H = ¯H(X) on ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (6) In (6), C represents the composition, H is the diffusive flux defined as H = −D∇C, (7) with D as the diffusivity, and H is the outward surface flux in the normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The discretized residual function (5) for steady-state diffusion is written as R = nelemA e=1 �� Ωe BT HdV − � Γe,H N T ¯H dS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (8) Diffusivity D ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0] has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Linear elasticity This problem also is posed as a linear elliptic PDE, but in terms of a vector field, u ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Following (2) we have: ∇ · σ = 0 on Ω, u(X) = ¯u(X) on Γu, T = ¯T (X) on ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (9) 10 A PREPRINT - JANUARY 31, 2023 In (9), u represents the displacement field, σ is the stress tensor, and T is the surface traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, σ is related to the infinitesimal strain ε = 1 2 � ∇u + (∇u)T � via the following constitutive relationship σ = λtr(ε)1 + 2µε (10) where λ and µ are the Lamé constants, and 1 is the second-order identity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The discretized residual function (5) for the linear elasticity problem is written as R = nelemA e=1 �� Ωe BT σdV − � Γe,T N T ¯T dS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (11) We used λ = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4231 and µ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='61538 in both DNSs with FEM for comparison with the (B)NN-PDE-S and in the PDE loss layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Nonlinear elasticity With the displacement u ∈ Rn as the vector field unknown, we write following (2): ∇ · P = 0 on Ω0, u(X) = ¯u(X) on Γu 0 , T = ¯T (X) on ΓT 0 , (12) for the nonlinear elasticity problem, with the subscript 0 indicating the reference configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In (12), P is the first Piola-Kirchhoff stress tensor, and T is the surface traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the nonlinear elasticity problem, the deformation gradient is defined as F = 1 + ∂u/∂X with 1 being the second-order identity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The right Cauchy-Green deformation tensor is written as C = F T F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The following compressible Neo-hookean hyperelastic free energy function is considered W = 1 2µ(tr(C)3 − 3 − 2 ln(J)) + λ1 2(J − 1)2, (13) with µ and λ as the Lamé constants and J = det(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The Piola stress tensor P is computed as P = ∂W ∂F = λ(J2 − J)F −T + µ(F − F −T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (14) The discretized residual function (5) for the nonlinear elasticity problem is written as R = nelemA e=1 �� Ωe BT P dV − � Γe,T N T ¯T dS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (15) The same Lamé constants were used as in linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Deterministic loss When using mini-batch optimization to train the NN-PDE-S over a dataset D, where each data point is a boundary value problem with information on problem domain and BCs, the batch loss Li is written in terms of the reduced total residual Rred tot , as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1, as Li = 1 N N � n=1 � Rred tot (Di, Θ) �2 , (16) for each mini-batch i = 1, 2, · · · , M with N indicating the number of data points (boundary value problems) in each mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The detailed training of NN-PDE-S is discussed in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Probabilistic loss In BNN-PDE-S, each model parameter is sampled from a posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We solve for the posterior distribution of model parameters with variational inference instead of Markov Chain Monte Carlo (MCMC) sampling, as the latter involves expensive iterative inference steps and is not suitable for systems with a large number of parameters [39, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our work, the likelihood function is constructed based on the discretized PDE residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' An additive noise is often applied to the NN predicted solution to represent the aleatoric uncertainty [42, 43, 10, 1, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, we also applied an additive noise to the solution to represent epistemic uncertainty stemming from model form error between BNN-PDE-S, whose perturbed solutions are still constrained by the PDEs, and the FEM solver that yields the DNS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The BNN model parameters Θ are stochastic and sampled from a posterior distribution P(Θ|D) instead of being represented by single values as in deterministic NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The posterior distribution P(Θ|D) is computed based on Bayes’ theorem P(Θ|D) = P(D|Θ)P(Θ) P(D) , (17) 11 A PREPRINT - JANUARY 31, 2023 where D denotes the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' observations (training data) and P represents the probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In (17), P(D|Θ) is the likelihood, P(Θ) is the prior probability, and P(D) is the evidence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The likelihood is the probability of D given Θ, which describes the probability of the observed data for given parameters Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A larger value of P(D|Θ) implies that Θ is more likely to yield D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The prior must be specified to begin Bayesian inference [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To compute the posterior distributions of Θ, one can use popular sampling-based methods, such as MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, MCMC involves expensive iterative inference steps and would be difficult to use when datasets are large or models are very complex [41, 45, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' An alternative is variational inference, which approximates the exact posterior distribution P(Θ|D) with a more tractable surrogate distribution Q(Θ) by minimizing the Kullback-Leibler (KL) divergence [45, 39, 46] Q∗ = arg min DKL(Q(Θ)||P(Θ|D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (18) Variational inference is faster than MCMC and easier to scale to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We therefore explore it in this work, even though it is less rigorously studied than MCMC [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The KL divergence is computed as DKL(Q(Θ)||P(Θ|D)) = EQ[log Q(Θ)] − EQ[log P(Θ, D)] + log P(D), (19) which requires computing the logarithm of the evidence, logP(D) in (17) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, computation of P(D) would require marginalization over all realizations of Θ–an intractable task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' It is also difficult to estimate P(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Consequently, it is challenging to directly evaluate the objective function in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Alternatively, we can optimize the so-called evidence lower bound (ELBO) which is equivalent to the KL-divergence up to an added constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Therefore, an optimal distribution determined using the ELBO is also optimal for the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ELBO(Q) = −DKL(Q(Θ)||P(Θ|D)) + log P(D) ELBO(Q) = EQ[log P(Θ, D)] − EQ[log Q(Θ)] = EQ[log P(D|Θ)] − (EQ[log Q(Θ)] − EQ[log P(Θ)]) = EQ[log P(D|Θ)] − DKL (Q(Θ)||P(Θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (20) So, the loss function for the BNN is written as L = DKL (Q(Θ)||P(Θ)) − EQ[log P(D|Θ)], (21) which consists of a prior-dependent but data-independent part and a data-dependent part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The former is the KL- divergence of the surrogate posterior distribution Q(Θ) and the prior P(Θ), and the latter is the negative log- likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For mini-batch optimization, the batch loss is written as Li = 1 M DKL (Q(Θ)||P(Θ)) − EQ[log P(Di|Θ(i))], (22) for each mini-batch i = 1, 2, · · · , M [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' With (22), we have L = � i Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Following Ref [47], Monte Carlo (MC) sampling is used to approximate the expectation in (22) as Li ≈ 1 M DKL (Q(Θ)||P(Θ)) − 1 N N � n=1 log P(Dn i |Θ(i)), (23) where N is the size of each mini-batch dataset, and Θ(i) denotes the ith batch sample drawn from the posterior distri- bution Q(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Even though only one set of parameters Θ(i) is drawn from Q(Θ) for each mini-batch, the perturbation approach proposed by Flipout (see SI) ensures that parameters are de-correlated for each individual example Dn i in calculating the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Probabilistic dense layers and convolutional layers with the Flipout weight perturbation technique have been implemented in the TFP Library 2 and are used to construct the BNNs in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Neural network structure and loss function Using modularized implementation of probabilistic layers in the TFP library it is easy to construct the BNN-PDE-S to have the encoder-decoder architecture shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1, which is similar to the NN-PDE-S but with all weights being drawn from probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The loss of the BNNs is given in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The probabilistic layers in the TFP library automatically calculate the prior-dependent KL-divergence and add it to the total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The data-dependent loss is accounted for by the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Assuming Gaussian noise ϵ ∼ N(0, Σ1I) with a zero- mean and a pre-specified constant covariance Σ1, the NN representation f(x, Θ) is augmented by noise to yield the output y: y = f(x, Θ) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (24) 2www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='org/probability/api_docs/python/tfp/layers 12 A PREPRINT - JANUARY 31, 2023 Forward solutions are obtained by seeking to drive the residual to zero, and correspond to satisfaction of the weak form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For BNN-PDE-S, the likelihood function is constructed from the residual value, rather than the NN predicted solutions, thus ensuring that the framework remains label free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' With the noise ϵ in (24) propagating through the residual calculation, the likelihood function is written as P(Rred tot (Di, Θ(i))|0, ΣI) = K � k=1 N � Rred,k tot |0, Σ2 � (25) where index k indicates the pixel number with K total pixels and Rred,k tot is the component of Rred tot at pixel k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For systems where nonlinear operations are involved in the residual calculation, the residual noise distribution is in general non Gaussian even if the noise in the BNN outputs is assumed to be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Under the conditions that Σ1 is small and the nonlinear operations are smooth, there exists a neighborhood of any point in which the linear approximation is valid (higher-order terms being negligible), we assume that Σ2, the noise distribution of the residual, is approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As it is challenging to directly calculate Σ2 via error propagation based on Σ1, we treat Σ2 as a learnable parameter to be optimized based on the NN loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In (25), Σ2 essentially serves as a convergence threshold for the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The batch-wise loss of the residual constrained BNNs has the following form Li ≈ 1 M DKL (Q(Θ)||P(Θ)) − 1 N N � n=1 K � k=1 log � N � Rred,k tot (Dn i , Θ(i))|0, Σ2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (26) The detailed training scheme for BNN-PDE-S is discussed in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Uncertainty quantification The BNNs allow us to quantify the epistemic uncertainty from model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' With the discretized residual constrained BNNs, the posterior predictive distribution P(y∗|x∗, D) of the BNN-predicted full field solution y∗ for a specific testing data point x∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' one boundary value problem with information on problem domain and BCs is [1, 42] P(y∗|x∗, D) = � P(y∗|x∗, Θ)P(Θ|D)dΘ ≈ � P(y∗|x∗, Θ)Q(Θ)dΘ, (27) which can be numerically evaluated via MC sampling as P(y∗|x∗, D) ≈ 1 S S � s=1 P(y∗|x∗, Θs) where Θs ∼ Q(Θ), (28) with s indicating each sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To represent the uncertainty, we compute the statistical moments of y∗ via the predictive expectation E[y∗|x∗, D] ≈ 1 S S � s=1 f(x∗, Θs) (29) and the predictive variance Var[y∗|x∗, D] = E[(y∗ + ϵ)2] − (E[y∗ + ϵ])2 ≈ 1 S S � s=1 � f(x∗, Θs)f T (x∗, Θs) + Σ1I � − � 1 S S � s=1 f(x∗, Θs) � � 1 S S � s=1 f(x∗, Θs) �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (30) Code Availability Our modularized code implementation is publicly available3, which will assist the extension to other PDE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Acknowledgements We gratefully acknowledge the support of Toyota Research Institute, Award #849910: “Computational framework for data-driven, predictive, multi-scale and multi-physics modeling of battery materials”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Computing resources were 3github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='com/mechanoChem/mechanoChemML 13 A PREPRINT - JANUARY 31, 2023 provided in part by the National Science Foundation, United States via grant 1531752 MRI: Acquisition of Conflux, A Novel Platform for Data-Driven Computational Physics (Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Monitor: Ed Walker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This work also used the Extreme Science and Engineering Discovery Environment (XSEDE) Comet at the San Diego Supercomputer Center and Stampede2 at The University of Texas at Austin’s Texas Advanced Computing Center through allocation TG- MSS160003 and TG-DMR180072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Author Contributions Competing Interests statement References [1] Yinhao Zhu and Nicholas Zabaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Bayesian deep convolutional encoder-decoder networks for surrogate mod- eling and uncertainty 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 32nd International Conference on Machine Learning, ICML 2015, 2:1613–1622, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 16 Learning solvers for elliptic partial differential equations with generalizability across boundary value problems - supplementary information Xiaoxuan Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Krishna Garikipati1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 ∗ 1Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States 2Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States 3Michigan Institute for Computational Discovery & Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' United States S1 Training NNs with stochastic weights - Flipout Different methods are available for training neural networks (NNs) with stochastic weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' such as weight perturba- tion [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' activation perturbation [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' reparameterization [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this work, we follow a specific weight perturbation method, the so-called Flipout, proposed in Ref [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Compared with other weight perturbation algo- rithms that suffer from high variance of the gradient estimates because the same perturbation is shared in a mini-batch for all training examples, Flipout is an efficient method, which decorrelates the gradients in a mini-batch by implic- itly sampling pseudo-independent weight perturbation for each example, and thus reduces the variance of NNs with stochastic weights [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This method can be efficiently implemented in a vectorized manner with unbiased stochastic gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A brief description of Flipout is summarized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Flipout assumes that the perturbations of different weights are independent, and the perturbation distribution is symmetric around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Under such assumptions, the perturbation distribution is invariant to element-wise multiplication by a random sign matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To minimize the loss L, the distribution of Q(Θ) can be described in terms of perturbations with W = W + ∆W, where W and ∆W are the mean and a stochastic perturbation for NN parameters Θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Flipout uses a base perturbation � ∆W shared by all samples (training data points) in a mini-batch, and arrives at the perturbation for an individual sample by multiplying � ∆W with a different rank-one sign matrix ∆Wn = � ∆W ◦ rnst n, (1) where the subscript n indicates an individual example in a mini-batch, the superscript t denotes the transpose operation, and rn and sn are entries of random vectors uniformly sampled from ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Using different perturbations for each sample in a mini-batch rather than an identical perturbation for all the example in a mini-batch ensures the reduction of the variance of the stochastic gradients in Flipout during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For BNNs, the W and � ∆W are the mean and standard deviation of the posterior distribution Q(Θ), which are obtained via backpropagation with stochastic optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S2 Efficient implementation of the residual calculation In this section, we describe the implementation details of the weak PDE loss layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We heavily utilize the convolu- tional operation, and the vector/matrix/tensor operations to achieve numerical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to our source code for additional details2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='b, the weak PDE loss layers take both NN inputs (BCs infor- mation) and outputs (NN predicted solution) as their inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The data structure to represent the BCs is discussed in detail in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Schematics of the major implementation steps of the bulk residual calculation and the residual calculation of Neumann BCs in the weak PDE loss layers are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We choose a steady state diffusion problem with a scalar unknown at each node for the purpose of illustration, with Dirichlet BCs being applied on the left boundary, non-zero Neumann BCs being applied on the bottom and right boundaries, and zero Neumann BCs on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To fix ideas, we consider an example such that the output of the NN shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1 is a 5 × 5 matrix (an image with 5 × 5 pixels), denoted as M NN 5,5 with the value of each entry being ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' E-mail address: krishna@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='edu January 31, 2023 2github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='com/mechanoChem/mechanoChemML arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='13165v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='NA] 30 Dec 2022 A PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 (a) nodal solution (b) selected nodes by each kernel (c) element like nodal solution (e) element like nodal residual (f) assemble residual (unfold) (g) Neumann BCs representation (h) selected nodes by each kernel (i) reduced total residual (d) vectorized representation M5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 M5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 kB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 kB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 kB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 kB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 M25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 R25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 bulk R5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 bulk (II) kII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 kII,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 residual from Neumann BCs (I) kI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 kI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 R5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 bulk I5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 D I5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 Neu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='II I5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 Neu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='I Figure S1: Illustration of the implementation steps of computing the residual in the weak PDE loss layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Paddings of zeros are not shown in (c,d,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' the actual concentration, M NN 5,5 is equivalent to the nodal solution on a domain, which is discretized by a 4 × 4 block of elements, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 The implementation procedure is summarized in the Algorithm Box 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Dirichlet BCs The channel of NN inputs with Dirichlet BCs information is denoted as I5,5 D , as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To enforce the Dirichlet BCs, we replace the nodal values of M NN 5,5 at the location of Dirichlet boundary with the actual values of I5,5 D 3In the source code, M NN 5,5 is stored as M NN 5,5,1 with a third dimension of 1, which indicates the DOF per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For elasticity problems, the third dimension has a size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here, we drop the “1” to simplify the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2 A PREPRINT - JANUARY 31, 2023 to obtain a new matrix, denoted as M 5,5, as indicated by the green color in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The Dirichlet BCs are then automatically incorporated into the residual vector during the bulk residual calculation discussed in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 Bulk residual The matrix representation of the nodal solution automatically contains the element connectivity information of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To compute the bulk residual, we first apply convolutional operations to M 5,5 with the following kernels kB,1 = � 1 0 0 0 � , kB,2 = � 0 1 0 0 � , kB,3 = � 0 0 1 0 � , kB,4 = � 0 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (2) Each convolutional operation results in a matrix with a size of 5 × 5,4 which corresponds to the selected nodes, as highlighted with colors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' With these four convolutional operations, we now have a matrix with a size of 5 × 5 × 4 (M 5,5,4), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We then reshape the matrix to an array 25 × 4 (M 25,4), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Each row of M 25,4 corresponds to the local nodal solution vector inside one finite element, the subdomain Ωe in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ), which can then be used to efficiently evaluate the residual via the matrix-vector operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' But it leaves one blank element at each of rows 5, 10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To evaluate the residual of the steady-state diffusion problem with 2×2 Gauss quadrature points, the B-operator matrix in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') has a size of 4 × 2 × 4 (# of Gauss quadrature points × spatial dimensions × # of nodes), denoted as B4,2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Its transpose with respect to its last two slots is denoted as BT 4,4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The bulk residual at each Gauss quadrature point i is evaluated as (R25,4 bulk )i = ωiDM 25,4Bi,4,2Bi,2,4 (3) with ωi denoting the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The total bulk residual is computed as R25,4 bulk = nquad � i=1 Ri bulk, (4) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' R25,4 bulk is then reshaped to R5,5,4 bulk , and stored in the element-like form, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Next, we use the tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll function to shift the element-like residual to the correct nodal position, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(f), with R5,5,0:1 bulk = R5,5,0:1 bulk R5,5,1:2 bulk = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll(R5,5,1:2 bulk , [1], [2]) R5,5,2:3 bulk = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll(R5,5,2:3 bulk , [1], [1]) R5,5,3:4 bulk = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll(R5,5,3:4 bulk , [1, 1], [1, 2]) (5) where the “=” sign represents an assignment operation, and the “:” sign represents the slicing operation in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to TensorFlow documentation for the usage of tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The assemble operation in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') for the bulk integration is now achieved by the tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='reduce_sum(R5,5,4 bulk ) function without looping over all the elements to get R5,5,1 bulk as done traditionally in the FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to our source code for the implementation of the linear/nonlinear elasticity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 Neumann BCs One channel of the inputs that contains purely Neumann BCs, denoted as I5,5 Neu, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(g), where the matrix contains only non-zero entries at the non-zero Neumann boundary locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The Neumann residual needs to be evaluated within surface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar to computing the bulk residual, we apply convolutional operations to I5,5 Neu to construct surface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Two sets of kernels are used to construct two groups of surface elements, with group I for computing the residual contributions on edges with a surface normal in the X-direction (zero padding is required), and group II for edges with a surface normal in the Y-direction (zero padding is required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We use the following two kernels kI,1 = � 1 0 0 0 � , kI,2 = � 0 0 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (6) 4The resulting matrix size is 4 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Zero paddings are used to ensure the resulted matrix with a dimension of 5 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Keeping the matrix size unchanged during the convolutional operations is not necessary and might require a small amount of extra floating-point operations, but it is less prone to errors if we handle matrices with a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3 A PREPRINT - JANUARY 31, 2023 Algorithm 1 Residual calculation for the steady-state diffusion example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Bulk residual with applied Dirichlet BCs: Rtot 1: Apply Dirichlet BCs to NN predicted solutions M NN 5,5 by replacing the nodal values at the corresponding locations to obtain M 5,5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2: Convert M 5,5 from nodal value representation to a four-node element representation M 5,5,4 by convolutional operations with kernels kB,1, kB,2, kB,3, and kB,4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1b, S1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For two-dimensional elasticity problems, NN predicted solutions have two channels to represent both the components of the displacement vector u = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The same four kernels are applied to both channels, resulting in the element representation M with a third dimension of 8 instead of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3: Get the vectorized representation M 25,4 with each row being the local nodal solutions for one element (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4: Compute bulk residual R25,4 for each element (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to our source code for details of the bulk residual calculation of linear/nonlinear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5: Switch back to matrix representation of element-like nodal residual R5,5,4 bulk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 6: Assemble bulk residual R5,5,1 bulk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Residual contributions from Neumann BCs: RNeu 1: Use kernels kI,1, kI,2 and kII,1, kII,2 to construct two groups of two-node surface elements I5,5,2 Neu,I and I5,5,2 Neu,II corresponding to the two edges with Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For two-dimensional elasticity problems, we have four groups of surface elements with two each for the traction vector T = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2: Get the vectorized representation of surface elements I25,2 Neu,I and I25,2 Neu,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3: Compute residuals R25,2 Neu,I and R25,2 Neu,II from Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4: Switch back to matrix representation of element-like nodal residual R5,5,2 Neu,I and R5,5,2 Neu,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5: Assemble residual at Neumann BCs R5,5,1 Neu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Reduced total residual: Rred tot 1: Create a mask matrix M 5,5 bulk based on I5,5 D to represent the pixel locations with valid bulk residual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The entries of M 5,5 bulk are zero for the components of I5,5 D with a value of −1, which indicates the margins between the true problem domain and the background grid (for more details see Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the steady-state diffusion examples, all entries of M 5,5 bulk are one, since the problem domain matches the background grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2: Create a reverse mask matrix M 5,5 D,rev based on I5,5 D to represent the pixel locations that are not at the Dirichlet boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The entries of M 5,5 D,rev are zero corresponding to the elements of I5,5 D with a value larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3: Compute total residual Rtot based on (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4: Multiply (element-wise) Rtot with M 5,5 D,rev and M 5,5 bulk to get Rred tot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' to construct surface elements I5,5,2 Neu,I for the first group, with the selected nodal information being shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(h-I), and the following kernels kII,1 = � 1 0 0 0 � , kII,2 = � 0 1 0 0 � , (7) to construct surface elements I5,5,2 Neu,II for the second group, with the selected nodal information being shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(h-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar to the bulk residual calculation, we form two matrices, I25,2 Neu,I and I25,2 Neu,II, to compute the Neumann residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We use two Gauss quadrature points for surface integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The shape function N in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') has a size of 2 × 2 (# of Gauss quadrature points × # of nodes), and is denoted by N 2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We evaluate the Neumann residual at each Gauss quadrature point i via (R25,2 Neu,I)i = ωiI25,2 Neu,IN i,2N i,2 and (R25,2 Neu,II)i = ωiI25,2 Neu,IIN i,2N i,2 (8) with ωi denoting the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The total Neumann residual is computed as R25,2 Neu,I = nsq � i=1 Ri Neu,I and R25,2 Neu,II = nsq � i=1 Ri Neu,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (9) 4 A PREPRINT - JANUARY 31, 2023 where nsq is the number of surface quadrature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Again, we use the tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll function to unfold the element-like residual to the correct nodal position, similar to those shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(f), for group I R5,5,0:1 Neu,I = R5,5,0:1 Neu,I R5,5,1:2 Neu,I = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll(R5,5,1:2 Neu,I , [1], [1]) (10) and for group II R5,5,0:1 Neu,II = R5,5,0:1 Neu,II R5,5,1:2 Neu,II = tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='roll(R5,5,1:2 Neu,II , [1], [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (11) The assemble operation in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') for the surface integration is now achieved by the tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='reduce_sum(R5,5,2 Neu,I) and tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='reduce_sum(R5,5,2 Neu,II) without looping over elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We obtain the final residual contributions from the Neu- mann BCs: R5,5,1 Neu = RNeu,I + RNeu,II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (12) The total residual Rtot in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=', is computed as R5,5,1 tot = R5,5,1 bulk − R5,5,1 Neu (13) by applying the Neumann residual to the bulk residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To construct the deterministic loss in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') and the likelihood function in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ), the reduced residual Rred tot obtained by excluding the residual at the Dirichlet boundary location from Rtot is used, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' It is worth mentioning that additional auxiliary matrix/vector/tensor operations have been introduced, which are not included in the description, to complete this efficient residual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are invited to refer to our code for the detailed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3 Data representation and numerical aspects To demonstrate the performance of our methods, we investigate different definitions of BVPs for the three considered physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We prepared both small datasets, which contain one or multiple BVPs, and large datasets, which could contain hundreds of thousands BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN inputs corresponding to BVPs in these datasets are synthetically generated to train the discretized residual-constrained NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To compare the solution accuracy between NNs and DNSs, we also solve these BVPs with mechanoChemFEM,5 which is a publicly available multiphysics code developed by us based on the deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='II library [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In general, for small datasets, the number of unique sets of BCs is much smaller than the number of parameters of the NNs that represent the PDE solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We therefore augment the unique sets of BVPs by duplicating them multiple times to form an augmented dataset during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the remaining part of this section, we present details on the data structure of NN inputs, domain/boundary detection, and the NN training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Data structure of NN inputs Since the discretized residual constrained NNs do not require labels for training, the NN inputs are synthetically generated with only information on problem domains and the applied BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We consider a fixed square background grid of [0, 1] × [0, 1], with nx and ny total pixels in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For both diffusion and elasticity problems, each input data point is a three-dimensional matrix Inx,ny,3×DOF to represent a set of BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The first two indices of I indicate the pixels locations in X- and Y- directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For steady-state diffusion problem with one scalar DOF per node, there are three channels in the third dimension, which contain information of Dirichlet BCs, Neumann BCs on the edge with a surface normal in the X-direction, and Neumann BCs on the edge with a surface normal in the Y- direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For elasticity problems, there are six channels in the third dimension with the first two channels containing Dirichlet BCs in X- and Y- directions, the third and fourth channels containing Neumann BCs in X- and Y- directions on the edge with a surface normal in the X-direction, and the fifth and sixth channels containing Neumann BCs in X- and Y- directions on the edge with a surface normal in the Y-direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Data normalization between [−1, 1] is used to ensure that all the physically meaningful data in our study has a value greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The structure of the input data is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S2 with the diffusion problem as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our study, the problem domain does not necessarily occupy the whole background grid, which results in the margin region as shown in Figs S2 and S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In small datasets, for the channel(s) containing Dirichlet BCs, the problem domain is filled with −2 5Code available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='com/mechanoChem/mechanoChemFEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 6For elasticity problems, the inputs contain four channels, with the first two representing Dirichlet BCs for ¯ux and ¯uy and the last two presenting Neumann BCs for ¯Tx and ¯Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5 A PREPRINT - JANUARY 31, 2023 (c) Neumann BCs (II) with jII = jn*n2 (a) Dirichlet BCs (b) Neumann BCs (I) with jI = jn*n1 (= 0) (= 0) (= 0) (= -1) (= -1) (= -1) (> 0) (> 0) (> 0) jn = j⋅n jI jII (= -2) (= 0) (= 0) (= -1) (= -1) (= -1) (> 0) (> 0) (> 0) jn = j⋅n jI jII (= -2) (= 0) (= 0) (= -1) (= -1) (= -1) (> 0) (> 0) (> 0) jn = j⋅n jI jII Small Large Figure S2: Illustration of the data structure of NN inputs for a steady-state diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN inputs contain three channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) the Dirichlet BCs (red), (b,c) the Neumann BCs (blue) on edges with a surface normal in the X- and Y-direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 Only the boundary locations have values that are greater than 0, which is physically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Top row: input structure for small datasets, which contain one or multiple BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Bottom row: input structure for large datasets, which could contain hundreds of thousands BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the first channel, the problem domain (gray color) is filled with a value of −2 (top) or 0 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' If the value is −2, the region is filled with random numbers during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The margin (white region) is filled with a value of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The residual contribution from this region is excluded when computing Rred tot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the second and third channel, the problem domain is filled with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similarly, the margin is filled with a value of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We use j = [jI, jII] and n = [n1, n2] to denote the surface flux and surface normal, with jI and jII being the projected values in X- and Y-direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' except the Dirichlet boundary values, which is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The auxiliary number −2 exists in augmented datasets and serves as an indicator to be filled with random numbers during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the margin region, which represents the space between the background grid and the problem domain, if there is any, is filled with −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The auxiliary number7 −1 serves as an indicator to evaluate Rred tot with the residual in this region being excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In large datasets, for the channel(s) containing Dirichlet BCs, the problem domain is filled with 0 except the Dirichlet boundary values, which is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the channel(s) containing Neumann BCs, the problem domain is filled with a value of 0 except the Neumann boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' When convolutional kernels operate on the problem domain, only the boundary makes a non-zero contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similarly, the margin is filled with a value of −1 for assisting the calculation of Rred tot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Examples of the actual inputs for steady state diffusion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S8(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 Domain/Boundary detection As discussed in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1, a fixed value of −1 is assigned to the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' When calculating the residual, a mask matrix is created for domain detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This mask matrix is created based on the information on Dirichlet BCs from the inputs and ensures that only the residual inside the actual problem domain is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to Refs [7, 8, 9] and many others for approaches to map complex and irregular domains onto a regular mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such geometric transformations can be easily taken into account in the proposed PDE loss layers with the parent to physical domain mapping commonly used in the FEM via basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The proposed approach, using a mask matrix for domain detection, should still be applicable to other parametric domain representations, though it is not the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our study, each input data point represents a unique BVP for a specific problem domain and set of applied BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To detect the Dirichlet BCs in small datasets, during the NN training, the input augmented data is first passed to a customized Keras layer, called LayerFillRandomNumber,which fills the pixel locations with values of −2 in the Dirichlet BCs channel with uniformly generated random numbers in the range of [0, 1] to ensure that all the data 7The auxiliary numbers −1 and −2 are arbitrary choices with no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Users can choose different values to assign to the margin and the problem domain for the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 6 A PREPRINT - JANUARY 31, 2023 Figure S3: Illustration of the deterministic NNs predicted solution at different epochs for a diffusion BVP setup with domain id 5 and BCs id 2 (flux loading from the right edge), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Top: without zero initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Bottom: with zero initialization for the first 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Algorithm 2 Training procedure for deterministic NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1: Load NN inputs with each data point being a unique set of BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2: Use a large dataset D or an augmented dataset D by by duplicating the small dataset multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3: Split D into training, validation, and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4: Setup the encoder-decoder deterministic NN structure, with the first layer being a customized layer to fill the locations that have values of −2 in D with uniform random numbers between [0, 1] to ensure D is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5: for epoch < total_epochs do 6: Batch train the NNs 7: if use zero initialization and epoch < total_zero_initialization_epochs then 8: Use dummy labels with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5, which is equivalent to an actual zero before data normalization, to form the MSE loss to train the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 9: else 10: Use Rred tot to form the deterministic loss to train the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 11: end if 12: end for 13: Make prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' points are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As the problem domain is filled with random numbers, the convolutional kernels iteratively learn the actual Dirichlet boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The data structure in the Neumann BCs channel ensures that the kernels learn to recognize the information on the Neumann BCs, as the problem domain is filled with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For large datasets, the interior domain of the Dirichlet BCs channel is filled with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The convolutional kernels will learn the actual problem domain, boundary locations, and boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 Neural network training For deterministic NNs, a fixed learning rate is used to batch optimize the loss function (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') and solve the PDE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In our study, we found that pure Dirichlet problems are learned (converge) faster than Dirichlet-Neumann problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In the latter case, the solution the NNs fail to learn the solution in some instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This observation holds for all three PDEs considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This is mainly because, for Dirichlet-Neumann problems, it is the gradient of the unknown field(s) that drives the loss instead of the field(s) itself (themselves) as in the case of the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We demonstrate this by showing the NN predicted solution at different epochs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3 for a diffusion BVP with domain id 5 and BCs id 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9a) with zero concentration on the left edge and non-zero flux on the right edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3 shows that even though the NN predicted concentration changes, it does so very slowly via a front progressing from the left edge (zero Dirichlet BCs) to the right edge (flux BCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This indicates that the solution has not yet been learned with an accuracy that is comparable to the DNS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This difficulty arises mainly because the parameters of NNs are randomly initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As a result, the NN predicted solutions at early training stages are random numbers close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Since data normalization is used, the NN solution of zero corresponds to an actual value of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such random outputs in early stages can violate the governing equations, potentially in catastrophic manner e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' resulting in a deformation gradient with negative determinant in nonlinear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In order to rectify this inconsistency, we draw from the conventional initialization of the solution vector to zero in the FEM, and adopt the same approach for the NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the first few epochs, we train NNs with dummy labels 7 DNS epochs 10 epochs 100 epochs 500 epochs 1000 epochs 1500 epochs 1900 epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0DNS epochs 50 epochs 100 epochs 200 epochs 300 epochs 400 epochs 500 epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0A PREPRINT - JANUARY 31, 2023 Algorithm 3 Training procedure for BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 1: Load NN inputs with each data point being a unique set of BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2: Use a large dataset D or an augmented dataset D by by duplicating the small dataset multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 3: Split D into training, validation, and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4: Setup the encoder-decoder probabilistic NN structure, with the first layer being a customized layer to fill the locations that have values of −2 in D with uniform random numbers between [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 5: if use optimal parameter initialization then 6: Load the optimized parameters from the deterministic NNs to initialize the mean of the posterior distribution of BNN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 7: else 8: Use random initialization for the posterior distribution of BNN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 9: end if 10: for epoch < total_epochs do 11: Batch train the NNs 12: if use zero initialization and epoch < total_zero_initialization_epochs and (not use optimal parameter initial- ization) then 13: Use dummy labels with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5, which is equivalent to an actual zero before data normalization, to form the MSE loss to train the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 14: else 15: Use Rred tot to form the likelihood loss to train the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 16: end if 17: end for 18: MC sampling for UQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 (equivalent to an actual value of 0) without enforcing the PDE constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We call this as the zero initialization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This process helps to improve the initialization of NN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' After the zero initialization procedure is completed, the PDE constraints are enabled to train the NNs to solve the PDE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We found this remedy to drastically improve the learned solution as well as speed up the training, as shown in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3, where the NN predicted solutions approach the DNS results at 500 epochs, much faster than the case without the zero initialization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training process for deterministic NNs is summarized in the Algorithm Box 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For probabilistic NNs, we can use the proposed approach successfully solve a single BVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, when we try to solve multiple BVPs, we notice that the BNNs converge faster to purely Dirichlet problems (boundary id 1, 3 for the diffusion problem and boundary id 1, 4 for elasticity problems) than those with non-zero Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Once the BNN parameters stagnate at sub-optimal solutions, it is very difficult to optimize them further for other BVPs with Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To overcome this challenge, we first train deterministic NNs with identical architectures as the BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Once the deterministic NNs have converged to a desired tolerance, we then initialize the mean of the posterior distribution of parameters in the BNNs with the optimized parameters from the deterministic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We refer to this as the optimal parameter initialization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' During the subsequent training of the BNNs we use a small learning rate to explore the local parameter space around these optimized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A similar approach has been adopted in Ref [10], The training process for BNNs is summarized in the Algorithm Box 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Here an epoch corresponds to a single application of an augmented dataset or a large dataset D as inputs to predict a NN solution by training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For augmented dataset, when splitting D into training, validation, and testing groups, each group could potentially contain all the unique BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The evaluation of the NN results based on such validation and testing dataset only indicates how well the NNs solve the exposed sets of BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To test the predictability of the trained NNs in Sections S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2, the unseen testing sets of BCs were set aside before data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 Data generation for large dataset We use the scikit-geometry package to generate the Polygons that were used for the large dataset study, where the vertices of polygons are randomly sampled approximately along a circle with applied perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The detailed code implementation is available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To choose the edges to impose the BCs, we first generate all the possible com- binations of two non-adjacent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We then apply the numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='shuffle() to the combinations and select the first few of them from the combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the boundary values, we use the numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='uniform()function to sample control points from the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' If the boundary value has a constant distribution, a single value is sam- pled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' If the boundary value has a linear distribution, two extremes are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' And interpolated values are imposed 8 A PREPRINT - JANUARY 31, 2023 to each pixel on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' One can refer to our code on Github for details to impose BCs with a quadratic or sinusoidal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4 Numerical results In this section, we provide additional information for the examples presented in the main text along with extra examples to support the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Steady state diffusion - small dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Single octagon domain with Dirichlet BCs - cold start versus warm start (a) (b) Figure S4: Illustration of the setup of BVPs with (a) purely Dirichlet BCs and (b) mixed BCs (a) cold start (b) warm start Figure S5: Comparison of BNN results between (a) cold start and (b) warm start with purely Dirichlet BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results are very similar, except the uncertainty level in (a) is higher than (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In this example, we use the proposed method to solve the steady-state diffusion problem on an octagon domain with purely Dirichlet BCs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The BNN is trained in two cases with (i) cold start and (ii) warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' One can see that the BNNs with cold start can also find the correct solution for this specific BVP, comparable with the results from the warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This confirms that the loss of the BNN is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' However, in our study, we found that it is very challenging, if not impossible, for BNNs to directly (cold start) solve multiple BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' All the other BNNs results in this work are solved based on the warm start approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 Single octagon domain with mixed BCs In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a-g), we use the proposed PDE constrained NNs to solve steady-state diffusion on an octagonal domain with mixed BCs, resulting in a solution with spatially varying gradients along both X- and Y- directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NN structure information for the octagonal domain simulation are summarized in Table 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN hyperparameters are manually tuned to achieve a desired performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training losses for both deterministic and probabilistic NNs are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S7(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Depends on the choice of the initial value of Σ2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S7(b) could have a negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The negative value is reasonable because the total 9 Mean ± 2 Std 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 DNS Mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 DNS Mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateDNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) Mean (BNN) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000Mean ± 2 Std 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 DNS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='95 value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateMean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 DNS Mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateDNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) Mean (BNN) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000A PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 32 ReLU Dense DenseFlipout units = 32 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4] Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 1 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 1: Details of both deterministic and probabilistic NNs for solving diffusion BVPs on the octagon domain with an output resolution of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Description Deterministic Probabilistic Total parameters 16,049 31,970 Size of D 1 × Aug: 212 1 × Aug: 211 Epochs 20,000 100 Zero initialization epochs 100 Optimizer Nadam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 2: Training related parameters for solving steady-state diffusion on the octagonal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Aug: data augmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' loss of BNNs in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') consists two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The first term in (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') is non-negative, whereas the second term could be either positive or negative depending on values of both Rred tot and Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The evolution of Σ2 from the BNN is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S7(c), which converges to a sharp value during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The evolution of Σ2 is correlated to the sign change of the BNN loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The logarithm of the probability density function converges to the maximum, or the negative of it converges the minimum, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S7(b), when N (0, Σ2) best represents Rred tot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such behavior is expected as the BNN is initialized with optimal parameters from the deterministic NNs and is trained with a very small learning rate to only explore the local parameter space around the optimized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 Single octagon domain with mixed BCs and different material parameters In this example, we use the proposed method to solve steady-state diffusion problem on an octagon domain with mixed BCs and varying diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S6 with heterogeneous inputs, which consists of images and scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The image-type input contains the information of problem domains and BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The scalar(s) represent the material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 Multiple rectangular domains with different BCs We also use the proposed PDE constrained NNs to simultaneously solve 20 steady-state diffusion BVPs with a resolu- tion of 16×16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The architectures of both deterministic and probabilistic NNs and other training related NN parameters are summarized in Table 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN hyperparameters are manually tuned to achieve a desired per- 10 A PREPRINT - JANUARY 31, 2023 NN solution zero/non-zero Dirichlet BCs non-zero Neumann BCs (I) (II) NN Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='BC N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='BC a b Neural Networks Encoder Decoder (fill random numbers) Rbulk Rneu Weak PDE loss layers Rtot Rtot red Figure S6: Illustration of the NN architectures with heterogeneous data inputs to account for different material pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The image-type input contains the information of problem domains and BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The scalar(s) represent the material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 64 ReLU Dense DenseFlipout units = 64 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4] Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 1 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 3: Details of both deterministic and probabilistic NNs for solving 20 steady-state diffusion BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Readers are directed to TensorFlow documentation for detailed description of the functionality of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We follow the training procedures in Algorithm Boxes 2 and 3 to first train the deterministic NN with zero initialization, followed by training the BNNs with the optimal parameter initialization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S8, which confirm the accuracy of the proposed method an d demonstrate that the proposed method can simultaneously solve multiple BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The statistical moments of the BNN predictions are evaluated based on 50 MC samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 11 A PREPRINT - JANUARY 31, 2023 Description Deterministic Probabilistic Total parameters 33,209 66,202 Size of D 20 × Aug: 210 20 × Aug: 29 Epochs 20,000 5,000 Zero initialization epochs 100 Optimizer Nadam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 4: Training related parameters for solving 20 steady-state diffusion BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Aug: data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) deterministic loss (b) probabilistic loss Figure S7: NN training information for octagon problem with mixed BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) Loss from the deterministic NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) Loss from the BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c) Evolution of Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 Linear elasticity - small dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Multiple rectangular domains with different BCs In this section, we use the proposed PDE constrained NNs to simultaneously solve 30 linear elasticity BVPs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9(a), with a resolution of 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deformed problem domains from DNSs for three representative BVPs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar architectures of both deterministic and probabilistic NNs as summarized in Table 3 are used, except that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' the last layer has two filters, representing ux and uy, instead of one for the steady-state diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The other training related NN parameters are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We follow the procedures described in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 to train both types of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN results of three selected BVPs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S10(a), are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11The statistical moments of the BNN predictions are evaluated based on 50 MC samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11, BVP (i), (ii), and (iii) correspond to bc id 1 (non-zero Dirichlet loading), bc id 2 (non-zero Neumann loading), bc id 3 (mixed loading) applied to domain id 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The comparison of solutions between DNSs, the deterministic NN, and the BNN for these threeBVPs is shown qualitatively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11(a,c,e,g,i,k), with quantitative comparison of the solution distribution along the dashed lines between DNSs and the BNN given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11(b,d,f,h,j,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Such Description Deterministic Probabilistic Total parameters 34,010 67,803 Size of D 30 × Aug: 29 30 × Aug: 29 Epochs 20,000 100 Zero initialization epochs 100 Optimizer Nadam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 128 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 5: Training related parameters for solving 30 linear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Aug: data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 12 var(sigma2) 4×10-6 3×10-6 2 ×10-6 10~6 0 2500 5000 7500 10000 12500150001750020000 epoch101 loss val_loss 100 10~1 10~2 10-3 10-4, 10-5 10-6 0 2500 5000 7500 10000 12500150001750020000 epochA PREPRINT - JANUARY 31, 2023 Figure S8: NN results for steady-state diffusion BVPs on rectangle domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Comparison between the FEM solution (DNS), the deterministic NN solution, the mean and std of BNN solutions over 50 MC samplings, and solution distributions along the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' a comparison shows that the proposed method has successfully solved most of the BVPs with desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The results from NNs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11(i,k,l) are slightly worse than the DNSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This happens mainly because the deformation for linear elasticity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The scaled results have a narrow range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55], which is challenging for NNs to learn to distinguish, particularly for purely non-zero traction loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' For the mathematically more complex nonlinear elasticity BVPs, in which the deformation is large, are solved by the NNs more successfully, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 L-shape domain with solution interpolation NN architectures for the L-shaped domain simulation are summarized in Table 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We follow the procedures described in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 to train both types of NNs with an output resolution of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 Nonlinear elasticity - small dataset Even with the zero-initialization process, the NN outputs at early stages of training could violate the physics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' with a negative or zero determinant of the deformation gradient J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' To ensure that the residual can be evaluated and to prevent residuals from these “bad” pixels values from contributing to the final loss, we regularize the loss by omitting the residual contribution with J < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 and J > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As the training continues towards a later stage, the NN predicted solutions gradually fulfill the governing PDEs, and the regularization on J will cease to function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 13 DNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) Mean (BNN) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00100 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00000Mean ± 2 Std DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 Mean lue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateDNS Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) Mean (BNN) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0008 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0000Mean ± 2 Std 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0000Mean ± 2 Std DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00000Mean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 DNS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateA PREPRINT - JANUARY 31, 2023 y (a) five simulation domains (b) six sets of BCs for linear/nonlinear elasticity x 1 2 3 4 5 1 2 3 4 5 6 Figure S9: Illustration of the setup of 30 BVPs on different domains for linear/nonlinear elasticity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In these drawings, red represents a zero Dirichlet BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Green represents a non-zero Dirichlet BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Blue represents a non-zero Neumann BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' No color is assigned to Zero Neumann BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a) Setup of five rectangle simulation domains of different sizes and locations on a fixed background grid with different applied BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b) For linear/nonlinear elasticity, 6 sets of BCs are assigned to each simulation domain, leading to 30 linear/nonlinear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BVP (i) BVP(ii) BVP (iii) (a) three selected BVPs on rectangle domains Figure S10: Illustration of the deformed shape of selected linear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The wireframe and the gray region indicate the undeformed and deformed problem domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Three selected BVPs of out the 30 BVPs solved in section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 Multiple rectangular domains with different BCs In this section, we use the proposed PDE constrained NNs to simultaneously solve 30 nonlinear elasticity BVPs, as show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9(a), with a resolution of 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deformed problem domains from DNSs for three representative setups are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The architectures of both deterministic and probabilistic NNs and the training related NN parameters used in this section are identical to those used in section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 for solving linear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We follow the procedures described in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 to train both types of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The NN results of three selected BVPs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S12, are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='The statistical moments of the BNN predictions are evaluated based on 50 MC samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13, BVP (i), (ii), and (iii) correspond to bc id 1 (non-zero Dirichlet loading), bc id 2 (non-zero Neumann loading), bc id 3 (mixed loading) applied to domain id 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The comparison of solutions between DNSs, the deterministic NN, and the BNN for these three BVPs is shown qualitatively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13(a,c,e,g,i,k), with quantitative comparison of the solution distribution along the dashed lines between DNSs and the BNN given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13(b,d,f,h,j,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We draw attention to the improved accuracy of the BNN for BVP (iii) seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S13(i,j,k,l) as a consequence of larger deformation (strain) in comparison to the same BVP with inifinitesimal strain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S11(i,j,k,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The BNN is able to better learn nonlinear than linear elasticity because of the stronger expression of physics in the nonlinear solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The comparison demonstrates that the proposed method has successfully solved multiple BVPs with desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 Rectangular domain with solution interpolation and extrapolation In this section, we explore the interpolating and extrapolating capacity of the proposed framework for the BVP setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S12, case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Both the DNS and NN solution have resolutions of 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The architectures of both deterministic and probabilistic NNs and the training related NN parameters used in this section are identical to those 14 A PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(a) ux results for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(b) ux UQ for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(c) uy results for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(d) uy UQ for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(e) ux results for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(f) ux UQ for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(g) uy results for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(h) uy UQ for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(i) ux results for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(j) ux UQ for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(k) uy results for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(l) uy UQ for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='Figure S11: Results of three selected BVPs out of the 30 linear elasticity BVPs with varying domains and different ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='applied BCs simultaneously solved by a single deterministic or probabilistic NN with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BVP (i), (ii), (iii) correspond to bc id 1, 2, and 3 for domain id 5, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a, c, e, g, i, k) Solutions from DNS, deterministic (det) NNs, and BNNs (Mean, Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') for different BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b, d, f, h, j, l) Quantitative comparison of the solution distribution between DNS and BNNs along the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 15 DNS (X) Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) (X) Mean (BNN) (X) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) (X) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='54 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='52 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10Mean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 DNS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='54 value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='52 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10Mean ± 2 Std DNS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='520 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='515 value 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10Mean ± 2 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='520 DNS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='515 lue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateA PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 32 ReLU Dense DenseFlipout units = 128 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8] Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 2 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 6: Details of both deterministic and probabilistic NNs for solving linear elasticity L-shape BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Description Deterministic Probabilistic Total parameters 41,346 82,435 Size of D 5 × Aug: 210 5 × Aug: 29 Epochs 10,000 100 Zero initialization epochs 100 Optimizer Adam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 7: Training related parameters for solving linear elasticity on L-shaped BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Aug: data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BVP (i) BVP(ii) BVP (iii) Figure S12: Illustration of the deformed shape of the three selected nonlinear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The wireframe and the gray region indicate the undeformed and deformed problem domains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' used in section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 for solving linear elasticity BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Additional interpolated and extrapolated NN prediction results for case (i) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S14 and S15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 Steady state diffusion - large dataset In this section, three large training datasets (D1, D2, D3) and two small testing datasets (T1, T2) with a mesh resolution of 64 × 64 are prepared for the steady-state diffusion problem to test the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The script to synthetically generate the BVPs in these datasets are provided in the source code on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 16 A PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(a) ux results for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(b) ux UQ for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(c) uy results for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(d) uy UQ for BVP (i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(e) ux results for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(f) ux UQ for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(g) uy results for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(h) uy UQ for BVP (ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(i) ux results for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(j) ux UQ for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(k) uy results for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(l) uy UQ for BVP (iii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='Figure S13: Results of three selected BVPs out of the 30 nonlinear elasticity BVPs with varying domains and different ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='applied BCs simultaneously solved by a single deterministic or probabilistic NN with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BVP (i), (ii), (iii) correspond to bc id 1, 2, and 3 for domain id 5, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a, c, e, g, i, k) Solutions from DNS, deterministic (det) NNs, and BNNs (Mean, Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=') for different BVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (b, d, f, h, j, l) Quantitative comparison of the solution distribution between DNS and BNNs along the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 17 DNS (X) Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (det) (X) Mean (BNN) (X) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (BNN) (X) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 0.' metadata={'source': 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val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0 coordinateA PREPRINT - JANUARY 31, 2023 Figure S14: Additional interpolated NN prediction results for case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S15: Additional extrapolated NN prediction results for case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' A1 A2 regular extreme boundary value distribution constant linear quadratic sinusoidal Different geometries Figure S16: Illustration of dataset D1/T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' D1 contains regular pentagons, whose inner angles are all smaller than 160 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' T1 contains both regular and irregular pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The latter has one innger angle greater than 160 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary values could have a constant/linear/quadratic/sinusoidal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='1 D1/T1: BVPs on regular pentagon domains D1 contains 176K unique BVPs, which covers 1100 regular pentagons, whose inner angles are all smaller than 160 degree, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The shape of the pentagons are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We apply boundary conditions to two non-adjacent edges, which are randomly sampled from all the possible locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary values are randomly generated, which could have a constant/linear/quadratic/sinusoidal distribution along the edges, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The testing dataset T1 contains 1000 BVPs, which covers both regular shapes and extreme shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The latter has one inner angle greater than 160 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Same strategies to generate the BCs for D1 are used to generate BCs for T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training loss is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NN architectures and training details are summarized in Table 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Selected NN predicted results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (a,b), with the L2 error of all the training and testing results given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' We can observe that, if the boundary locations and the extremes of boundary values in the training dataset could be more uniformly distributed, the training and prediction are in generally good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Also, the predictions are more accurate if extremes of the testing dataset are not near the extremes of training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This suggests that if we know a targeting range to use the NNs to make predictions, we can then design the NN training dataset to have a wider range, so the accuracy of NN predicted solution can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' In another word, make interpolated prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' This study also shows that a single NN could simultaneously solve hundreds of thousands BVPs with reasonable accuracy and make prediction for unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 18 DNS (Y) Sol.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 +0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70A PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 32 ReLU Dense DenseFlipout units = 128 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8] Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 2 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 8: Details of both deterministic and probabilistic NNs for solving D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Description Deterministic Probabilistic Total parameters 41,346 82,435 Size of D 5 × Aug: 210 5 × Aug: 29 Epochs 10,000 100 Zero initialization epochs 100 Optimizer Adam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 9: Training related parameters for solving D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S17: The training loss of the residual constrained NNs to solve all BVPs in D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 D2/T2: BVPs on quadrilateral/pentagon/hexagon D2 contains 192K unique BVPs, which consists of 32K BVPs on quadrilateral, 64K BVPs on pentagons, and 96K BVPs on hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' These BVPs covers 400 quadrilaterals, 400 pentagons, and 400 hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The shape of these polygons are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As for D1/T1, we apply boundary conditions to two non-adjacent edges, which are randomly sampled from all the possible locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary values are randomly generated, which could have a constant/linear/quadratic/sinusoidal distribution along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The testing dataset T2 contains 320 BVPs, which 19 Training 104 Validation 103 102 SSOT 101 100 10-1 10-2 0 10000 20000 30000 40000 50000 60000 epoch105 Training 104 Validation 103 102 101 0 100 101 102 103 104, 0 2500 5000 7500 10000 12500 epoch10-5 0 2500 5000 7500 10000 12500 epochA PREPRINT - JANUARY 31, 2023 Figure S18: Poor results from NN predictions for T1 from the NN trained over D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S19: Statistics of NN results for different training to solve D1 to confirm the repeatability of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8 9 10 11 4 5 6 7 Figure S20: Illustration of dataset D2/T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' D2 contains BVPs that covers quadrilateral, pentagons, and hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' T2 contains polygons with the total number of edges ranging from four to eleven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar as for D1/T1, the boundary values could have a constant/linear/quadratic/sinusoidal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' covers polygons with the total number of edges range from four to eleven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training loss is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NN architectures and training details are summarized in Table 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Selected NN predicted results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (c), with the L2 error of all the training and testing results given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='3 D3/T2: BVPs on quadrilateral/pentagon/hexagon/nonagon D3 contains 288K unique BVPs, which consists of 32K BVPs on quadrilateral, 64K BVPs on pentagons, 96K BVPs on hexagon and 96K BVPs on nonagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' These BVPs covers 400 quadrilaterals, 400 pentagons, 400 hexagons, and 400 nonagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The shape of these polygons are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' As for other large datasets, we apply boundary conditions to two non-adjacent edges, which are randomly sampled from all the possible locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The boundary values are randomly generated, which could have a constant/linear/quadratic/sinusoidal distribution along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The same testing dataset T2 are used for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The training loss is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' NN architectures and training details are summarized in Table 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Selected NN predicted results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S26(a), with the L2 error of all the training and testing results given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' S26(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 20 Dirichlet BC Neumann BC (x) Neumann BC (y) DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Mear Pointwise Error Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (actual range) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='6Dirichlet BC Neumann BC (x) Neumann BC (y) DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Mear Pointwise Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='955Dirichlet BC Neumann BC (x) Neumann BC (y) DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Mean Pointwise Error Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (actual range) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='9 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='000 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 regular extreme train test0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 all BCs w Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='15 w/o Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 regular extreme train test0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 all BCs w Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='15 w/o Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 regular extreme train test0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='20 all BCs w Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='15 w/o Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' BCs error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 regular extreme train testA PREPRINT - JANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2023 Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 32 ReLU Dense DenseFlipout units = 128 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8] Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 2 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 10: Details of both deterministic and probabilistic NNs for solving D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Description Deterministic Probabilistic Total parameters 41,346 82,435 Size of D 5 × Aug: 210 5 × Aug: 29 Epochs 10,000 100 Zero initialization epochs 100 Optimizer Adam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 11: Training related parameters for solving D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Deterministic Probabilistic Size Layer arguments Input Input LayerFillRandomNumber LayerFillRandomNumber Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 8 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU MaxPooling2D MaxPooling2D kernel (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same Flatten Flatten Dense DenseFlipout units = 32 ReLU Dense DenseFlipout units = 128 ReLU Reshape Reshape [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 8] Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU UpSampling2D UpSampling2D size (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='2) Conv2D Convolution2DFlipout filters = 16 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Conv2D Convolution2DFlipout filters = 2 kernel (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' padding: same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' ReLU Table 12: Details of both deterministic and probabilistic NNs for solving D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 21 A PREPRINT - JANUARY 31, 2023 Figure S21: The training loss of the residual constrained NNs to solve all BVPs in D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S22: Poor results from NN predictions for T2 from the NN trained over D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' References [1] Alex Graves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Practical variational inference for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 24: 25th Annual Conference on Neural Information Processing Systems 2011, NIPS 2011, pages 1–9, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [2] Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Weight uncertainty in neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 32nd International Conference on Machine Learning, ICML 2015, 2:1613–1622, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [3] Yeming Wen, Paul Vicol, Jimmy Ba, Dustin Tran, and Roger Grosse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Flipout: Efficient pseudo-independent weight perturbations on mini-batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 6th International Conference on Learning Representations, ICLR 2018 - Conference Track Proceedings, pages 1–16, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [4] Sergey Ioffe and Christian Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' arXiv preprint arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='03167, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [5] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Auto-encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 2nd International Conference on Learning Representations, ICLR 2014 - Conference Track Proceedings, pages 1–14, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 22 Training 104 Validation 103 102 SSOT 101 100 10-1 10-2 0 10000 20000 30000 40000 50000 60000 epoch105 Training 104 Validation 103 102 101 0 100 101 102 103 104, 0 2500 5000 7500 10000 12500 epoch10-5 0 2500 5000 7500 10000 12500 epochDirichlet BC Neumann BC (x) Neumann BC (y) DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (actual range) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='65Dirichlet BC Neumann BC (x) Neumann BC (y) DNS Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pointwise Error Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' (actual range) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 1.' metadata={'source': 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ranging from four to eleven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Similar as other large datasets, the boundary values could have a constant/linear/quadratic/sinusoidal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S25: The training loss of the residual constrained NNs to solve all BVPs in D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 23 Training 104 Validation 103 102 SSOT 101 100 10-1 10-2 0 10000 20000 30000 40000 50000 60000 epoch105 Training 104 Validation 103 102 101 0 100 101 102 103 104, 0 2500 5000 7500 10000 12500 epoch10-5 0 2500 5000 7500 10000 12500 epoch0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} 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+page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='00 4 5 6 4 5 6 7 8 9 10 11 train testA PREPRINT - JANUARY 31, 2023 Description Deterministic Probabilistic Total parameters 41,346 82,435 Size of D 5 × Aug: 210 5 × Aug: 29 Epochs 10,000 100 Zero initialization epochs 100 Optimizer Adam Nadam Learning Rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='5e-4 1e-8 Batch Size 256 64 Σ1 1e-8 Initial value of Σ2 1e-8 Table 13: Training related parameters for solving D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S26: Selected good NN predictions for T2 (newly randomly generated polygons with different total number of edges) from the NN trained over D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Figure S27: Poor results from NN predictions for T2 from the NN trained over D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [6] G Alzetta, D Arndt, Wolfgang Bangerth, V Boddu, B Brands, D Davydov, R Gassmoeller, T Heister, L Heltai, K Kormann, M Kronbichler, M Maier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Pelteret, B Turcksin, and D Wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' The deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='II Library, Version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Journal of Numerical Mathematics, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [7] Saakaar Bhatnagar, Yaser Afshar, Shaowu Pan, and Karthik Duraisamy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Prediciton of Aerodynamic Flow Fields 24 Dirichlet BC Neumann BC (x) 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Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=', 5:1–30, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [8] Angran Li, Ruijia Chen, Amir Barati Farimani, and Yongjie Jessica Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Reaction diffusion system prediction based on convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=', 10:1–9, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [9] Han Gao, Luning Sun, and Jian Xun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parametric PDEs on Irregular Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' arXiv, pages 1–45, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' [10] Nicholas Geneva and Nicholas Zabaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=', 403:109056, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFPT4oBgHgl3EQfxDUf/content/2301.13165v1.pdf'} +page_content=' 25 0.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..47dffc80a81caabbc9507f676a10e32b4d3bf957 --- /dev/null +++ b/ddFAT4oBgHgl3EQf6h5C/content/tmp_files/2301.08739v1.pdf.txt @@ -0,0 +1,1761 @@ +FlatFormer: Flattened Window Attention for +Efficient Point Cloud Transformer +Zhijian Liu1,∗ +Xinyu Yang1,2,∗ +Haotian Tang1 +Shang Yang1,3 +Song Han1 +1MIT +2Shanghai Jiao Tong University +3Tsinghua University +Abstract +Transformer, as an alternative to CNN, has been proven +effective in many modalities (e.g., texts and images). For 3D +point cloud transformers, existing efforts focus primarily on +pushing their accuracy to the state-of-the-art level. However, +their latency lags behind sparse convolution-based models +(3× slower), hindering their usage in resource-constrained, +latency-sensitive applications (such as autonomous driving). +This inefficiency comes from point clouds’ sparse and irreg- +ular nature, whereas transformers are designed for dense, +regular workloads. This paper presents FlatFormer to close +this latency gap by trading spatial proximity for better com- +putational regularity. We first flatten the point cloud with +window-based sorting and partition points into groups of +equal sizes rather than windows of equal shapes. This effec- +tively avoids expensive structuring and padding overheads. +We then apply self-attention within groups to extract local +features, alternate sorting axis to gather features from differ- +ent directions, and shift windows to exchange features across +groups. FlatFormer delivers state-of-the-art accuracy on +Waymo Open Dataset with 4.6× speedup over (transformer- +based) SST and 1.4× speedup over (sparse convolutional) +CenterPoint. This is the first point cloud transformer that +achieves real-time performance on edge GPUs and is faster +than sparse convolutional methods while achieving on-par +or even superior accuracy on large-scale benchmarks. Code +to reproduce our results will be made publicly available. +1. Introduction +Transformer [73] has become the model of choice in nat- +ural language processing (NLP), serving as the backbone +of many successful large language models (LLMs) [2,17]. +Recently, its impact has further been expanded to the vision +community, where vision transformers (ViTs) [18, 44, 72] +have demonstrated on-par or even superior performance com- +pared with CNNs in many visual modalities (e.g., image and +video). 3D point cloud, however, is not yet one of them. +∗ indicates equal contributions. +Mean mAPH L2 +64 +66 +68 +70 +72 +74 +Latency (ms) +0 +13 +26 +39 +52 +65 +FlatFormer +(Ours) +CenterPoint +SST +(Center) +SST +VoxSet +CenterPoint++ +4x faster +3x faster +Transformer +Convolution +Point Cloud Transformers +Sparse Convolutional Models +Figure 1. Previous point cloud transformers (⋆) are 3-4× slower +than sparse convolution-based models (•) despite achieving similar +detection accuracy. FlatFormer is the first point cloud transformer +that is faster than sparse convolutional methods with on-par accu- +racy. Latency is measured on an NVIDIA Quadro RTX A6000. +Different from images and videos, 3D point clouds are +intrinsically sparse and irregular. Most existing point cloud +models [92] are based on 3D sparse convolution [24] which +computes convolution only on non-zero features. They re- +quire dedicated system support [14,69,89] to realize high +utilization on parallel hardware (e.g., GPUs). +Many efforts have been made toward point cloud trans- +formers (PCTs) to explore their potential as an alternative to +sparse convolution. Global PCTs [25] benefit from the reg- +ular computation pattern of self-attention but suffer greatly +from the quadratic computational cost (w.r.t. the number of +points). Local PCTs [49,96] apply self-attention to a local +neighborhood defined in a similar way to point-based mod- +els [56] and are thus bottlenecked by the expensive neighbor +gathering [46]. These methods are only applicable to single +objects or partial indoor scans (with <4k points) and cannot +be efficiently scaled to outdoor scenes (with >30k points). +Inspired by Swin Transformer [44], window PCTs [19] +compute self-attention at the window level, achieving im- +pressive accuracy on large-scale 3D detection benchmarks. +arXiv:2301.08739v1 [cs.CV] 20 Jan 2023 + +Despite being spatially regular, these windows could have +drastically different number of points (which differ by more +than 80×) due to the sparsity. This severe imbalance results +in redundant computation with inefficient padding and par- +titioning overheads. As a result, window PCTs can be 3× +slower than sparse convolutional models (Figure 1), limiting +their applications in resource-constrained, latency-sensitive +scenarios (e.g., autonomous driving, augmented reality). +This paper introduces FlatFormer to close this huge la- +tency gap. Building on top of window PCTs [19], FlatFormer +trades spatial proximity for better computational regularity +by partitioning 3D point cloud into groups of equal sizes +rather than windows of equal shapes. It applies self-attention +within groups to extract local features, alternates sorting axis +to aggregate features from different orientations, and shifts +windows to exchange features across groups. Benefit from +the regular computation pattern, FlatFormer achieves 4.6× +speedup over (transformer-based) SST and 1.4× speedup +over (sparse convolutional) CenterPoint while delivering the +state-of-the-art accuracy on Waymo Open Dataset. +To the best of our knowledge, FlatFormer is the first point +cloud transformer that achieves on-par or superior accuracy +than sparse convolutional methods with lower latency. It is +also the first to achieve real-time performance on edge GPUs. +With better hardware support for transformers (e.g., NVIDIA +Hopper), point cloud transformers will have a huge potential +to be the model of choice in 3D deep learning. We believe +our work will inspire future research in this direction. Also, +we will make the code available to facilitate reproduction. +2. Related Work +Deep Learning on Point Clouds. +Early research converts +point clouds from 3D sensors to dense voxel grids and ap- +plies 3D CNNs [15,50,55,85] on the volumetric inputs. How- +ever, the compute and memory consumption of volumetric +CNNs grows cubically w.r.t. to the input resolution, limiting +the scalability of these methods. To overcome this bottleneck, +later research [27,34,54,56,58,71,81,83] directly performs +feature extraction on point sets, while [59, 74, 75] convert +point clouds to octrees and [11,14,24,42,89] performs sparse +convolution on sparse voxels. Recently, researchers also ex- +plore point+voxel [46–48,87] or point+sparse voxel [48,62, +63,70] hybrid representations for efficient 3D deep learning. +3D Object Detection. +Extensive attention has been paid to +3D object detection [3,23,68], a crucial task for the percep- +tion systems of autonomous vehicles. Early research [53,82] +generates object proposals on 2D images and refines the +predictions in the lifted 3D frustums. VoxelNet [99] leads +another stream of research that directly detects 3D ob- +ject without 2D proposals. Similar to VoxelNet, PointPil- +lars [33], SECOND [89,101], 3DSSD [90] and MVF [98] +are single-stage detectors using anchor heads. +Center- +Point [92,94], AFDet [22,28], HotspotNet [5], MVF++ [57], +RangeDet [21], PolarStream [6], ObjectDGCNN [80], Pillar- +Net [61], LiDARMultiNet [91] are single-stage anchor-free +detectors. PointRCNN [64], Fast PointRCNN [12], Part- +A2Net [65], PVRCNN [62,63], LiDAR R-CNN [38], Cen- +terFormer [100], FSD [20], MPPNet [8] add a second stage +that refines the proposals from the region proposal network +(RPN) in the 3D space. There are also recent explorations on +multi-sensor [1,7,10,37,41,45,60,93] 3D object detection. +Vision Transformers. +Motivated by the huge success of +transformers [17, 73] in natural language processing, re- +searchers start to adapt transformers to various vision tasks +recently [31]. ViT [32] first demonstrates that an image +can be viewed as 16×16 words and processed by multi- +head attention layers. DeiT [72] further shows that ViTs +can be trained in a data-efficient manner without pretrain- +ing on JFT [67]. T2T-ViT [95], Pyramid ViT [76,77] and +CrossFormer [78] introduce hierarchical modeling capabil- +ity to ViTs. Swin Transformer [43,44] limits self-attention +computation to non-overlapping windows and enables cross- +window information exchange via window shifting. There +are also task-specific ViTs such as ViTDet [36] for object +detection, SETR [97] and SegFormer [86] for semantic seg- +mentation. Instead of adopting a fully-attentional backbone, +another line of research, such as DETR [4], Deformable +DETR [102], MaskFormer [13], PanopticSegFormer [40], +DETR3D [79], BEVFormer [39], apply self-attention only to +the task-specific heads and still uses CNNs for the backbone. +Point Cloud Transformers. +Recently, fully-attentional +architectures begin to emerge in point cloud deep learn- +ing. Similar to ViT, PCT [25] calculates self-attention glob- +ally on the entire point cloud. PCT falls short in scala- +bility since its computation comlexity scales quadratically +as number of input points grows. PointASNL [88], Point- +Transformer [84, 96], Fast Point TransFormer [52], Point- +Former [51], VoTr [49], VoxSet [26] applies transformer- +based architecture on the local neighborhood of each point. +The efficiency of these local point cloud transformers is lim- +ited by neighborhood query and feature restructuring. Most +related to our work is the window-based point cloud trans- +former, SST [19]. Inspired by Swin Transformer, it projects +the point cloud into bird’s eye view and divides the BEV +space into non-overlapping windows with the same spatial +sizes (but different number of points). Window shifting is +used to communicate information across windows. SST suf- +fers from large computation in window partition and padding +overhead due to regional grouping, and achieves only one- +sixth utilization compared with sparse convolutional models. +In this paper, we only refer to those methods that adopt +a transformer-based architecture in the backbone as point +cloud transformers. As CenterFormer [100], FUTR3D [9] +and UVTR [35] apply sparse convolutional backbones and + +only use transformer in their detection heads, we still cate- +gorize them as sparse convolutional methods. +3. Why are Point Cloud Transformers Slow? +Although point cloud transformers (PCTs) start to catch +up with the accuracy of sparse convolutional detectors, there +is still a 3× latency gap between the fastest PCT (SST [19]) +and sparse convolutional CenterPoint [92] (Figure 1). In this +section, we dissect the efficiency bottleneck of PCTs, which +lays a solid foundation for our FlatFormer design. +3.1. Global PCTs +1k +2k +4k +8k +16k +32k +64k +Number of Points +1.1 ms +Outdoor +Scene + Latency (ms) +967 ms +Single +Object +1 +10 +1k +Figure 2. Latency of global PCTs scale quadratically with respect +to number of input points and cannot scale up to outdoor scenes +Inspired by ViT [32], the most simple and straightforward +design for transformers on point cloud is global PCTs [25], +where each point is a token. They leverage multi-head self- +attention (MHSA) [73] globally across the entire point cloud. +While being effective on small-scale 3D objects, global PCTs +fall short in scaling to large-scale scenes due to its O(N 2D) +complexity, where N is the number of tokens and D is the +number of channels. From Figure 2, the runtime of global +PCTs [25] grows quadratically as the number of input points +grows. For example, with 32k input points*, the model takes +almost one second to execute on an NVIDIA A6000 GPU, +66× slower than CenterPoint [92]. +3.2. Local PCTs +PT (1 Layer) +VoTr +CenterPoint +0 +15 +30 +45 +60 +Neighbor Preparation (Query & Gathering) +MHSA +4x slower +Latency (ms) +Figure 3. Local PCTs suffer from large neighborhood query and +data restructuring overhead. +Researchers have proposed local PCTs [49,51,52,84,88, +96] to solve the scalability issue of global PCTs. They apply +*32k is the number of points left after 0.32m×0.32m BEV projection in +a single-frame Waymo [68] scene. +MHSAs to the neighborhood of each point rather than the +entire point cloud. Hence, their computational complexity +is O(NK2D), where N is the number of points, K is the +number of neighbors for each point, and D is the number of +channels. As N ranges from 20k to 100k for real workloads +and K is less than 64 for local PCTs, their theoretical cost is +much lower than global PCTs. +Local PCTs, however, suffer greatly from neighbor prepa- +ration overheads. As point cloud is sparse and irregular, we +have to first find the neighbors of each point, and then re- +restructure the data from the N×D format to the N×K×D +format on which MHSAs can be applied. These two steps +are slow, taking 22 ms (i.e., 36% of the total runtime) for +VoTr [49] to execute for a single scene on Waymo, which is +already slower than the entire CenterPoint model. For Point +Transformer (PT) [96], the cost for preparing neighbors takes +up to 70% of the runtime. Such overhead in a single layer is +already larger than the total runtime of CenterPoint! +3.3. Window PCTs +CDF +0% +5% +10% +15% +20% +1 +11 +21 +31 +41 +51 +61 +71 +81 +PDF +Percentage +Cum. Probability +0% +25% +50% +75% +100% +Group Size (# of Points in Window) +Figure 4. In SST [19], the number of points in each window has +a large variance; padding is necessary and leads to significant +overhead for MHSA computation. +The great success of Swin Transformers [43,44] in vari- +ous visual recognition tasks motivates the design of window +PCTs, among which, SST [19] is a representative work. It +first projects the point cloud into the bird’s-eye-view (BEV) +space, then divides the BEV space into equally-shaped, non- +overlapping windows, and applies MHSA within each win- +dow. Similar to Swin Transformer, SST uses window shifting +to enable information exchange across windows. +Different from images, point clouds are sparse and non- +uniformly distributed over the space. As a result, the number +of point within each window is not the same and can differ +by two orders of magnitude (Figure 4). As the vanilla MHSA +kernel cannot efficiently support variable sequence lengths, +SST [19] batches windows with similar sizes together and +pad all windows in each batch to the largest group size within +the batch (Figure 5l). It then applies MHSA within each +batch separately. In practice, such padding introduces a 1.7× +computation overhead on Waymo. Worse still, partitioning +points to equal windows also introduce significant latency +overhead: it takes 18 ms per scene on Waymo, even slower + +\ +A +C +E +H +B +I +G +F +D +C +A +E +H +B +G +I +F +D +A +C +E +H +B +I +G +F +D +Equal-Size Grouping (FlatFormer) +C +A +E +H +B +G +I +F +D +MHSA +\ +\ +MHSA +MHSA +MHSA +MHSA +MHSA +A +C +E +H +B +I +G +F +D +C +A +E +H +B +G +I +F +D +MHSA +MHSA +Batched +Equal-Window Grouping (SST) +(Window Shape: 2x2, Group Size: 3) +Window & Sorting Axis: +A +C +E +H +B +I +G +F +D +C +0 +H +B +G +A +E +I +F +D +0 +(Window Shape: 2x2, Group Size: Varied) +Figure 5. FlatFormer partitions the point cloud into groups of equal sizes (right), rather than windows of equal shapes (left). This effectively +trades spatial proximity for better computational regularity. It then applies self-attention within each group to extract local features, alternates +sorting axis to aggregate features from different directions, and shift windows to exchange features across groups. +than the total runtime of CenterPoint. To sum up, the padding +and partitioning overheads make SST less hardware-friendly +compared with sparse convolutional methods. +4. FlatFormer +With all the lessons learned in Section 3, we will design +our point cloud transformer to be scalable and efficient. +4.1. Overview +The basic building block of FlatFormer is Flattened Win- +dow Attention (FWA). As in Figure 5r, FWA adopts window- +based sorting to flatten the point cloud and partitions it to +groups of equal sizes rather than windows of equal shapes. +This naturally resolves the group size imbalance problem and +avoids the padding and partitioning overheads. FWA then +applies self-attention within groups to extract local features, +alternates sorting axis to aggregate features from different +orientations, and shifts windows to exchange features across +groups. Finally, we provide an implementation of FWA that +further improves its efficiency and minimizes the overheads. +4.2. Flattened Window Attention (FWA) +4.2.1 +Sorting & Grouping +Window-Based Sorting. +With a point cloud {(x, y)}†, we +first quantize the coordinate of each point (x, y) to +� +⌊x/wx⌋, ⌊y/wy⌋ +� +�� +� +window coordinates +, x − ⌊x/wx⌋ · wx, y − ⌊y/wy⌋ · wy +� +�� +� +local coordinates within window +� +, +(1) +where (wx, wy) is the window shape. Next, we sort all points +first by window coordinates and then by local coordinates +within the window. This step turns the unordered point cloud +into an ordered one, where points within the same window +will be next to each other. +†We assume that the point cloud is in 2D for ease of notation, while our +method applies to 3D or higher-dimension point clouds. +Equal-Size Grouping. +Conventional window PCTs [19] +will then group the points within the same window together. +However, as discussed in Section 3, each group can have +drastically different numbers of points due to inherent spar- +sity. To overcome the padding overheads, we partition the +point cloud into groups of equal sizes based on the sorted +sequence. This step allows the subsequent group attention to +enjoy a perfectly regular workload. From the implementa- +tion perspective, our grouping only involves a simple tensor +reshaping (which is free since it does not change the layout) +and is more efficient than window partitioning in SST [19]. +Alternate Sorting Axis. +Between the two axes, x has a +higher priority in sorting. Thus, points with identical ⌊x/wx⌋ +will be next to each other while points with the same ⌊y/wy⌋ +can be very far away from each other in the sorted sequence, +breaking the geometric locality. To solve this inequity, we +alternate the sorting axis between x and y in different FWA +blocks. This is very similar to spatially separable convolution +that decomposes a 3×3 kernel into 3×1 and 1×3 kernels. +Stacking FWA blocks with different sorting axes enables the +model to aggregate local features from different directions. +Equal Size vs. Equal Window. +The key design choice we +made is to partition the point cloud into groups of equal sizes +rather than windows of equal shapes. There is a trade-off: +equal-window grouping maintains perfect spatial proximity +(i.e., each group has the same radius) but breaks the computa- +tional regularity, while equal-size grouping ensures balanced +computation workload (i.e., each group has the same number +of points) but cannot guarantee the geometric locality. We +show in Section 5.3 that computation regularity is more im- +portant since spatial irregularity can be partially addressed +by our algorithm design: i.e., window-based sorting offers +a fairly good spatial ordering, and self-attention is robust to +outliers (i.e., distant point pairs). + +4.2.2 +Group Attention +With points partitioned, we then apply self-attention [73] +within each group to extract local features. For each group +of points with coordinates C and features F, we follow the +standard transformer block design: +F′ = F + MHSA(LN(F), PE(C)), +F′′ = F′ + FFN(LN(F′)), +(2) +where MHSA(·), FFN(·) and LN(·) correspond to multi-head +self-attention, feed-forward layer, and layer normalization, +respectively. Different from SST [19], PE(·) gives global ab- +solute positional embedding. Here, we use the most standard +softmax attention formulation for MHSA(·). Our method will +benefit from other more efficient attention variants, such as +linear attention [29], which we leave for future work. +Window Shift. +Benefit from the non-overlapping design, +window-based attention typically has a larger receptive field +than convolution (e.g., 69 neighbors in our FWA vs. ≤27 +neighbors in a sparse convolution of kernel size 3). However, +its modeling power is limited as there is no feature exchange +across groups. Similar to Swin Transformer [19,43,44], we +adopt the shifted window approach that alternates the sorting +configuration in consecutive FWA blocks. Specifically, we +translate the coordinates of all points by (wx/2, wy/2) for +sorting in shifted FWA blocks. This mechanism introduces +cross-group feature communication while effectively main- +taining workload independence between groups. Note that +alternating sorting axis also enables feature exchange. +4.3. Efficient Implementation +Besides the algorithm design, we also provide an imple- +mentation that improves the efficiency of MHSA and FFN +and minimizes the sorting and masking overheads. All these +optimizations are specialized for our point cloud transformer +design and are not applicable to sparse convolution models. +Efficient MHSA. +Within MHSA, query Q, key K, and +value V will first be transformed with separate linear layers. +We pack these three linear projections into a batched matrix +multiplication (since Q, K and V have the same shape in our +FWA) to improve the parallelism. In addition, standard atten- +tion implementations materialize QKT and softmax(QKT). +We leverage a recent efficient functional-preserving imple- +mentation (FlashAttention [16]) that uses tiling to reduce the +number of memory reads/writes, achieving better efficiency. +Efficient FFN. +FFN consists of two linear layers with a +GELU activation in the middle. We implement a fused linear +kernel (in Triton) that absorbs the activation into the layer +before to avoid writing the intermediate results to DRAM. +We also observe that our linear kernel (optimized by Triton) +is even more efficient than cuBLAS, which is probably due to +the unconventional tall-and-skinny workload. +Reuse Sorting. +Sorting the coordinates of all points is a +non-negligible overhead. As the coordinates remain identical +(w/o downsampling), we reuse the sorting results (i.e., ranks +of each point) with the same axis and window. In practice, +this reduces the sorting overhead in our model by 50%. +Drop Residual. +The size of the input point cloud might +not be divisible by the group size, generating a group with +fewer points after partition. This minor irregularity will still +result in some overheads in self-attention since we need to +introduce masking to correctly zero them out. Instead, we +directly drop the final non-full group. This only corresponds +to less than 0.1% of all points, having a negligible impact on +the model’s performance (<0.1%). +5. Experiments +5.1. Setup +Dataset. +We carry out our experiments on the large-scale +Waymo Open Dataset (WOD) [68] with 1150 LiDAR point +cloud sequences. Each sequence has 200 frames, collected +by a 360◦ FoV LiDAR sensor at 10 frames per second. There +are four foreground classes, three of which (vehicles, pedes- +trians and cyclists) are used for detection metric evaluation. +Metrics. +We follow the official metrics on Waymo to cal- +culate the standard 3D mAP and heading-weighted 3D mAP +(mAPH) of all methods. The matching IoU thresholds for +vehicle, pedestrian and cyclist are set to default values (0.7, +0.5 and 0.5). Objects are divided into two difficulty levels, +where objects with fewer than five laser points or annotated +as hard are categorized into level 2 (L2) and other objects +are defined as level 1 (L1). We mainly report L2 metrics in +the main paper and provide detailed metrics in the appendix. +Model. +Based on FWA, we provide an instantiation of Flat- +Former for 3D object detection. We follow the design of +PointPillars [33] to first voxelize the point cloud into sparse +BEV pillars (with MLPs) at a resolution of 0.32m×0.32m. +We then apply eight consecutive FWA blocks with alternat- +ing sorting axes (i.e., x or y) and window shifting configura- +tions (i.e., on or off). All FWA blocks have a window shape +of 9×9 and a group size of 69. Following SST [19], we do +not apply any spatial downsampling, which is beneficial for +small objects. Finally, we apply regular BEV encoder and a +center-based detection head following CenterPoint [92,94]. +5.2. Main Results +5.2.1 +Single-Stage Detectors +Baseline. +We compare our FlatFormer with state-of-the-art +sparse convolutional [61,92,94] and transformer-based [19, +26,49] single-stage 3D detectors. All models apply anchor- +or center-based detection heads [89, 92, 94]. We compare +models with different numbers of input frames separately. + +#Frames +#MACs +Latency +Speedup +Mean L2 +Vehicle L2 +Pedestrian L2 +Cyclist L2 +(G) +(ms) +(w.r.t. [92]) +(mAPH) +(mAP/APH) +(mAP/APH) +(mAP/APH) +SECOND [89]3 +1 +– +– +– +57.2 +63.9 / 63.3 +60.7 / 51.3 +58.3 / 57.0 +PointPillars [33]3 +1 +– +– +– +57.8 +63.6 / 63.1 +62.8 / 50.3 +61.9 / 59.9 +◦ CenterPoint [92]1 +1 +126.9 +14.6 +1.0× +65.5 +66.7 / 66.2 +68.3 / 62.6 +68.7 / 67.6 +• VoTr-SSD [49] +1 +110.3 +59.1∗ +0.2× +– +60.2 / 59.7 +– +– +• SST [19]2 +1 +204.9 +45.5 +0.3× +64.8 +64.8 / 64.4 +71.7 / 63.0 +68.0 / 66.9 +• SST-Center [19] +1 +226.4 +35.1 +0.4× +66.3 +66.6 / 66.2 +72.4 / 65.0 +68.9 / 67.6 +• VoxSet [26] +1 +189.4 +39.5 +0.4× +66.2 +66.0 / 65.6 +72.5 / 65.4 +69.0 / 67.7 +◦ PillarNet [61] +1 +138.3 +11.1 +1.3× +67.2 +70.4 / 69.9 +71.6 / 64.9 +67.8 / 66.7 +• FlatFormer (Ours) +1 +177.2 +10.8 +1.4× +67.2 +69.0 / 68.6 +71.5 / 65.3 +68.6 / 67.5 +◦ CenterPoint [92]1 +2 +137.6 +16.4 +1.0× +68.4 +67.7 / 67.2 +71.0 / 67.5 +71.5 / 70.5 +◦ PillarNet [61] +2 +148.8 +11.6 +1.4× +70.0 +71.6 / 71.1 +74.5 / 71.4 +68.3 / 67.5 +• FlatFormer (Ours) +2 +186.6 +11.9 +1.4× +71.2 +70.8 / 70.3 +73.8 / 70.5 +73.6 / 72.6 +◦ CenterPoint [92] +3 +144.7 +18.3 +1.0× +– +– +– +– +◦ CenterPoint++ [94]1 +3 +113.0 +13.6 +1.3× +71.6 +71.8 / 71.4 +73.5 / 70.8 +73.7 / 72.8 +• SST [19]2 +3 +250.0 +57.8 +0.3× +70.4 +66.5 / 66.1 +76.2 / 72.3 +73.6 / 72.8 +• SST-Center [19]† +3 +243.1 +40.5 +0.5× +71.2 +68.8 / 68.2 +75.8 / 71.8 +74.4 / 73.3 +• FlatFormer (Ours) +3 +193.2 +12.7 +1.4× +72.0 +71.4 / 71.0 +74.5 / 71.3 +74.7 / 73.7 +Table 1. Results of single-stage 3D detectors on Waymo Open Dataset (validation set). FlatFormer achieves 1.4× speedup over CenterPoint +and 4.6× speedup over SST while being more accurate. We refer the readers to the appendix for detailed metrics (e.g., L1 mAP/mAPH). +Markers ◦ and • refer to sparse convolutional models and point cloud transformers, respectively. Methods with <60 L2 mAPH are marked +gray. (†: reproduced by us, 1: from CenterPoint authors, 2: from SST authors, 3: from FSD paper, *: projected latency) +#Frames +Latency +Mean L2 +(ms) +(mAPH) +◦ LiDAR R-CNN [38]† +1 +– +61.3 +◦ PV-RCNN [62]† +1 +– +63.3 +◦ Part-A2 [66]† +1 +– +63.8 +◦ PV-RCNN++ [63]† +1 +– +64.9 +◦ CenterFormer [100] +1 +33.8 +69.0 +◦ FSD-SpConv [20] +1 +47.8 +70.8 +• FlatFormer+FSD (Ours) +1 +39.3 +70.5 +◦ CenterFormer [100] +2 +53.5 +72.8 +• FlatFormer+FSD (Ours) +2 +51.8 +73.8 +◦ CenterFormer [100] +4 +85.8 +73.2 +◦ MPPNet [8] +4 +– +74.2 +• FlatFormer+FSD (Ours) +3 +60.6 +74.8 +Table 2. Results of two-stage 3D detectors on Waymo Open Dataset +(validation set). FlatFormer achieves on-par or even higher accu- +racy compared with sparse convolutional two-stage detectors. We +refer the readers to the appendix for detailed metrics (e.g., per- +class L1/L2 mAP/mAPH). Markers ◦ and • refer to SpConv-based +models and point cloud transformers. (†: from FSD paper) +Latency. +We measure the latency on an NVIDIA Quadro +RTX A6000 GPU using FP16 precision. We adopt SpConv +v2.2.3 [89], the state-of-the-art 3D sparse convolution library, +to execute the 3D encoder of all sparse convolutional detec- +tors. For transformer-based detectors, we use their official +implementation to measure the runtime. All modules after +the 3D encoder (e.g., BEV encoder and detection head) are +executed with TensorRT 8.4. We execute all the methods on +the first 1,000 samples for 50 runs (with 10 warmup runs). +We report the average latency (with outliers excluded). We +do not include the data loading and post-processing time. +Results. +As in Table 1, our FlatFormer achieves consistent +performance improvements over both sparse convolutional +and transformer-based detectors with much better efficiency. +For one-frame models, FlatFormer is 4.2×, 3.3× and 3.7× +faster than SST, SST-Center and the recent VoxSet [26]. It +also compares favorably with strong sparse convolutional +baselines: 1.4× faster than CenterPoint with 1.7 higher L2 +mAPH and performs on par with PillarNet [61]. The accu- +racy advantage magnifies in the two-frame setting. Specifi- +cally, FlatFormer is 1.4× faster than CenterPoint with 2.8 L2 +mAPH higher accuracy, and outperforms PillarNet by 1.3 L2 +mAPH with a similar latency. With three input frames, Flat- +Former is 4.6× and 3.2× faster than SST and SST-Center, +respectively. It also achieves better latency-accuracy trade- +off (1.1× faster and 0.4% higher accuracy) compared with +CenterPoint++ [94]. Remarkably, FlatFormer requires 1.7× +more MACs than CenterPoint++ while it is still faster. This +indicates that our design is more hardware-friendly than the +sparse convolutional baselines. +Deployment. +We deploy our FlatFormer on an NVIDIA +Jetson AGX Orin. This is a resource-constrained edge GPU + +CenterPoint +SST +SST (Center) +FlatFormer +0 +5 +10 +15 +20 +13.7 +16.1 +5.4 +3.4 +Frames Per Second (FPS) +Real Time +(>10 FPS) +Figure 6. Measured latency on NVIDIA Jetson AGX Orin. Flat- +Former is the first point cloud transformer that achieves real-time +performance on edge GPUs. +platform that is widely used in real-world self-driving cars. +From Figure 6, FlatFormer runs at 16 FPS, which is 1.2× +faster than CenterPoint [92] and 3× faster than SST-Center. +To the best of our knowledge, FlatFormer is the first point +cloud transformer that achieves real-time inference (i.e., >10 +FPS, which is the LiDAR sensor frequency) on edge GPUs. +We believe that it paves the way for efficient LiDAR-centric +perception in real-world applications. +5.2.2 +Two-Stage Detectors +Model. +To verify the generalizability, we replace the 3D +backbone in FSD [20], a state-of-the-art two-stage detector, +and compare its results with previous two-stage models. We +keep the same grid resolution, window shape and group size +as in our single-stage experiments. +Baseline & Latency. +We compare our model against state- +of-the-art two-stage detectors in Table 2. We follow the same +latency measurement protocol. For CenterFormer [100], we +adapt the official implementation to support SpConv v2.2.3 +backend in FP16 precision for a fair comparison. +Results. +All existing high-performing two-stage detectors +are sparse convolutional, while our FlatFormer is the only +transformer-based method that achieves state-of-the-art level +accuracy. It also shows better scalability with respect to the +number of input frames compared with CenterFormer [100]. +Note that our paper focuses on optimizing the latency of +3D backbone. However, two-stage detectors [20, 100] are +usually bottlenecked by the second stage in runtime, which +is out of our scope. We expect that the latency of FlatFormer +could benefit from a more efficient second-stage design. +5.3. Analysis +In this section, we present analyses to validate the effec- +tiveness of our design choices. All experiments are based on +our single-frame model trained with 20% data. +5.3.1 +Flattened Window Attention +In Figure 7, we visualize the learned attention weights in +our FWA. The color represents the scale of attention weights, +(a) moving vehicles +(b) turning vehicles +High +(c) parked vehicles +Low +Figure 7. Visualization of attention weights in FlatFormer for +vehicles that are moving straight ahead, turning and parking. High +attention weights corresponds to the detected object. +Sorting Strategy +Mean L2 Veh. L2 +Ped. L2 +Cyc. L2 +(mAPH) (mAPH) (mAPH) (mAPH) +Ours +61.7 +63.3 +57.9 +63.9 +w/o Quantization +60.4 +61.9 +57.6 +61.7 +w/o Axis Alternation +61.1 +63.1 +57.3 +62.9 +w/o Window Shift +61.2 +62.9 +57.8 +62.8 +Random Order +57.8 +58.8 +55.1 +59.4 +SST [19] +60.7 +62.3 +56.7 +63.1 +Table 3. Window-based sorting in FlatFormer provides even better +performance than equally-shaped window partition in SST [19] and +outperforms other sorting strategies. +where warmer color means larger attention weights. Black +points correspond to query points, and gray points are those +with weights smaller than a threshold. For vehicles moving +straight ahead, turning and parked, query points on the vehi- +cle are always highly attended to nearby points on the same +car, while faraway points have very small learned attention +weights. Such an observation can partially explain the ef- +fectiveness of FWA: i.e., even if equal-size grouping does +not create spatially regular windows, the model can learn to +suppress the importance of outlier points in the background +and focus on important foreground points within each group. +5.3.2 +Ablation Studies on Model Design +Sorting Strategy. +We first analyze the effectiveness of our +window-based, axis-alternating sorting strategy in Table 3. +Randomly grouping points together without any spatial sort- +ing will give ∼ 4% worse performance compared with Flat- +Former. Furthermore, due to spatial discontinuities on the +boundary regions, directly sorting the points by xyz coor- +dinates or window sorting along single axis both provide +sub-optimal results. We also notice that window shifting +brings about 0.5% improvement to the final performance. +Interestingly, despite the fact that our sorting strategy does + +DVIDIA00 +0008 +00008 +000 +2008 +2800000 +00080 +9008 +090000 +000 +9000 +200 +2000 +000000 +000 +00008 +Q000 +00000 +0000 +0000 +800008 +00 +20000000 +09008 +000 +9000 +8000000 +00000 +9000 +000660 +0008 +00008 +0000 +0000000 +88 +00000 +000 +2000 +0000000000o +000 +00 +8800008 +8000 +00008 +000 +00000 +0000 +0008 +U +00 +00 +88 +0000 +0000 +88 +00 +8000 +0000 +00000 +60. +000 +000 +o +00 ++ +0606 +00000 +000 +800080 +20000 +200000 +60000 +000 +8 +000 +9000 +000 +o0 +000 +0000 +0 +2900000 +: +00 +0000 +900 +00 +88 +900000 +00 +00.00 +o +000 +00000 +o +0090888800 +000 +000 +00 +000 +000 +008 +8000 +8 +600 +80 +00 +0.900 +00000000000 +0.00 +5000 +0000 +88 +00 +0000 +000 +00000 +000 +000 +o +088888 +00 +0000000 +0_00 +000000 +000 +0000 +0000000000 +0000 +00 +0000 +00 +00000 +00000 +0000 +0000 +88 +o00 +00 +80008 +o +00000000 +00 +900 +9000 +o +0000 +2000 +60000000.000 +900000 +0000000 +00000 +.000000 +00 +0009 +0000 +00 +0000 +00.001.0 +0.8 +0.6 +0.4 +0.2 +0.0800 +000000 +000000 +0.00000 +00000000 +o +00 +009. +88 +50000000 +0000000 +9000 +00 +000 +00 +000 +o0 +000 +o +0000000o +000 +0000 +88 +00 +900 +00 +00 +o +0000 +000 +00000 +000 +0000000Window Shape +Mean L2 +Veh. L2 +Ped. L2 +Cyc. L2 +(mAPH) +(mAPH) +(mAPH) +(mAPH) +6×6 +61.1 +62.9 +57.0 +63.4 +9×9 +61.7 +63.3 +57.9 +63.9 +13×13 +61.3 +63.3 +57.9 +62.9 +Table 4. FlatFormer is not sensitive to the choice of window shapes. +Group Size +Mean L2 +Veh. L2 +Ped. L2 +Cyc. L2 +(mAPH) +(mAPH) +(mAPH) +(mAPH) +81×50% +60.7 +62.7 +56.7 +62.7 +81×85% +61.7 +63.3 +57.9 +63.9 +81×125% +60.9 +63.1 +57.0 +62.5 +Table 5. Choosing a group size that is slightly smaller than the +window shape (9×9) provides the best accuracy on Waymo. +Grid Resolution +Mean L2 +Veh. L2 +Ped. L2 +Cyc. L2 +(mAPH) +(mAPH) +(mAPH) +(mAPH) +0.36m +60.7 +63.0 +56.7 +62.4 +0.32m +61.7 +63.3 +57.9 +63.9 +0.28m +61.7 +63.1 +57.8 +64.1 +Table 6. Ablation on input resolution in FlatFormer: 0.32m×0.32m +is the best design choice that balances efficiency and accuracy. +not guarantee the windows to be geometrically regular as in +SST [19], FlatFormer still consistently outperforms SST in +all three classes. +Window Shape. +FlatFormer achieves robust performance +under different window shapes. We choose the window +shape of 9×9 (2.88m×2.88m, which is the size of a vehicle) +in all experiments according to the results in Table 4, where +we always fix the group size to be 85% of the window shape. +Group Size. +We further study the choice of group sizes +in Table 5. We fix the window shape to be 9×9 according +to the results in Table 4 and sweep the group size in 50%, +85% and 125% of the window shape. The results show that +setting group size to be 85% of the window shape gives the +best performance. Intuitively, if the group size is too small, +FlatFormer will not be able to have a large enough receptive +field (e.g., group size = 1, FWA will degenerate to MLP). +When the group size is too large (say, group size = the entire +point cloud), there will be a large number of outliers within +each group and FlatFormer will behave like a global PCT, +which is not desired. +Input Resolution. +From Table 6, 0.32m×0.32m input res- +olution is the sweet spot in the latency-accuracy tradeoff for +FlatFormer while further increasing the input size will only +hurt the efficiency with no performance improvements. +Baseline +Flash Attn. +Packed MM +Efficient FFN +Drop Res. +CenterPoint +0 +5 +10 +15 +20 +1.7x faster +Backbone Latency (ms) +2.2x faster +2.9x faster +2.7x faster +Figure 8. Improvement breakdown for system optimizations. We +accelerate the backbone latency of FlatFormer by 2.9×, making it +2.3× faster than CenterPoint. +5.3.3 +Breakdown for System Optimizations +In Figure 8, we analyze the effectiveness of our system +optimizations proposed in Section 4.3. An efficient MHSA +implementation (FlashAttention) brings 1.7× improvement +to our latency. Note that FlashAttention does not improve +the latency of SST. This is because SST requires attention +masks to deal with varied group sizes, which is harder to be +optimized on GPUs. Packing the computation for Q, K, V +in a single linear kernel results in 1.3× speedup. Fusing the +linear and activation layers (in FFN) brings another 1.2× +speedup. Finally, dropping the non-full window improves +our latency by 1.1×. To sum up, our system optimizations +improve the latency of our FlatFormer by 2.9×, making its +backbone 2.3× faster than CenterPoint [92]. +Discussions. +CenterPoint is backed by SpConv [89], a +highly-optimized sparse convolution inference library built +upon CUTLASS [30]. Nevertheless, our FlatFormer can still +achieve the best efficiency on NVIDIA GPUs. We partially +attribute our efficiency advantage to the equally-sized groups +in FlatFormer which not only gives us the best computation +regularity but also eliminates the computation overhead. Sp- +Conv, on the other hand, implements 3D sparse convolution +with a masked implicit GEMM algorithm, which inevitably +introduces computation overhead when points within one +thread block do not have exactly the same neighbor patterns. +As such, FlatFormer can beat sparse convolutional models +on GPUs despite their heavy system optimizations. +6. Conclusion +This paper introduces FlatFormer to bridge the huge effi- +ciency gap between point cloud transformers and sparse con- +volutional models. It partitions the point cloud with equal- +size grouping rather than equal-window grouping, trading +spatial proximity for computational regularity. FlatFormer +achieves state-of-the-art accuracy on Waymo Open Dataset +with 4.6× speedup over previous point cloud transformers. +We hope that FlatFormer can inspire future research on de- +signing efficient and accurate point cloud transformers. + +Acknowledgement. +We would like to thank Tianwei Yin, +Lue Fan and Ligeng Mao for providing detailed results of +CenterPoint, SST/FSD and VoTr, and Yue Wang and Yukang +Chen for their helpful discussions. This work was supported +by National Science Foundation, MIT-IBM Watson AI Lab, +NVIDIA, Hyundai and Ford. 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In ICLR, 2021. 2 + diff --git a/ddFAT4oBgHgl3EQf6h5C/content/tmp_files/load_file.txt b/ddFAT4oBgHgl3EQf6h5C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..07d8c4aa32e89d3f4340916916f9e7c0309958b7 --- /dev/null +++ b/ddFAT4oBgHgl3EQf6h5C/content/tmp_files/load_file.txt @@ -0,0 +1,1050 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf,len=1049 +page_content='FlatFormer: Flattened Window Attention for Efficient Point Cloud Transformer Zhijian Liu1,∗ Xinyu Yang1,2,∗ Haotian Tang1 Shang Yang1,3 Song Han1 1MIT 2Shanghai Jiao Tong University 3Tsinghua University Abstract Transformer, as an alternative to CNN, has been proven effective in many modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', texts and images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For 3D point cloud transformers, existing efforts focus primarily on pushing their accuracy to the state-of-the-art level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' However, their latency lags behind sparse convolution-based models (3× slower), hindering their usage in resource-constrained, latency-sensitive applications (such as autonomous driving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This inefficiency comes from point clouds’ sparse and irreg- ular nature, whereas transformers are designed for dense, regular workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This paper presents FlatFormer to close this latency gap by trading spatial proximity for better com- putational regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We first flatten the point cloud with window-based sorting and partition points into groups of equal sizes rather than windows of equal shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This effec- tively avoids expensive structuring and padding overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We then apply self-attention within groups to extract local features, alternate sorting axis to gather features from differ- ent directions, and shift windows to exchange features across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer delivers state-of-the-art accuracy on Waymo Open Dataset with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6× speedup over (transformer- based) SST and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× speedup over (sparse convolutional) CenterPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This is the first point cloud transformer that achieves real-time performance on edge GPUs and is faster than sparse convolutional methods while achieving on-par or even superior accuracy on large-scale benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Code to reproduce our results will be made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Introduction Transformer [73] has become the model of choice in nat- ural language processing (NLP), serving as the backbone of many successful large language models (LLMs) [2,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Recently, its impact has further been expanded to the vision community, where vision transformers (ViTs) [18, 44, 72] have demonstrated on-par or even superior performance com- pared with CNNs in many visual modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', image and video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3D point cloud, however, is not yet one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' ∗ indicates equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Mean mAPH L2 64 66 68 70 72 74 Latency (ms) 0 13 26 39 52 65 FlatFormer (Ours) CenterPoint SST (Center) SST VoxSet CenterPoint++ 4x faster 3x faster Transformer Convolution Point Cloud Transformers Sparse Convolutional Models Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Previous point cloud transformers (⋆) are 3-4× slower than sparse convolution-based models (•) despite achieving similar detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer is the first point cloud transformer that is faster than sparse convolutional methods with on-par accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Latency is measured on an NVIDIA Quadro RTX A6000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Different from images and videos, 3D point clouds are intrinsically sparse and irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Most existing point cloud models [92] are based on 3D sparse convolution [24] which computes convolution only on non-zero features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' They re- quire dedicated system support [14,69,89] to realize high utilization on parallel hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Many efforts have been made toward point cloud trans- formers (PCTs) to explore their potential as an alternative to sparse convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Global PCTs [25] benefit from the reg- ular computation pattern of self-attention but suffer greatly from the quadratic computational cost (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' the number of points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Local PCTs [49,96] apply self-attention to a local neighborhood defined in a similar way to point-based mod- els [56] and are thus bottlenecked by the expensive neighbor gathering [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' These methods are only applicable to single objects or partial indoor scans (with <4k points) and cannot be efficiently scaled to outdoor scenes (with >30k points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Inspired by Swin Transformer [44], window PCTs [19] compute self-attention at the window level, achieving im- pressive accuracy on large-scale 3D detection benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='08739v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='CV] 20 Jan 2023 Despite being spatially regular, these windows could have drastically different number of points (which differ by more than 80×) due to the sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This severe imbalance results in redundant computation with inefficient padding and par- titioning overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As a result, window PCTs can be 3× slower than sparse convolutional models (Figure 1), limiting their applications in resource-constrained, latency-sensitive scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', autonomous driving, augmented reality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This paper introduces FlatFormer to close this huge la- tency gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Building on top of window PCTs [19], FlatFormer trades spatial proximity for better computational regularity by partitioning 3D point cloud into groups of equal sizes rather than windows of equal shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It applies self-attention within groups to extract local features, alternates sorting axis to aggregate features from different orientations, and shifts windows to exchange features across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Benefit from the regular computation pattern, FlatFormer achieves 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6× speedup over (transformer-based) SST and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× speedup over (sparse convolutional) CenterPoint while delivering the state-of-the-art accuracy on Waymo Open Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To the best of our knowledge, FlatFormer is the first point cloud transformer that achieves on-par or superior accuracy than sparse convolutional methods with lower latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It is also the first to achieve real-time performance on edge GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' With better hardware support for transformers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', NVIDIA Hopper), point cloud transformers will have a huge potential to be the model of choice in 3D deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We believe our work will inspire future research in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Also, we will make the code available to facilitate reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Related Work Deep Learning on Point Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Early research converts point clouds from 3D sensors to dense voxel grids and ap- plies 3D CNNs [15,50,55,85] on the volumetric inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' How- ever, the compute and memory consumption of volumetric CNNs grows cubically w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' to the input resolution, limiting the scalability of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To overcome this bottleneck, later research [27,34,54,56,58,71,81,83] directly performs feature extraction on point sets, while [59, 74, 75] convert point clouds to octrees and [11,14,24,42,89] performs sparse convolution on sparse voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Recently, researchers also ex- plore point+voxel [46–48,87] or point+sparse voxel [48,62, 63,70] hybrid representations for efficient 3D deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3D Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Extensive attention has been paid to 3D object detection [3,23,68], a crucial task for the percep- tion systems of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Early research [53,82] generates object proposals on 2D images and refines the predictions in the lifted 3D frustums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' VoxelNet [99] leads another stream of research that directly detects 3D ob- ject without 2D proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Similar to VoxelNet, PointPil- lars [33], SECOND [89,101], 3DSSD [90] and MVF [98] are single-stage detectors using anchor heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Center- Point [92,94], AFDet [22,28], HotspotNet [5], MVF++ [57], RangeDet [21], PolarStream [6], ObjectDGCNN [80], Pillar- Net [61], LiDARMultiNet [91] are single-stage anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' PointRCNN [64], Fast PointRCNN [12], Part- A2Net [65], PVRCNN [62,63], LiDAR R-CNN [38], Cen- terFormer [100], FSD [20], MPPNet [8] add a second stage that refines the proposals from the region proposal network (RPN) in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' There are also recent explorations on multi-sensor [1,7,10,37,41,45,60,93] 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Vision Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Motivated by the huge success of transformers [17, 73] in natural language processing, re- searchers start to adapt transformers to various vision tasks recently [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' ViT [32] first demonstrates that an image can be viewed as 16×16 words and processed by multi- head attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' DeiT [72] further shows that ViTs can be trained in a data-efficient manner without pretrain- ing on JFT [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' T2T-ViT [95], Pyramid ViT [76,77] and CrossFormer [78] introduce hierarchical modeling capabil- ity to ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Swin Transformer [43,44] limits self-attention computation to non-overlapping windows and enables cross- window information exchange via window shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' There are also task-specific ViTs such as ViTDet [36] for object detection, SETR [97] and SegFormer [86] for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Instead of adopting a fully-attentional backbone, another line of research, such as DETR [4], Deformable DETR [102], MaskFormer [13], PanopticSegFormer [40], DETR3D [79], BEVFormer [39], apply self-attention only to the task-specific heads and still uses CNNs for the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Point Cloud Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Recently, fully-attentional architectures begin to emerge in point cloud deep learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Similar to ViT, PCT [25] calculates self-attention glob- ally on the entire point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' PCT falls short in scala- bility since its computation comlexity scales quadratically as number of input points grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' PointASNL [88], Point- Transformer [84, 96], Fast Point TransFormer [52], Point- Former [51], VoTr [49], VoxSet [26] applies transformer- based architecture on the local neighborhood of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The efficiency of these local point cloud transformers is lim- ited by neighborhood query and feature restructuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Most related to our work is the window-based point cloud trans- former, SST [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Inspired by Swin Transformer, it projects the point cloud into bird’s eye view and divides the BEV space into non-overlapping windows with the same spatial sizes (but different number of points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Window shifting is used to communicate information across windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' SST suf- fers from large computation in window partition and padding overhead due to regional grouping, and achieves only one- sixth utilization compared with sparse convolutional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In this paper, we only refer to those methods that adopt a transformer-based architecture in the backbone as point cloud transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As CenterFormer [100], FUTR3D [9] and UVTR [35] apply sparse convolutional backbones and only use transformer in their detection heads, we still cate- gorize them as sparse convolutional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Why are Point Cloud Transformers Slow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Although point cloud transformers (PCTs) start to catch up with the accuracy of sparse convolutional detectors, there is still a 3× latency gap between the fastest PCT (SST [19]) and sparse convolutional CenterPoint [92] (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In this section, we dissect the efficiency bottleneck of PCTs, which lays a solid foundation for our FlatFormer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Global PCTs 1k 2k 4k 8k 16k 32k 64k Number of Points 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 ms Outdoor Scene Latency (ms) 967 ms Single Object 1 10 1k Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Latency of global PCTs scale quadratically with respect to number of input points and cannot scale up to outdoor scenes Inspired by ViT [32], the most simple and straightforward design for transformers on point cloud is global PCTs [25], where each point is a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' They leverage multi-head self- attention (MHSA) [73] globally across the entire point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' While being effective on small-scale 3D objects, global PCTs fall short in scaling to large-scale scenes due to its O(N 2D) complexity, where N is the number of tokens and D is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' From Figure 2, the runtime of global PCTs [25] grows quadratically as the number of input points grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For example, with 32k input points*, the model takes almost one second to execute on an NVIDIA A6000 GPU, 66× slower than CenterPoint [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Local PCTs PT (1 Layer) VoTr CenterPoint 0 15 30 45 60 Neighbor Preparation (Query & Gathering) MHSA 4x slower Latency (ms) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Local PCTs suffer from large neighborhood query and data restructuring overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Researchers have proposed local PCTs [49,51,52,84,88, 96] to solve the scalability issue of global PCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' They apply 32k is the number of points left after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m BEV projection in a single-frame Waymo [68] scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' MHSAs to the neighborhood of each point rather than the entire point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Hence, their computational complexity is O(NK2D), where N is the number of points, K is the number of neighbors for each point, and D is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As N ranges from 20k to 100k for real workloads and K is less than 64 for local PCTs, their theoretical cost is much lower than global PCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Local PCTs, however, suffer greatly from neighbor prepa- ration overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As point cloud is sparse and irregular, we have to first find the neighbors of each point, and then re- restructure the data from the N×D format to the N×K×D format on which MHSAs can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' These two steps are slow, taking 22 ms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', 36% of the total runtime) for VoTr [49] to execute for a single scene on Waymo, which is already slower than the entire CenterPoint model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For Point Transformer (PT) [96], the cost for preparing neighbors takes up to 70% of the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Such overhead in a single layer is already larger than the total runtime of CenterPoint!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Window PCTs CDF 0% 5% 10% 15% 20% 1 11 21 31 41 51 61 71 81 PDF Percentage Cum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Probability 0% 25% 50% 75% 100% Group Size (# of Points in Window) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In SST [19], the number of points in each window has a large variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' padding is necessary and leads to significant overhead for MHSA computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The great success of Swin Transformers [43,44] in vari- ous visual recognition tasks motivates the design of window PCTs, among which, SST [19] is a representative work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It first projects the point cloud into the bird’s-eye-view (BEV) space, then divides the BEV space into equally-shaped, non- overlapping windows, and applies MHSA within each win- dow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Similar to Swin Transformer, SST uses window shifting to enable information exchange across windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Different from images, point clouds are sparse and non- uniformly distributed over the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As a result, the number of point within each window is not the same and can differ by two orders of magnitude (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As the vanilla MHSA kernel cannot efficiently support variable sequence lengths, SST [19] batches windows with similar sizes together and pad all windows in each batch to the largest group size within the batch (Figure 5l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It then applies MHSA within each batch separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In practice, such padding introduces a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7× computation overhead on Waymo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Worse still,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' partitioning points to equal windows also introduce significant latency overhead: it takes 18 ms per scene on Waymo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' even slower \\ A C E H B I G F D C A E H B G I F D A C E H B I G F D Equal-Size Grouping (FlatFormer) C A E H B G I F D MHSA \\ \\ MHSA MHSA MHSA MHSA MHSA A C E H B I G F D C A E H B G I F D MHSA MHSA Batched Equal-Window Grouping (SST) (Window Shape: 2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Group Size: 3) Window & Sorting Axis: A C E H B I G F D C 0 H B G A E I F D 0 (Window Shape: 2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Group Size: Varied) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer partitions the point cloud into groups of equal sizes (right), rather than windows of equal shapes (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This effectively trades spatial proximity for better computational regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It then applies self-attention within each group to extract local features, alternates sorting axis to aggregate features from different directions, and shift windows to exchange features across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' than the total runtime of CenterPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To sum up, the padding and partitioning overheads make SST less hardware-friendly compared with sparse convolutional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer With all the lessons learned in Section 3, we will design our point cloud transformer to be scalable and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Overview The basic building block of FlatFormer is Flattened Win- dow Attention (FWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As in Figure 5r, FWA adopts window- based sorting to flatten the point cloud and partitions it to groups of equal sizes rather than windows of equal shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This naturally resolves the group size imbalance problem and avoids the padding and partitioning overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FWA then applies self-attention within groups to extract local features, alternates sorting axis to aggregate features from different orientations, and shifts windows to exchange features across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Finally, we provide an implementation of FWA that further improves its efficiency and minimizes the overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Flattened Window Attention (FWA) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 Sorting & Grouping Window-Based Sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' With a point cloud {(x, y)}†, we first quantize the coordinate of each point (x, y) to � ⌊x/wx⌋, ⌊y/wy⌋ � �� � window coordinates , x − ⌊x/wx⌋ · wx, y − ⌊y/wy⌋ · wy � �� � local coordinates within window � , (1) where (wx, wy) is the window shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Next, we sort all points first by window coordinates and then by local coordinates within the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This step turns the unordered point cloud into an ordered one, where points within the same window will be next to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' †We assume that the point cloud is in 2D for ease of notation, while our method applies to 3D or higher-dimension point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Equal-Size Grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Conventional window PCTs [19] will then group the points within the same window together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' However, as discussed in Section 3, each group can have drastically different numbers of points due to inherent spar- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To overcome the padding overheads, we partition the point cloud into groups of equal sizes based on the sorted sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This step allows the subsequent group attention to enjoy a perfectly regular workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' From the implementa- tion perspective, our grouping only involves a simple tensor reshaping (which is free since it does not change the layout) and is more efficient than window partitioning in SST [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Alternate Sorting Axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Between the two axes, x has a higher priority in sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Thus, points with identical ⌊x/wx⌋ will be next to each other while points with the same ⌊y/wy⌋ can be very far away from each other in the sorted sequence, breaking the geometric locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To solve this inequity, we alternate the sorting axis between x and y in different FWA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This is very similar to spatially separable convolution that decomposes a 3×3 kernel into 3×1 and 1×3 kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Stacking FWA blocks with different sorting axes enables the model to aggregate local features from different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Equal Size vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Equal Window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The key design choice we made is to partition the point cloud into groups of equal sizes rather than windows of equal shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' There is a trade-off: equal-window grouping maintains perfect spatial proximity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', each group has the same radius) but breaks the computa- tional regularity, while equal-size grouping ensures balanced computation workload (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', each group has the same number of points) but cannot guarantee the geometric locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 that computation regularity is more im- portant since spatial irregularity can be partially addressed by our algorithm design: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', window-based sorting offers a fairly good spatial ordering, and self-attention is robust to outliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', distant point pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 Group Attention With points partitioned, we then apply self-attention [73] within each group to extract local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For each group of points with coordinates C and features F, we follow the standard transformer block design: F′ = F + MHSA(LN(F), PE(C)), F′′ = F′ + FFN(LN(F′)), (2) where MHSA(·), FFN(·) and LN(·) correspond to multi-head self-attention, feed-forward layer, and layer normalization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Different from SST [19], PE(·) gives global ab- solute positional embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Here, we use the most standard softmax attention formulation for MHSA(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Our method will benefit from other more efficient attention variants, such as linear attention [29], which we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Window Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Benefit from the non-overlapping design, window-based attention typically has a larger receptive field than convolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', 69 neighbors in our FWA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' ≤27 neighbors in a sparse convolution of kernel size 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' However, its modeling power is limited as there is no feature exchange across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Similar to Swin Transformer [19,43,44], we adopt the shifted window approach that alternates the sorting configuration in consecutive FWA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Specifically, we translate the coordinates of all points by (wx/2, wy/2) for sorting in shifted FWA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This mechanism introduces cross-group feature communication while effectively main- taining workload independence between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Note that alternating sorting axis also enables feature exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Efficient Implementation Besides the algorithm design, we also provide an imple- mentation that improves the efficiency of MHSA and FFN and minimizes the sorting and masking overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All these optimizations are specialized for our point cloud transformer design and are not applicable to sparse convolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Efficient MHSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Within MHSA, query Q, key K, and value V will first be transformed with separate linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We pack these three linear projections into a batched matrix multiplication (since Q, K and V have the same shape in our FWA) to improve the parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In addition, standard atten- tion implementations materialize QKT and softmax(QKT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We leverage a recent efficient functional-preserving imple- mentation (FlashAttention [16]) that uses tiling to reduce the number of memory reads/writes, achieving better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Efficient FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FFN consists of two linear layers with a GELU activation in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We implement a fused linear kernel (in Triton) that absorbs the activation into the layer before to avoid writing the intermediate results to DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We also observe that our linear kernel (optimized by Triton) is even more efficient than cuBLAS, which is probably due to the unconventional tall-and-skinny workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Reuse Sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Sorting the coordinates of all points is a non-negligible overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As the coordinates remain identical (w/o downsampling), we reuse the sorting results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', ranks of each point) with the same axis and window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' In practice, this reduces the sorting overhead in our model by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Drop Residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The size of the input point cloud might not be divisible by the group size, generating a group with fewer points after partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This minor irregularity will still result in some overheads in self-attention since we need to introduce masking to correctly zero them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Instead, we directly drop the final non-full group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This only corresponds to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1% of all points, having a negligible impact on the model’s performance (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Setup Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We carry out our experiments on the large-scale Waymo Open Dataset (WOD) [68] with 1150 LiDAR point cloud sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Each sequence has 200 frames, collected by a 360◦ FoV LiDAR sensor at 10 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' There are four foreground classes, three of which (vehicles, pedes- trians and cyclists) are used for detection metric evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We follow the official metrics on Waymo to cal- culate the standard 3D mAP and heading-weighted 3D mAP (mAPH) of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The matching IoU thresholds for vehicle, pedestrian and cyclist are set to default values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Objects are divided into two difficulty levels, where objects with fewer than five laser points or annotated as hard are categorized into level 2 (L2) and other objects are defined as level 1 (L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We mainly report L2 metrics in the main paper and provide detailed metrics in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Based on FWA, we provide an instantiation of Flat- Former for 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We follow the design of PointPillars [33] to first voxelize the point cloud into sparse BEV pillars (with MLPs) at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We then apply eight consecutive FWA blocks with alternat- ing sorting axes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', x or y) and window shifting configura- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', on or off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All FWA blocks have a window shape of 9×9 and a group size of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Following SST [19], we do not apply any spatial downsampling, which is beneficial for small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Finally, we apply regular BEV encoder and a center-based detection head following CenterPoint [92,94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Main Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 Single-Stage Detectors Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We compare our FlatFormer with state-of-the-art sparse convolutional [61,92,94] and transformer-based [19, 26,49] single-stage 3D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All models apply anchor- or center-based detection heads [89, 92, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We compare models with different numbers of input frames separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' #Frames #MACs Latency Speedup Mean L2 Vehicle L2 Pedestrian L2 Cyclist L2 (G) (ms) (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' [92]) (mAPH) (mAP/APH) (mAP/APH) (mAP/APH) SECOND [89]3 1 – – – 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 / 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 PointPillars [33]3 1 – – – 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 / 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 CenterPoint [92]1 1 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0× 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 VoTr-SSD [49] 1 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2× – 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 / 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 – – SST [19]2 1 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 SST-Center [19] 1 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 VoxSet [26] 1 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 PillarNet [61] 1 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 / 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 FlatFormer (Ours) 1 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 CenterPoint [92]1 2 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0× 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 PillarNet [61] 2 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 / 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 FlatFormer (Ours) 2 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 CenterPoint [92] 3 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0× – – – – CenterPoint++ [94]1 3 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 SST [19]2 3 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 SST-Center [19]† 3 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5× 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 / 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 FlatFormer (Ours) 3 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 / 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Results of single-stage 3D detectors on Waymo Open Dataset (validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× speedup over CenterPoint and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6× speedup over SST while being more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We refer the readers to the appendix for detailed metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', L1 mAP/mAPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Markers ◦ and • refer to sparse convolutional models and point cloud transformers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Methods with <60 L2 mAPH are marked gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' (†: reproduced by us, 1: from CenterPoint authors, 2: from SST authors, 3: from FSD paper, *: projected latency) #Frames Latency Mean L2 (ms) (mAPH) LiDAR R-CNN [38]† 1 – 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 PV-RCNN [62]† 1 – 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 Part-A2 [66]† 1 – 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 PV-RCNN++ [63]† 1 – 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 CenterFormer [100] 1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 FSD-SpConv [20] 1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 FlatFormer+FSD (Ours) 1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 CenterFormer [100] 2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 FlatFormer+FSD (Ours) 2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 CenterFormer [100] 4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 MPPNet [8] 4 – 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 FlatFormer+FSD (Ours) 3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Results of two-stage 3D detectors on Waymo Open Dataset (validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer achieves on-par or even higher accu- racy compared with sparse convolutional two-stage detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We refer the readers to the appendix for detailed metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', per- class L1/L2 mAP/mAPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Markers ◦ and • refer to SpConv-based models and point cloud transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' (†: from FSD paper) Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We measure the latency on an NVIDIA Quadro RTX A6000 GPU using FP16 precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We adopt SpConv v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 [89], the state-of-the-art 3D sparse convolution library, to execute the 3D encoder of all sparse convolutional detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For transformer-based detectors, we use their official implementation to measure the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All modules after the 3D encoder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', BEV encoder and detection head) are executed with TensorRT 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We execute all the methods on the first 1,000 samples for 50 runs (with 10 warmup runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We report the average latency (with outliers excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We do not include the data loading and post-processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As in Table 1, our FlatFormer achieves consistent performance improvements over both sparse convolutional and transformer-based detectors with much better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For one-frame models, FlatFormer is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2×, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7× faster than SST, SST-Center and the recent VoxSet [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It also compares favorably with strong sparse convolutional baselines: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× faster than CenterPoint with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 higher L2 mAPH and performs on par with PillarNet [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The accu- racy advantage magnifies in the two-frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Specifi- cally, FlatFormer is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4× faster than CenterPoint with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 L2 mAPH higher accuracy, and outperforms PillarNet by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 L2 mAPH with a similar latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' With three input frames, Flat- Former is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6× and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2× faster than SST and SST-Center, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It also achieves better latency-accuracy trade- off (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1× faster and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4% higher accuracy) compared with CenterPoint++ [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Remarkably, FlatFormer requires 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7× more MACs than CenterPoint++ while it is still faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This indicates that our design is more hardware-friendly than the sparse convolutional baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We deploy our FlatFormer on an NVIDIA Jetson AGX Orin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This is a resource-constrained edge GPU CenterPoint SST SST (Center) FlatFormer 0 5 10 15 20 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 Frames Per Second (FPS) Real Time (>10 FPS) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Measured latency on NVIDIA Jetson AGX Orin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Flat- Former is the first point cloud transformer that achieves real-time performance on edge GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' platform that is widely used in real-world self-driving cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' From Figure 6, FlatFormer runs at 16 FPS, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2× faster than CenterPoint [92] and 3× faster than SST-Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To the best of our knowledge, FlatFormer is the first point cloud transformer that achieves real-time inference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', >10 FPS, which is the LiDAR sensor frequency) on edge GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We believe that it paves the way for efficient LiDAR-centric perception in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 Two-Stage Detectors Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To verify the generalizability, we replace the 3D backbone in FSD [20], a state-of-the-art two-stage detector, and compare its results with previous two-stage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We keep the same grid resolution, window shape and group size as in our single-stage experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Baseline & Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We compare our model against state- of-the-art two-stage detectors in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We follow the same latency measurement protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For CenterFormer [100], we adapt the official implementation to support SpConv v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 backend in FP16 precision for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All existing high-performing two-stage detectors are sparse convolutional, while our FlatFormer is the only transformer-based method that achieves state-of-the-art level accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It also shows better scalability with respect to the number of input frames compared with CenterFormer [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Note that our paper focuses on optimizing the latency of 3D backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' However, two-stage detectors [20, 100] are usually bottlenecked by the second stage in runtime, which is out of our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We expect that the latency of FlatFormer could benefit from a more efficient second-stage design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Analysis In this section, we present analyses to validate the effec- tiveness of our design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' All experiments are based on our single-frame model trained with 20% data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 Flattened Window Attention In Figure 7, we visualize the learned attention weights in our FWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The color represents the scale of attention weights, (a) moving vehicles (b) turning vehicles High (c) parked vehicles Low Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Visualization of attention weights in FlatFormer for vehicles that are moving straight ahead, turning and parking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' High attention weights corresponds to the detected object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Sorting Strategy Mean L2 Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Ped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Cyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 (mAPH) (mAPH) (mAPH) (mAPH) Ours 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 w/o Quantization 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 w/o Axis Alternation 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 w/o Window Shift 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 Random Order 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 SST [19] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Window-based sorting in FlatFormer provides even better performance than equally-shaped window partition in SST [19] and outperforms other sorting strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' where warmer color means larger attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Black points correspond to query points, and gray points are those with weights smaller than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' For vehicles moving straight ahead, turning and parked, query points on the vehi- cle are always highly attended to nearby points on the same car, while faraway points have very small learned attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Such an observation can partially explain the ef- fectiveness of FWA: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', even if equal-size grouping does not create spatially regular windows, the model can learn to suppress the importance of outlier points in the background and focus on important foreground points within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 Ablation Studies on Model Design Sorting Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We first analyze the effectiveness of our window-based, axis-alternating sorting strategy in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Randomly grouping points together without any spatial sort- ing will give ∼ 4% worse performance compared with Flat- Former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Furthermore, due to spatial discontinuities on the boundary regions, directly sorting the points by xyz coor- dinates or window sorting along single axis both provide sub-optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We also notice that window shifting brings about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5% improvement to the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Interestingly, despite the fact that our sorting strategy does DVIDIA00 0008 00008 000 2008 2800000 00080 9008 090000 000 9000 200 2000 000000 000 00008 Q000 00000 0000 0000 800008 00 20000000 09008 000 9000 8000000 00000 9000 000660 0008 00008 0000 0000000 88 00000 000 2000 0000000000o 000 00 8800008 8000 00008 000 00000 0000 0008 U 00 00 88 0000 0000 88 00 8000 0000 00000 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 000 000 o 00 + 0606 00000 000 800080 20000 200000 60000 000 8 000 9000 000 o0 000 0000 0 2900000 : 00 0000 900 00 88 900000 00 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='00 o 000 00000 o 0090888800 000 000 00 000 000 008 8000 8 600 80 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='900 00000000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='00 5000 0000 88 00 0000 000 00000 000 000 o 088888 00 0000000 0_00 000000 000 0000 0000000000 0000 00 0000 00 00000 00000 0000 0000 88 o00 00 80008 o 00000000 00 900 9000 o 0000 2000 60000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='000 900000 0000000 00000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='000000 00 0009 0000 00 0000 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0800 000000 000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='00000 00000000 o 00 009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 88 50000000 0000000 9000 00 000 00 000 o0 000 o 0000000o 000 0000 88 00 900 00 00 o 0000 000 00000 000 0000000Window Shape Mean L2 Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Ped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Cyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 (mAPH) (mAPH) (mAPH) (mAPH) 6×6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 9×9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 13×13 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer is not sensitive to the choice of window shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Group Size Mean L2 Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Ped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Cyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 (mAPH) (mAPH) (mAPH) (mAPH) 81×50% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 81×85% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 81×125% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='5 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Choosing a group size that is slightly smaller than the window shape (9×9) provides the best accuracy on Waymo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Grid Resolution Mean L2 Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Ped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 Cyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' L2 (mAPH) (mAPH) (mAPH) (mAPH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='36m 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='28m 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Ablation on input resolution in FlatFormer: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m is the best design choice that balances efficiency and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' not guarantee the windows to be geometrically regular as in SST [19], FlatFormer still consistently outperforms SST in all three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Window Shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer achieves robust performance under different window shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We choose the window shape of 9×9 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='88m×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='88m, which is the size of a vehicle) in all experiments according to the results in Table 4, where we always fix the group size to be 85% of the window shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Group Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We further study the choice of group sizes in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We fix the window shape to be 9×9 according to the results in Table 4 and sweep the group size in 50%, 85% and 125% of the window shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' The results show that setting group size to be 85% of the window shape gives the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Intuitively, if the group size is too small, FlatFormer will not be able to have a large enough receptive field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=', group size = 1, FWA will degenerate to MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' When the group size is too large (say, group size = the entire point cloud), there will be a large number of outliers within each group and FlatFormer will behave like a global PCT, which is not desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Input Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' From Table 6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='32m input res- olution is the sweet spot in the latency-accuracy tradeoff for FlatFormer while further increasing the input size will only hurt the efficiency with no performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Baseline Flash Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Packed MM Efficient FFN Drop Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' CenterPoint 0 5 10 15 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7x faster Backbone Latency (ms) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2x faster 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9x faster 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7x faster Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Improvement breakdown for system optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We accelerate the backbone latency of FlatFormer by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9×, making it 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× faster than CenterPoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3 Breakdown for System Optimizations In Figure 8, we analyze the effectiveness of our system optimizations proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' An efficient MHSA implementation (FlashAttention) brings 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='7× improvement to our latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Note that FlashAttention does not improve the latency of SST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' This is because SST requires attention masks to deal with varied group sizes, which is harder to be optimized on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Packing the computation for Q, K, V in a single linear kernel results in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Fusing the linear and activation layers (in FFN) brings another 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='2× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Finally, dropping the non-full window improves our latency by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='1×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' To sum up, our system optimizations improve the latency of our FlatFormer by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='9×, making its backbone 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='3× faster than CenterPoint [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' CenterPoint is backed by SpConv [89], a highly-optimized sparse convolution inference library built upon CUTLASS [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Nevertheless, our FlatFormer can still achieve the best efficiency on NVIDIA GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We partially attribute our efficiency advantage to the equally-sized groups in FlatFormer which not only gives us the best computation regularity but also eliminates the computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Sp- Conv, on the other hand, implements 3D sparse convolution with a masked implicit GEMM algorithm, which inevitably introduces computation overhead when points within one thread block do not have exactly the same neighbor patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' As such, FlatFormer can beat sparse convolutional models on GPUs despite their heavy system optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Conclusion This paper introduces FlatFormer to bridge the huge effi- ciency gap between point cloud transformers and sparse con- volutional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' It partitions the point cloud with equal- size grouping rather than equal-window grouping, trading spatial proximity for computational regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' FlatFormer achieves state-of-the-art accuracy on Waymo Open Dataset with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content='6× speedup over previous point cloud transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We hope that FlatFormer can inspire future research on de- signing efficient and accurate point cloud transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' We would like to thank Tianwei Yin, Lue Fan and Ligeng Mao for providing detailed results of CenterPoint, SST/FSD and VoTr, and Yue Wang and Yukang Chen for their helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} +page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFAT4oBgHgl3EQf6h5C/content/2301.08739v1.pdf'} diff --git a/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/2301.11700v1.pdf.txt b/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/2301.11700v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..80f7c9eba3cc140dda230b79f479a0def5e86f27 --- /dev/null +++ b/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/2301.11700v1.pdf.txt @@ -0,0 +1,1437 @@ +arXiv:2301.11700v1 [math.DG] 27 Jan 2023 +The Schwarzian derivative and the degree +of a classical minimal surface +thomas mettler and lukas poerschke +Abstract. Using the Schwarzian derivative we construct a sequence +{Pℓ}ℓ⩾2 of meromorphic differentials on every non-flat oriented minimal +surface in Euclidean 3-space. +The differentials {Pℓ}ℓ⩾2 are invariant +under all deformations of the surface arising via the Weierstrass rep- +resentation and depend on the induced metric and its derivatives only. +A minimal surface is said to have degree n if its n-th differential is a +polynomial expression in the differentials of lower degree. We observe +that several well-known minimal surfaces have small degree, including +Enneper’s surface, the helicoid/catenoid and the Scherk – as well as the +Schwarz family. Furthermore, it is shown that locally and away from +umbilic points every minimal surface can be approximated by a sequence +of minimal surfaces of increasing degree. +1. Introduction +In [2], a meromorphic quadratic differential P2 is introduced on every non- +flat oriented minimal surface M in Euclidean 3-space E3. The differential P2 +arises as a conservation law for a certain curvature entropy functional and is +hence called the entropy differential. In this note we show that P2 is the first +element in a geometrically natural sequence {Pℓ}ℓ⩾2 of meromorphic differen- +tials on M, where Pℓ has degree ℓ, that is, Pℓ is a section of the ℓ-th tensorial +power of the canonical bundle of M. The entropy differentials {Pℓ}ℓ⩾2 arise +as certain higher order Schwarzian derivatives of the stereographically pro- +jected Gauss map G : M → C ∪ {∞}; where we compute the Schwarzian +derivative and its higher order descendants with respect to the Levi-Civita +connection of the flat metric +√ +−Kg. Here g denotes the induced metric +on M and K ⩽ 0 its Gauss curvature. The differentials {Pℓ}ℓ⩾2 satisfy the +following properties: +Theorem 1.1. Let X : M → E3 be a non-flat minimal immersion of the +oriented surface M. Denoting by {Pℓ}ℓ⩾2 the induced differentials, we have: +(i) the differential Pℓ is holomorphic away from umbilic points and ex- +tends meromorphically to all of M with a pole of order ℓ at the umbilic +points; +(ii) the differential Pℓ depends on the induced metric g and its derivatives +up to order (ℓ+4) only. In particular, the differentials {Pℓ}ℓ⩾2 are the +same for all members of the associated family of minimal immersions +of X : M → E3; +Date: 27th January 2023. +1 + +2 +T. METTLER AND L. POERSCHKE +(iii) the differentials {Pℓ}ℓ⩾2 are invariant under Goursat transforms of +the minimal immersion X : M → E3; +(iv) the differentials {Pℓ}ℓ⩾2 are invariant under (constant) rescaling of +X : M → E3. +Having the infinite sequence {Pℓ}ℓ⩾2 at hand, we may say that a minimal +immersion X : M → E3 has degree n if Pn vanishes identically or if Pn is a +polynomial expression in the lower order differentials, that is, +Pn = ψn(P2, . . . , Pn−1), +where ψn is a weighted-homogeneous polynomial of degree n which we call +the algebraic type of the immersion. We observe that several classical min- +imal surfaces are surfaces of low degree. Using [2], it follows that – up to +Euclidean motion and the deformations listed in Theorem 1.1 – open sub- +sets of Enneper’s surface are the only surfaces of degree 2 and open subsets +of the helicoid and catenoid are the only surfaces of degree 3. A complete +description of degree four surfaces is not feasible. However, we prove that if +a degree four minimal surface admits an umbilic point, then it satisfies an +equation of the form P4 + an(P2)2 = 0, where +an = 12(n + 2)2 +(3n2 + 4n), +n ∈ N. +The equation P4 + an(P2)2 = 0 is satisfied by the Enneper surfaces with +higher dihedral symmetry group Dn+1. Furthermore, the sequence an con- +verges to 4 as n goes to infinity and we show that at this limit coefficient there +exists a 1-parameter family of immersed singly periodic degree four minimal +deformations of the plane which keep the lines +� +z = 0, y = kπ +2 | k ∈ Z +� +fixed +and so that the induced metric admits a Killing vector field that is the real +part of a holomorphic vector field. Likewise, a degree five minimal surface +admitting an umbilic point satisfies P5 + 2anP3P2 = 0. We show that the +smallest admissible coefficient 216/7 is realized by the Schwarz family (includ- +ing the P- and D-surface and the Gyroid). Furthermore, the limit coefficient +8 is realized by the Scherk family. Also, at degree seven we encounter the +k-noids of Jorge & Meeks [9]. +We also prove that locally and away from umbilic points a minimal surface +can be approximated by a sequence of minimal surfaces of increasing degree. +Theorem 1.2. Let X : M → E3 be a non-flat minimal immersion of an +oriented surface M. Then for every point p in the complement M′ of the +umbilic locus there exists a neighbourhood Up and a sequence of minimal +immersions Xn : Up → E3 with Xn having degree n so that limn→∞ ∥Xn − +X∥gEucl = 0 locally uniformly. +The Schwarzian derivative arises most naturally in the context of project- +ive differential geometry. In Appendix A we discuss the relation between +the definition of the Schwarzian derivative used in this note and the usual +definition of projective differential geometry. + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +3 +Acknowledgements +This article is based on the doctoral thesis of L.P.. The authors were par- +tially supported by the priority programme SPP 2026 “Geometry at Infinity” +of DFG. The authors are grateful to Jacob Bernstein for several helpful dis- +cussions regarding the content of this article. +2. Preliminaries +2.1. Weyl connections +Let (M, [g]) be an oriented surface equipped with a conformal structure +[g]. Recall that a Weyl connection for [g] (or [g]-conformal connection) is a +torsion-free connection ∇ on TM preserving [g], that is, its parallel transport +maps are angle preserving with respect to the conformal structure [g]. A +torsion-free connection ∇ on TM preserves [g] if and only if for some – and +hence any – Riemannian metric g ∈ [g], there exists a 1-form β ∈ Ω1(M), so +that +(2.1) +∇g = 2β ⊗ g. +Conversely, it follows from Koszul’s identity that for every pair (g, β) there +exists a unique affine torsion-free connection satisfying (2.1), which is +(g,β)∇ = g∇ + g ⊗ β# − β ⊗ Id − Id ⊗ β, +where g∇ denotes the Levi-Civita connection of the metric g and β# denotes +the g-dual vector field to β. Note that for every smooth function u on M, the +pair (e2ug, β + du) determines the same Weyl connection as the pair (g, β) +(exp(2u)g,β+du)∇ = (g,β)∇, +as can easily be verified using the identity (c.f. [3, Theorem 1.159]) +e2ug∇ = g∇ − g ⊗ g∇u + du ⊗ Id + Id ⊗ du. +We will use the notation [g]∇ to denote a Weyl connection for [g] and simply +write ∇ when the conformal structure is clear from the context. +Let J denote the complex structure on M induced by [g] and the ori- +entation. +Clearly, a torsion-free connection preserves [g] if and only if it +preserves J and we may therefore also think of the Weyl connections for [g] +as torsion-free J-linear connections, that is, torsion-free connections on TM +whose parallel transports maps are J-linear. Consequently, a Weyl connec- +tion ∇ induces connections on all tensorial powers of the canonical bundle +L = T 1,0M∗ of (M, J). By standard abuse of notation, we will denote these +connections by ∇ as well, so that for all ℓ ∈ Z we have first order differential +operators +∇ : Γ(Lℓ) → Ω1(M, Lℓ), +sending smooth sections of Lℓ to Lℓ-valued 1-forms on M. Since Lℓ is a +complex vector bundle, the Lℓ-valued 1-forms on M decompose +Ω1(M, Lℓ) = Ω1,0(M, Lℓ) ⊕ Ω0,1(M, Lℓ) + +4 +T. METTLER AND L. POERSCHKE +into (1,0) and (0,1) parts. Note that we may canonically identify Ω1,0(M, Lℓ) +with Γ(Lℓ+1) so that the (1,0) part of ∇ may be thought of as a differential +operator +∇1,0 : Γ(Lℓ) → Γ(Lℓ+1). +For a smooth section σ ∈ Γ(Lℓ), we write σ′ = ∇1,0σ as well as σ′′ = (σ′)′ +and likewise for higher order derivatives. +For what follows it will be convenient to have local coordinate expres- +sions for the differential operators ∇1,0. To this end let z : U → C be a +local holomorphic coordinate on M. Extending ∇ complex-linearly to the +complexified tangent bundle TM ⊗ C, it follows from the J-linearity of ∇ +that +(2.2) +∇∂z∂¯z = 0 +and furthermore that there exists a unique complex-valued function γ on U +such that +(2.3) +∇∂z∂z = γ ∂z. +Conversely, a torsion-free connection satisfying (2.2) and (2.3) in every holo- +morphic coordinate system is J-linear and hence a Weyl connection. +Suppose σ : U → Lℓ is a smooth section, we write σ = s dzℓ for some +smooth complex-valued function s on U. Then we have +(2.4) +∇1,0s dzℓ = (∂zs − ℓγs) dzℓ+1. +If w : V → C is another local holomorphic coordinate with U ∩ V ̸= ∅, then +writing σ = ˆsdwℓ and ∇∂w∂w = ˆγ ∂w for complex-valued functions ˆs, ˆγ on V , +we have +(2.5) +ˆs = (∂wz)ℓs, +ˆγ = (∂wz) γ + (∂zw) ∂2 +wwz, +on U ∩ V . Using (2.5) it is easy to check that the right hand side of (2.4) +does not depend on the chosen coordinates. The reader less familiar with +complex geometry may therefore take (2.4) as the definition of the operators +∇1,0. Note in particular that ∇1,0 agrees with the usual “del-operator” ∂ on +functions. +2.2. Higher order Schwarzian derivatives +Let (M, [g]) be a Riemann surface and ∇ a Weyl connection for [g]. Let +f be a holomorphic function on M satisfying f ′ = ∂f ̸= 0. We define the +Schwarzian derivative of f with respect to ∇ to be the quadratic differential +S∇(f) = f ′′′ +f ′ − 3 +2 +�f ′′ +f ′ +�2 +. +In a local holomorphic coordinate system z : U → C with ∇∂z∂z = γ ∂z for +some complex-valued function γ on U, we obtain with (2.4) +(2.6) +S∇(f) = {f, z} dz2 + +�1 +2γ2 − ∂zγ +� +dz2, +where +{f, z} = ∂3 +zzzf +∂zf +− +�∂2 +zzf +∂zf +�2 + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +5 +denotes the classical Schwarzian derivative of f with respect to the coordinate +z. Taking the Levi-Civita connection of the spherical metric +g1 = +� +2 +1 + |z|2 +�2 +dz ◦ d¯z +on the Riemann sphere ˆC gives +g1∇∂z∂z = − +2¯z +1 + |z|2 ∂z. +Consequently, we have γ = −2¯z/(1 + |z|2), so that +1 +2γ2 − ∂zγ = 0, +and (2.6) simplifies to give the classical coordinate definition of the Schwar- +zian derivative. +Recall that the classical definition of the Schwarzian de- +rivative extends to the set of locally injective meromorphic functions and +consequently by using (2.6), so does our definition. Furthermore, a locally +injective meromorphic function a defined on some domain Ω ⊂ ˆC satisfies +(2.7) +S +g1∇(a) = 0 +if and only if a is the restriction of a Möbius transformation. +A crucial +property of the Schwarzian derivative is that it defines a cocycle. If f is a +locally injective meromorphic function on M and a a local biholomorphism +on ˆC so that a ◦ f is well defined, then one may easily verify that +(2.8) +S∇(a ◦ f) = S∇(f) + f ∗ � +S +g1∇(a) +� +. +As a consequence of this identity and (2.7) we see that the Schwarzian de- +rivative is invariant under post-composition by a Möbius transformation. +The Schwarzian derivative is the first in a sequence of differential operators +enjoying invariance under post-composition by a Möbius transformation: +Definition 2.1. Let (M, [g]) be a Riemann surface equipped with a Weyl +connection ∇. For ℓ ⩾ 2 we define the ℓ-th Schwarzian derivative to be the +(ℓ + 1)-th order differential operator defined by +S∇ +ℓ+1(f) = +� +S∇ +ℓ (f) +�′ +where +S∇ +2 (f) = f ′′′ +f ′ − 3 +2 +�f ′′ +f ′ +�2 +. +The ℓ-th Schwarzian derivative maps a locally injective meromorphic function +f on M to a smooth section of Lℓ, that is, a smooth differential on M of +degree ℓ. +Remark 2.2. By construction, the operators S∇ +ℓ +are invariant under post- +composition by a Möbius transformation. However, none of the higher order +Schwarzian derivatives defines a cocycle. +Local coordinate definitions of +higher order Möbius invariant Schwarzian differential operators previously +appeared in [1] (see also [22]). +Every non-vanishing holomorphic quadratic differential Q on a Riemann +surface (M, [g]) gives rise to a flat [g]-conformal connection A∇ where A∇ +denotes the Levi-Civita connection of the flat Lorentzian metric A = Re(Q). + +6 +T. METTLER AND L. POERSCHKE +Indeed, in a local holomorphic coordinate z : U → C on M so that Q = qdz2 +for some holomorphic function q on U, we have A∇∂z∂¯z = 0 and +A∇∂z∂z = γ ∂z, +with +(2.9) +γ = ∂zq +2q , +showing that A∇ is [g]-conformal. If f is a locally injective meromorphic +function on U, we obtain with (2.4) +(2.10) +S +A∇ +2 +(f) = +� +{f, z} + 5 +8 +�∂zq +q +�2 +− 1 +2 +∂2 +zzq +q +� +dz2. +3. The entropy differentials and their invariance properties +3.1. The entropy differentials +Let M be an oriented smooth surface and X = (Xi) : M → E3 a smooth +immersion into Euclidean 3-space. Let g and A denote the induced metric +and second fundamental form on M +g = dX · dX, +A = −dN · dX, +where N = (Ni) : M → S2 ⊂ E3 denotes the (orientation compatible) +Gauss map of X. +The Weingarten shape operator is the endomorphism +S : TM → TM satisfying +A(X, Y ) = g(S(X), Y ) = g(X, S(Y )) +for all X, Y ∈ Γ(TM). The operator S is g-symmetric and hence (pointwise) +diagonalizable with real eigenvalues κ1, κ2, called the principal curvatures +of X. Recall that X is called minimal if its mean curvature H = 1 +2 tr S = +1 +2(κ1 + κ2) vanishes identically. A point p ∈ M where κ1 = κ2 is an umbilic +point, which – in the minimal case – amounts to κ1 = κ2 = 0. Consequently, +the umbilic points are precisely those points where the Gauss curvature K = +κ1κ2 = −(κ1)2 vanishes. The umbilic locus is the set +U = {p ∈ M | κ1(p) = κ2(p) = 0} +and we use M′ to denote its complement in M, i.e., M′ = M \ U. The +minimality of X is well-known to be equivalent to the meromorphicity of +the stereographically projected Gauss-map. +More precisely, we have the +following lemma whose proof may be found in [11] or most standard texts +on minimal surfaces. +Lemma 3.1. Let X : M → E3 be a smooth immersion with stereographically +projected Gauss map G = (N1+iN2)/(1−N3) : M → ˆC. Then X is minimal +if and only if G is meromorphic with respect to the complex structure on M +induced by g and the orientation. +Another key property of minimal surfaces is that the induced metric sat- +isfies the so-called Ricci condition. + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +7 +Theorem 3.2. The induced metric g of a minimal immersion X : M → E3 +has non-positive Gauss curvature K and whenever K < 0, the metric g +satisfies +(3.1) +∆ log(−K) = 4K. +Conversely, if (M, g) is a simply connected Riemannian 2-manifold of strictly +negative Gauss curvature satisfying (3.1), then (M, g) can be immersed iso- +metrically and minimally into E3. +Remark 3.3. Here ∆ denotes the Laplace-Beltrami operator with respect to +g. For a proof the reader may consult [5]. +An immediate consequence of (3.1) is the following result. +Corollary 3.4. Let g denote the induced metric of a minimal immersion +X : M → E3 and K its Gauss curvature. Then on M′ ⊂ M, the complement +of the umbilic locus, g0 = +√ +−Kg defines a flat metric. +Proof. The proof is an immediate consequence of the well-known and easily- +derived formula for the dependence of the Gauss curvature on a conformal +change of the metric. For u ∈ C∞(M), we have +Ke2ug = e−2u (Kg − ∆u) . +Consequently, on M′ we obtain +Kg0 = +1 +√ +−K +� +K − 1 +2∆ log +�√ +−K +�� += +1 +√ +−K +� +K − 1 +4∆ log (−K) +� += 0. +□ +Clearly, the Levi-Civita connection g0∇ of g0 is a [g]-conformal connection. +Combining Lemma 3.1 and Corollary 3.4 we immediately see that we obtain +a sequence of differentials Pℓ on the complement M′ of the umbilic locus of +a minimal immersion X : M → E3. Indeed, since the Gauss map is a locally +injective meromorphic function on M′, the differentials +Pℓ = S +g0∇ +ℓ +(G) +are well defined on M′. +On M′ we have another flat metric given by the second fundamental form. +Lemma 3.5. On M′ the Levi-Civita connection g0∇ of the flat Riemannian +metric g0 and the Levi-Civita connection A∇ of the flat Lorentzian metric A +agree. +Proof. Denote by J the complex structure on M induced by g and the orient- +ation. Then the Hopf differential Q = A + iAJ is a holomorphic quadratic +differential [8], where we write +AJ(X, Y ) = A(JX, Y ), +for X, Y ∈ Γ(TM). +In a neighbourhood of every point p ∈ M′ we may +find local holomorphic coordinates z = x + iy such that Q = dz2. We will +henceforth call such coordinates Q-adapted. In such a coordinate system we +have +g = e2u dz ◦ d¯z = e2u � +dx2 + dy2� +, +and +A = Re(dz2) = dx2 − dy2. + +8 +T. METTLER AND L. POERSCHKE +for some real-valued function u satisfying Liouville’s equation +(3.2) +4∂2 +z¯zu = e−2u. +It follows that g has Gauss curvature K = −e−4u and hence +g0 = +√ +−Kg = dx2 + dy2. +Therefore, with respect to the coordinate z, the Christoffel symbols of both +g0∇ and A∇ vanish identically. +Covering M′ with local holomorphic Q- +adapted coordinates implies that g0∇ = A∇. +□ +The differentials Pℓ have the property of being holomorphic away from +umbilic points and extending meromorphically across umbilic points. +Proposition 3.6. Let X : M → E3 be a non-flat minimal immersion with in- +duced differentials {Pℓ}ℓ⩾2 on the complement M′ of the umbilic locus. Then +the differentials {Pℓ}ℓ⩾2 are holomorphic on M′ and extend meromorphically +to all of M with Pℓ having a pole of order ℓ at the umbilic points. Furthermore, +the residue of Pℓ at an umbilic point p ∈ M is +Resp(Pℓ) = +� +−1 +2 +�ℓ+1 +(ℓ − 1)! (n + 2)ℓ−2(3n2 + 4n), +where n denotes the order of vanishing of the Hopf differential at p. +Remark 3.7. Suppose that P is a meromorphic differential of degree ℓ ∈ N +having a pole of order ℓ at some point p, so that there exists a local p-centred +holomorphic coordinate z satisfying +P(z) = f(z) +zℓ dzℓ, +where f is a holomorphic function near 0 with f(0) ̸= 0. Then the residue +of P at p is defined as +Resp(P) = f(0), +which is clearly well-defined, that is, independent of the choice of local p- +centred holomorphic coordinate. +Proof of Proposition 3.6. Let p ∈ M′ and let w : Up → C be a local holo- +morphic coordinate defined in a neighbourhood of p so that Q = dw2. There- +fore, in such a coordinate system (2.6) becomes +(3.3) +P2 = {G, w} dw2 +and +(3.4) +Pℓ+1 = ∂wpℓdwℓ+1 +where we write Pℓ = pℓdwℓ. Since the Schwarzian derivative maps mero- +morphic locally injective functions to holomorphic quadratic differentials, it +follows that P2 and hence the differentials Pℓ on M′ are holomorphic. +Now suppose z is any local holomorphic coordinate defined in some p- +neighbourhood Vp so that Q = qdz2 for some holomorphic function q on +Vp. Note that the umbilic points are isolated since they are precisely the +points where the holomorphic quadratic differential Q vanishes. Hence, at an + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +9 +umbilic point p ∈ M we may choose a local p-centred holomorphic coordinate +z so that X3 = Re(z) and +G = 1 + azn+1 + bzn+2 + O(zn+2), +where a ∈ C∗ and b ∈ C. From this we compute that +Q = −a(n + 1)zndz2 + O(zn), +so that (2.10) yields +(3.5) +P2 = − +��3n2 + 4n +8 +� +z−2 + n(n + 2) +(n + 1) +b +az−1 +� +dz2 + O(1). +Hence we may write +P2 = f2,n(z) +z2 +dz2, +where the complex-valued function f2,n is holomorphic near 0 and satisfies +f2,n(0) = −3n2 + 4n +8 +. +Using (3.4) and (3.5) together with a straightforward inductive argument we +see that +Pℓ = +�ℓ−1 +� +k=0 +cℓ,k zk−ℓ +� b +a +�k� +dzℓ + O(1) +for some coefficients cℓ,k where +cℓ,0 = +� +−1 +2 +�ℓ+1 +(ℓ − 1)! n(n + 2)ℓ−2(3n + 4). +It follows that we may write +Pℓ = fℓ,n(z) +zℓ +dzℓ +where the complex-valued function fℓ,n is holomorphic near 0 and satisfies +Resp(Pℓ) = fℓ,n(0) = cℓ,0, +thus completing the proof. +□ +Remark 3.8. In [2] it was shown that the real part of P2 is a conservation +law for critical points of a certain curvature entropy functional. For this +reason we call {Pℓ}ℓ⩾2 the entropy differentials. In fact, it was shown that if +we write g = exp(2u)dz ◦ d¯z for some real-valued function u and some local +Q-adapted holomorphic coordinate z so that Q = dz2, then P2 is given by +(3.6) +P2 = −2 +� +∂2 +zzu + (∂zu)2� +dz2. +3.2. Invariance properties +The entropy sequence enjoys certain invariance properties which we will now +discuss. First we observe (see also [2]): +Proposition 3.9. The entropy sequence is intrinsic, i.e. Pℓ+2 depends on the +induced metric and its derivatives up to order ℓ + 4 only. In particular, Pℓ+2 +is invariant under post-composing X : M → E3 with an ambient isometry. + +10 +T. METTLER AND L. POERSCHKE +Proof. We work in a local Q-adapted local coordinate z so that Q = dz2 and +g = e2udz ◦ d¯z. Consider the metric ˆg = (−K)3/4g with Gauss curvature +Kˆg = 1 +2|Kg|1/4 +where Kg denotes the Gauss curvature of g. Now define +T = ˆg˚∇2 log(Kˆg) = 1 +4 +ˆg˚∇2 log(−Kg) +where ˆg˚∇2 denotes the trace-free Hessian of ˆg. Clearly, the symmetric trace- +less covariant 2-tensor field T depends on g and its derivatives up to order +four. Using the identity u = − 1 +4 log(−Kg), we compute +T = −g0˚∇2u − du2 + 1 +2g0(g0∇u, g0∇u)g0 += −2 Re +�� +∂2 +zzu + (∂zu)2� +dz2� +, +which agrees with the real part of (3.6). It follows that P2 depends on the +induced metric only. Since Pℓ+2 is just the ℓ-th derivative of P2 with respect +to the Levi-Civita connection of the induced flat metric g0, we see that Pℓ+2 +depends on the induced metric and its derivatives up to order ℓ+4 only. +□ +In order to discuss the further invariance properties we first recall the +Weierstrass representation of minimal surfaces. We consider C3 and let φ +denote the natural complex inner product +φ(z, w) = z1w1 + z2w2 + z3w3, +where z = (zi) and w = (wi) are elements of C3. +Weierstrass observed +that if (M, J) is a Riemann surface and ˜X : M → C3 is a holomorphic null +immersion, i.e. ˜X∗φ = 0, then X = Re ◦ ˜X : M → R3 is a conformal and +minimal immersion of (M, J). +Conversely, every minimal immersion of a +simply connected surface M arises in this way. +Having a holomorphic null immersion ˜X : M → C3, the triple (M, G, η), +where G is the stereographically projected Gauss map of the minimal immer- +sion Re( ˜X) and η = d ˜X3 is called the Weierstrass data of the null immersion +˜X. Standard computations give (see for instance [10]) +(3.7) +Q = − 1 +GdG ◦ η, +where Q denotes the Hopf differential of Re( ˜X). +A (global) method of producing a holomorphic null immersion of a Riemann +surface (M, J) into C3 from Weierstrass data was obtained by Osserman [17]: +Theorem 3.10. Let G : M → ˆC be a meromorphic function and η a holo- +morphic 1-form on M. Suppose that: +(i) The zeroes of η coincide with the poles and zeros of G, with the same +order; +(ii) For any closed curve γ ⊂ M, +� +γ +Gη = +� +γ +η +G, +Re +� +γ +η = 0, +where ¯z denotes complex conjugation of z ∈ C; + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +11 +then +˜X(p) − ˜X(p0) = +� p +p0 +�1 +2(G−1 − G), i +2(G−1 + G), 1 +� +η, +yields a holomorphic null immersion ˜X : M → C3 so that Re( ˜X) has Weier- +strass data (M, G, η). +The natural left action of the linear conformal group C∗ × SO(3, C) on +C3 yields a left action on the space of holomorphic null immersions and +consequently on the space W(M) of minimal immersions of M arising via +the Weierstrass representation. For an element X : M → E3 ∈ W(M) we +call the deformations of X obtained by the C∗ × SO(3, C) action its Weier- +strass deformations. Next, following [12], we study the space of Weierstrass +deformations more carefully. +The action by the subgroup R+×SO(3, R) corresponds to similarity trans- +formations of the minimal surface associated to the null immersion. It follows +that the space of non-similar Weierstrass deformations is the homogeneous +space +(C∗ × SO(3, C)) / +� +R+ × SO(3, R) +� +≃ S1 × (SO(3, C)/SO(3, R)) . +The first circle factor yields the well-known associated family (or Bonnet +family) of minimal surfaces. The associated family of minimal surfaces has +the properties of being locally isometric and sharing a common Gauss map. +Furthermore, the Hopf differential Q changes by a complex phase. +The +latter factor gives rise to the so-called Goursat family of minimal surfaces. +The group SO(3, C) ≃ PSL(2, C) acts on the Gauss-map by Möbius trans- +formation and leaves the Hopf differential unchanged (c.f. [7, Lemma 5.3.1] +or [12]). +Since under a Bonnet-transform the induced metric is unchanged, so is +the induced flat metric. +It follows that the differentials {Pℓ}ℓ⩾2 are the +same for the whole S1-family of associated minimal surfaces. Furthermore, +since the Hopf differential is unchanged under a Goursat transform, so is +the Levi-Civita connection of the second fundamental form A∇. Lemma 3.5 +implies that the Levi-Civita connection g0∇ of the flat metric is invariant +under Goursat transform as well (this is noteworthy since the induced metric +itself does change non-trivially under Goursat transforms). Hence it follows +from the invariance of the Schwarzian derivative under post-composition +by a Möbius transformation that P2 and hence all differentials {Pℓ}ℓ⩾2 are +invariant under Goursat transforms. Concluding, we have: +Proposition 3.11. The meromorphic differentials {Pℓ}ℓ⩾2 are invariant un- +der Goursat – and Bonnet transforms. +Finally, note that scaling the immersion X : M → E3 by a constant +does neither change the Levi-Civita connection of the induced flat met- +ric, nor the Gauss map and hence leaves the sequence {Pℓ}ℓ⩾2 unchanged. +This fact together with the content of Proposition 3.6, Proposition 3.9 and +Proposition 3.11 is summarized in Theorem 1.1 of the introduction. + +12 +T. METTLER AND L. POERSCHKE +4. The degree of a minimal surface +The existence of an intrinsic sequence of meromorphic differentials on a min- +imal surface motivates the following definition: +Definition 4.1. Let M be an oriented smooth surface. A non-flat minimal +immersion X : M → E3 is said to have degree n ∈ N if Pn vanishes identically +or if there exists a weighted-homogeneous polynomial ψn : Cn−2 → C of +degree n with weights (2, 3, . . . , n − 1) such that +(4.1) +Pn = ψn(P2, . . . Pn−1). +Furthermore, we call ψn the algebraic type of the immersion X. Clearly, if +a non-flat minimal immersion has degree n, then it also has degree m for +all m ⩾ n. The degree will therefore always denote the smallest integer for +which a relation of the form (4.1) holds. +Remark 4.2. Recall, polynomial ψn(z1, z2, . . . , zn−1) which does not vanish +identically is called weighted-homogeneous of degree n if there exist positive +integers (w1, . . . , wn−1), called the weights of the variables, such that for +every λ ̸= 0 +ψn(λw1z1, λw2z2, . . . , λwn−1zn−1) = λnψn(z1, z2, . . . , zn−1). +Remark 4.3. A definition of degree for constant mean curvature surfaces +without umbilic points was previously given by Pinkall and Sterling [20]. +Example 4.4 (Enneper’s surface). Enneper’s surface is the minimal surface +with Weierstrass data M = C, G(z) = z and η(z) = zdz. From (3.7) we +compute +Q(z) = −dz2 +and hence using (2.10) we obtain +P2(z) = {z, z} dz2 = 0. +Therefore, Enneper’s surface has the lowest possible degree two. Conversely, +it was shown in [2] that a minimal surface satisfying P2 = 0 is – up to Euc- +lidean motion and scaling – an open subset of Enneper’s surface. Thus En- +neper’s surface and its Weierstrass deformations exhaust all degree 2 surfaces. +Note that Enneper’s Weierstrass deformations are just similarity transforms. +Remark 4.5. A characterisation of Enneper’s surface in terms of so-called +Chern–Ricci functions was given in [13]. +Example 4.6 (The helicoid & catenoid). The helicoid is the minimal surface +with Weierstrass data M = C, G(z) = ez and η(z) = idz. From (3.7) we +compute +Q(z) = −idz2 +and hence using (2.10) we obtain +P2(z) = {ez, z} dz2 = −1 +2dz2, +so that +P3(z) = ∂z +� +−1 +2 +� +dz3 = 0, + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +13 +showing that the helicoid is a degree 3 surface (and consequently so is the +catenoid, since it is a member of the helicoid’s associated family of minimal +surfaces). Conversely, it was proved in [2] that a minimal surface satisfying +P2 = λQ for some non-zero complex constant λ is – up to Euclidean mo- +tion, scaling and Goursat transform – an open subset of a member of the +helicoid/catenoid family. Since P2 = λQ precisely characterizes the degree 3 +surfaces, it follows that the helicoid and its Weierstrass deformations exhaust +all degree 3 surfaces. +4.1. Degree four surfaces +A minimal immersion with entropy differentials {Pℓ}ℓ⩾2 is of degree 4 if and +only if there exists a complex constant λ such that +(4.2) +P4 − 6λ(P2)2 = 0. +Therefore, we have a (complex) one-dimensional space of algebraic types of +degree 4 immersions and a complete classification is not feasible. However, +locally and away from umbilic points a somewhat explicit description is avail- +able. Let therefore X : M → E3 be a degree 4 immersions and let z : U → C +be a local holomorphic coordinate in which the entropy differential Q of X +satisfies Q = dz2. Writing P2 = pdz2 for some holomorphic function p on U, +(4.2) becomes +∂2 +zzp − 6λp2 = 0, +so that for λ ̸= 0, we obtain +p(z) = 1 +λ℘(z + c1; 0, c2) +for complex constants c1, c2 where ℘(z; g2, g3) denotes the Weierstrass elliptic +function with invariants g2 and g3. Consequently, the Schwarzian derivative +of the Gauss map of a degree 4 immersion, computed with respect to the +coordinate z, is a Weierstrass ℘-function or a polynomial of degree 1 in z +(provided λ = 0). +The space of algebraic types of degree four surfaces simplifies in the pres- +ence of umbilic points. +Proposition 4.7. Suppose X : M → E3 is a smooth minimal immersion of +degree four whose Hopf differential vanishes to order n ∈ N at some umbilic +point. Then the Hopf differential vanishes to order n at every umbilic point +and the immersion X satisfies +P4 + 12(n + 2)2 +(3n2 + 4n)(P2)2 = 0. +Proof. Let p ∈ M denote the umbilic point at which the Hopf differential +vanishes to order n. Then Proposition 3.6 implies that +Resp(P2) = −3n2 + 4n +8 +. +and hence +Resp((P2)2) = (Resp(P2))2 = +�3n2 + 4n +8 +�2 +. + +14 +T. METTLER AND L. POERSCHKE +Since X has degree four there exists a complex-constant λ such that P4 − +6λ(P2)2 = 0 and hence using Proposition 3.6 we must have +Resp(P4) = − 3 +16(n + 2)2(3n2 + 4n) = 6λ (Resp(P2))2 = 6λ +�3n2 + 4n +8 +�2 +, +which implies +P4 + 12(n + 2)2 +(3n2 + 4n)(P2)2 = 0, +as claimed. Furthermore, note that the sequence an = 12(n+2)2 +(3n2+4n) is injective +which excludes the possibility of having two umbilic points at which the Hopf +differential vanishes to different order. +□ +Example 4.8 (Enneper surfaces of higher dihedral symmetry). The minimal +surfaces arising from the Weierstrass data M = C, G(z) = zk and η(z) = +zkdzk for some integer k ⩾ 2 are called Enneper surfaces of higher dihedral +symmetry. From (3.7) we obtain +(4.3) +Q(z) = −kzk−1dz2, +showing that these surfaces have an umbilic point at which the Hopf differ- +ential vanishes to order k − 1. Therefore, by Proposition 4.7, if the Enneper +surfaces of higher dihedral symmetry have degree four, then they will have +to satisfy +P4 + 12 +(k + 1)2 +(k − 1)(3k + 1)(P2)2 = 0. +From (2.9) and (4.3) we get γ = (k−1) +2z , hence with (2.10) we obtain +P2(z) = +�� +zk, z +� +− (k − 1)(k + 3) +8z2 +� +dz2 += − +�(k − 1)(k + 1) +2z2 ++ (k − 1)(k + 3) +8z2 +� +dz2 += − 1 +8z2 (k − 1)(3k + 1)dz2. +Therefore, using (2.4) we have +P3(z) = +� +−1 +8(k − 1)(3k + 1)∂z +� 1 +z2 +� ++ 1 +4 +(k − 1) +2z +�(k − 1)(3k + 1) +z2 +�� +dz3 += +1 +8z3 (k − 1)(k + 1)(3k + 1)dz3, +and likewise +P4(z) = +� +− 3 +8z4 (k − 1)(k + 1)(3k + 1) +− 3 +2z (k − 1) 1 +8z3 (k − 1)(k + 1)(3k + 1) +� +dz4 += − +3 +16z4 (k + 1)2(k − 1)(3k + 1)dz4. +Hence we obtain +P4 + 12 +(k + 1)2 +(k − 1)(3k + 1)(P2)2 = 0, + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +15 +which agrees with Proposition 4.7. +The sequence +an = 12(n + 2)2 +(3n2 + 4n) +describing the admissible coefficients for degree four minimal immersions +with umbilic points satisfies limn→∞ λn = 4. It is therefore natural to ask +what (umbilic-free) minimal surfaces satisfy P4 + 4(P2)2 = 0. We will next +study a 1-parameter family of such surfaces. +Example 4.9 (Limit degree four surfaces). Let M = C with Gt(z) = 1 +t e−z +and ηt(z) = 2tezdz where t > 0 is a real parameter. From this Weierstrass +data we obtain a family of minimal immersions Xt : R2 → R3 given by +(x, y) �→ +� +x − t2 +2 e2x cos(2y), y + t2 +2 e2x sin(2y), −2tex cos(y) +� +where z = x + iy denotes the standard coordinate on C ≃ R2. Note that the +immersion X0 yields the plane. The induced metrics are +gt = +� +1 + t2e2x�2 � +dx2 + dy2� +, +and the second fundamental forms are +At = 2tex � +cos(y) +� +dx2 − dy2� +− 2 sin(y) (dx ◦ dy) +� +. +The metrics gt have Gauss curvature +Kt = − +4t2e2x +(1 + t2e2x)4 +Writing Σt = Xt(C), these surfaces are singly periodic with respect to the +action Σt × Z → Σt +(p, n) �→ p + + + +0 +2nπ +0 + + . +for n ∈ Z and p ∈ Σ. The fundamental domain Xt(R×[−π, π]) is symmetric +around y = 0 and contains two straight lines defined by y = ±(1/2)π. +Note moreover that ∂y is a Killing vector field for gt which is the imaginary +part of a holomorphic vector field. +From (3.7) we obtain +Q(z) = 2t ezdz2 +and from (2.9) we get γ = 1 +2, hence with (2.10) we obtain P2(z) = − 3 +8dz2. +Likewise we obtain P3(z) = 3 +8dz3 and P4 = − 9 +16dz4 so that +P4 + 4(P2)2 = 0. + +16 +T. METTLER AND L. POERSCHKE +Figure 1. A portion of the image of X1. +4.2. Degree five surfaces +A minimal immersion with entropy differentials {Pℓ}ℓ⩾2 is of degree 5 if and +only if there exists a complex constant λ such that +P5 + λP2P3 = 0. +Therefore, again, we have a (complex) one-dimensional space of algebraic +types of degree 5 immersions and a complete classification is not feasible. +However, in entirely similar fashion to the degree four case we can prove: +Proposition 4.10. Suppose X : M → E3 is a smooth minimal immersion +of degree five whose Hopf differential vanishes to order n ∈ N at some umbilic +point. Then the Hopf differential vanishes to order n at every umbilic point +and the immersion X satisfies +P5 + 24 (n + 2)2 +(3n + 4)nP3P2 = 0. +The sequence +an = 24 (n + 2)2 +(3n + 4)n +satisfies a1 = 216/7 and a∞ = limn→∞ an = 8. Both coefficients a1 and a∞ +are realized by well-known surfaces. +Example 4.11 (Schwarz family). The Schwarz primitive triply periodic sur- +face is the minimal surface with Weierstrass data +M = +� +(z, w) ∈ ˆC2 | w2 = z8 − 14z4 + 1 +� +and G(z, w) = z and η(z, w) = z +wdz. From (3.7) we compute +Q(z, w) = − 1 +wdz2 +and hence using (2.10) we obtain +P2(z, w) = −42z2(z4 + 1)2 +w4 +dz2. +Simple computations give +P3(z, w) = 84z(z16 + 34z12 − 34z4 − 1) +w6 +dz3, + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +17 +and +P4(z, w) = − 84 +w8 (z24 + 282z20 + 1887z16 − 884z12 + 1887z8 + 282z4 + 1)dz4, +as well as +P5(z, w) = 108864z3(z24 + 36z20 + 69z16 − 69z8 − 36z4 − 1) +w10 +dz5. +Consequently, we have +P5 + 216 +7 P2P3 = 0, +thus showing that the Schwarz family of minimal surfaces has degree 5 and +realises the first possible coefficient 216 +7 among degree 5 surfaces with umbilic +points. +The limit a∞ = 8 is also realized by a well-known family of minimal +surfaces. +Example 4.12 (Scherk family). The singly-periodic Scherk surface is the +minimal surface with Weierstrass data M = ˆC \ {±1, ±i}, G(z) = z and +η(z) = +iz +(z4−1)dz. From (3.7) we compute +Q(z) = − +i +z4 − 1dz2 +and hence using (2.10) we obtain +P2(z) = − +6z2 +(z4 − 1)2 . +Simple computations give +P3(z) = 12z(z4 + 1) +(z4 − 1)3 , +P4(z) = −12(z8 + 10z4 + 1) +(z4 − 1)4 +and +P5(z) = 576z3(z4 + 1) +(z4 − 1)5 +, +so that +P5 + 8P2P3 = 0, +showing that Scherk’s family of minimal surfaces has degree 5 and realizes +the limit coefficient a∞ = 8. +4.3. Degree seven surfaces +Example 4.13. Taking M = ˆC \ {z : zk = 1} and G(z) = zk−1 as well as +η(z) = +zk−1 +(zk−1)2 dz gives the k-noids of Jorge & Meeks [9]. With the help of a +computer algebra system one easily verifies that for k > 2 the k-noids satisfy +0 = P7 + akP5P2 + bkP4P3 + ckP3(P2)2 + +18 +T. METTLER AND L. POERSCHKE +and hence are surfaces of degree 7. The coefficients are +ak = 16(3k − 2)(k − 2) +3k2 − 16k + 8 +, +bk = 8(27k4 − 144k3 − 56k2 + 128k − 32) +(3k2 − 16k + 8)(k − 2)(3k − 2) +, +ck = +192k2 +3k2 − 16k + 8. +5. Approximating minimal surfaces +It is natural to ask if a minimal surface can be approximated by a sequence +of minimal surfaces of increasing degree. In this section we will show that +locally and away from umbilic points this is indeed the case. +Let therefore M be an oriented surface and X : M → E3 a non-flat +minimal immersion. For p ∈ M′ pick a simply connected neighbourhood Up +so that Up does not contain any umbilic points. After possibly shrinking Up +and applying an ambient rotation, we may assume that the stereographically +projected Gauss map G of X does not attain the value ∞ on Up. Let Q +denote the Hopf differential and P2 denote the first entropy differential of X. +We take a local p-centred holomorphic coordinate z so that Q = dz2. By +analytic continuation we may extend z to all of Up. We write +P2 = ρ +2dz2 +for some holomorphic function ρ on Up, where, by definition, we have +ρ +2 = {G, z} . +Recall the following classical fact. +Theorem 5.1. Let ρ be a holomorphic function on a simply connected do- +main Ω in the complex plane. +Then for every prescription of w(z0) and +(∂zw)(z0) at some point z0 ∈ Ω, there exists a unique holomorphic function +w on Ω satisfying Hill’s equation with potential ρ/4 +(5.1) +∂zzw + ρ +4w = 0. +For a proof see for instance [14, p. 53]. By a straightforward power series +coefficient comparison argument, we also obtain: +Lemma 5.2. Let D ⊂ C be the open unit disk. +For every holomorphic +function ρ on D and for all w0, w′ +0 ∈ C, the sequence wn : D → C with wn +being the unique solution of +∂2 +zzwn + ρn +4 wn = 0, +wn(0) = w0, +(∂zwn)(0) = w′ +0, +converges locally uniformly to the unique solution of +∂2 +zzw + ρw = 0, +w(0) = w0, +(∂zw)(0) = w′ +0, +where ρn denotes the Taylor series around 0 of ρ up to order n. + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +19 +Hill’s equation is linear and by Theorem (5.1) its space of solutions is +complex two-dimensional. Let therefore w1, w2 : Ω → C be a pair of linearly +independent solutions. Their Wronskian +W(w1, w2) = w1∂zw2 − w2∂zw1 +is a non-zero constant W by Abel’s identity, and w1 and w2 cannot vanish +simultaneously. Writing ˆG = w1 +w2, we obtain +∂z ˆG = w2∂zw1 − w1∂zw2 +(w2)2 += − W +(w2)2 +so that ˆG is a locally injective meromorphic map except possibly on the zero +locus of w2. Computing ∂z(1/ ˆG) we see that ˆG is locally injective away from +there zero locus of w1, and thus everywhere. A simple computation now +gives +{ ˆG, z} = −2∂2 +zzw2 +w2 += ρ +2, +where we have used the fact that w2 solves (5.1). From the cocycle prop- +erty (2.8) of the Schwarzian derivative we see that ˆG differs from the stereo- +graphically projected Gauss map by a Möbius transformation. In particular, +on Up, there exists a pair of solutions w1, w2 of (5.1) – unique up to res- +caling by a common constant – so that G = w1/w2. Since G does never +attain the value ∞ on Up the function w2 is non vanishing on Up. Fix such +a pair (w1, w2) and let w1,n and w2,n be the unique solutions on Up to Hill’s +equation with potential ρn/4 which satisfy +wi,n(p) = wi(p), +and +(∂zwi,n)(p) = (∂zwi)(p). +Here ρn denotes the power series of ρn around p of order (n−3) with respect +to the coordinate z. Note that it follows from (3.7) and Theorem 3.10 that we +may recover X : Up → E3 by taking the Weierstrass data (G, −G/(∂zG)dz) +on Up. However, we may also take (Gn, −Gn/(∂zGn)dz) as Weierstrass data +on Up where Gn are the locally injective meromorphic functions on Up defined +by +Gn = w1,n +w2,n +After possibly shrinking Up again, we can assume that the functions Gn are +holomorphic for sufficiently large n, and hence by applying Theorem 3.10 +again we obtain a sequence of minimal immersions +� +Xn : Up → E3� +n⩾n0 . +By construction, Xn has degree n, since ρn is just a polynomial of order +(n − 3). By Lemma 5.2, the functions wi,n converge locally uniformly to wi +as n tends to infinity and since w2 is non-vanishing, Gn converges locally +uniformly to G as n tends to infinity. Since Xn arises by integration from +the Weierstrass data (Gn, −Gn/(∂zGn)dz), it follows that ∥Xn − X∥gEucl +converges to zero locally uniformly as n tends to infinity. +This proves +Theorem 1.2. + +20 +T. METTLER AND L. POERSCHKE +Appendix A. Higher order Schwarzian derivatives +For background about the Schwarzian derivative we refer to reader to [18, +19]. We have defined the Schwarzian derivative with respect to a conformal +connection. +However, less geometric structure is required on a Riemann +surface (M, [g]) to define the Schwarzian derivative. All one needs is a Möbius +structure [4], which is a generalization of the classical notion of a complex +projective structure. A Möbius structure is a linear second order differential +operator which can be written as the sum of the symmetrized trace-free +Hessian of a [g]-conformal connection ∇ and the real part of a quadratic +differential Q +H = Sym0(Hess(∇)) + Re(Q). +The differential operator H acts on densities of weight −1/2 on M and takes +values in the symmetric and [g]-traceless covariant 2-tensor fields of weight +−1/2, i.e. +H : Γ +� +Λ2(T ∗M)−1/2� +→ Γ +� +S2 +0(T ∗M) ⊗ Λ2(T ∗M)−1/2� +. +The Schwarzian derivative of a local biholomorphism ϕ : (M, [g], H) → +(M′, [g]′, H′) between Riemann surfaces equipped with Möbius structures is +then defined to be ϕ∗H′−H, where the pullback is defined in the obvious way. +The classical definition of the Schwarzian derivative is recovered by taking +both (M, [g]) and (M′, [g]′) to be ˆC and H the Möbius structure associated +to the Levi-Civita connection of the spherical metric. +A Möbius structure H on (M, [g]) is called flat if in a neighbourhood of +every point of M there exists a local holomorphic coordinate z so that +(A.1) +H = Re(∂2 +zz) +with respect to the canonical local trivialisations of the vector bundles Λ2(T ∗M)−1/2 +and S2(T ∗M) ⊗ Λ2(T ∗M)−1/2 induced by z. We call such a coordinate sys- +tem adapted to the flat Möbius structure H. It is straightforward to check +that any two overlapping adapted holomorphic coordinates are related by a +Möbius transformation. +Recall that a complex projective structure on a Riemann surface (M, [g]) +is a (maximal) atlas of charts mapping open sets into CP1 so that trans- +ition functions are (restrictions) of Möbius transformations. +Therefore, a +flat Möbius structure induces a complex projective structure and conversely +every complex projective structure induces a flat Möbius structure by defin- +ing H as in (A.1). We refer the reader to [4] for additional details on Möbius +structures and to [6] for complex projective structures. +A crucial difference between having a conformal connection at hand versus +a Möbius structure only, is when it comes to defining higher order Schwarzian +derivatives. The operators SH +ℓ from Proposition A.1 below are well defined +on a Riemann surface (M, [g]) equipped with a (flat) Möbius structure. Al- +though they arise as derivatives of the Schwarzian derivative – and hence +may be thought of as higher order Schwarzian derivatives – for ℓ ⩾ 3 they all +do lack the invariance under post-composition by a Möbius transformation. +Proposition A.1. Let (M, [g]) be a Riemann surface equipped with a flat +Möbius structure H. Then for ℓ ⩾ 2 there exists a differential operator SH +ℓ + +THE DEGREE OF A CLASSICAL MINIMAL SURFACE +21 +of order ℓ + 1 mapping locally injective meromorphic functions on M to +holomorphic differentials of order ℓ. In a local H-adapted coordinate z on M +the operators are given by +SH +ℓ+1(f) = +� +∂z (sℓ(f)) − ℓsℓ(f)∂2 +zzf +∂zf +� +dzℓ+1 +with +SH +2 (f) = +� +∂3 +zzzf +∂zf +− +�∂2 +zzf +∂zf +�2� +dz2 +where we write SH +ℓ (f) = sℓ(f)dzℓ. +Remark A.2. These operators can also be defined in the case where H is not +flat, for the sake of brevity we will however not show this here. In local +coordinates, they appeared in [21]. +For instance, in a local H-adapted coordinate z we have +SH +3 (f) = +� +∂4 +zzzzf +∂zf +− 6 (∂3 +zzzf)(∂2 +zzf) +(∂zf)2 ++ 6 +�∂2 +zzf +∂zf +�3� +dz3. +The reader may easily verify that SH +3 (f) is not invariant under post-compo- +sition by a Möbius transformation. For comparison, we mention that if ∇ +is a [g]-conformal connection admitting a local holomorphic coordinate z in +which 1 +2γ2 − ∂zγ = 0, then we have +S∇ +3 (f) = +� +∂4 +zzzzf +∂zf +− 4 (∂3 +zzzf)(∂2 +zzf) +(∂zf)2 ++ 3 +�∂2 +zzf +∂zf +�3� +dz3. +Proof of Proposition A.1. We only need to show that the definition of the +operators SH +ℓ is invariant under coordinate changes of the form +(A.2) +w = az + b +cz + d, +with +� +a +b +c +d +� +∈ SL(2, C). +We have dw = τdz and ∂w = τ −1∂z with +τ = (ad − bc) +(cz + d)2 . +The proof is by induction. Recall the (well-known) fact that the Schwarzian +SH +2 as defined above is invariant under coordinate changes of the form (A.2). +Let us therefore assume that SH +ℓ is invariant under coordinate changes of the +form (A.2). We write SH +ℓ (f) = ˆsℓ(f)dwℓ = sℓ(f)dzℓ so that +ˆsℓ(f) = τ −ℓsℓ(f). +Now using ∂zτ = − +2c +(cz+d)τ, we obtain +(A.3) +∂w (ˆsℓ(f)) dwℓ+1 = +� +τ −(ℓ+1)∂z(sℓ(f)) + τ −1sℓ(f)∂zτ −l� +dwℓ+1 += ∂z(sℓ(f))dzℓ+1 + sℓ(f) +2c +(cz + d)ℓdzℓ+1, + +22 +T. METTLER AND L. POERSCHKE +as well as +(A.4) +ℓˆsℓ(f)∂w (∂wf) +∂wf +dwℓ+1 = ℓτ −ℓsℓ(f)∂z +� +τ −1∂zf +� +∂zf +τ ℓ+1dzℓ+1 += ℓsℓ(f)∂2 +zz(f) +∂zf dzℓ+1 + ℓτsℓ(f)τ −1 +2c +(cz + d)dzℓ+1. +Combining (A.3) and (A.4) proves the inductive step. +□ +Remark A.3. The Schwarzian derivative can be generalized to higher di- +mensions from the conformal viewpoint [16] as well as from the (complex) +projective viewpoint [15]. +References +[1] D. Aharonov, A necessary and sufficient condition for univalence of a meromorphic +function, Duke Math. J. 36 (1969), 599–604. MR 0249622 5 +[2] J. Bernstein, T. Mettler, Characterizing classical minimal surfaces via the en- +tropy differential, J. Geom. Anal. 27 (2017), 2235–2268. DOI 10.1007/s12220-017-9759-6 +MR 3667429 1, 2, 9, 12, 13 +[3] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebi- +ete (3) 10, Springer-Verlag, Berlin, 1987. DOI 10.1007/978-3-540-74311-8 MR 867684 +zbM 0613.53001 3 +[4] D. M. J. Calderbank, Möbius structures and two-dimensional Einstein-Weyl geo- +metry, J. Reine Angew. Math. 504 (1998), 37–53. DOI 10.1515/crll.1998.111 MR 1656822 +zbM 0909.53029 20 +[5] S. S. Chern, R. Osserman, Remarks on the Riemannian metric of a minimal sub- +manifold, in Geometry Symposium, Utrecht 1980 (Utrecht, 1980), Lecture Notes in Math. +894, Springer, Berlin-New York, 1981, pp. 49–90. MR 655419 7 +[6] D. 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DOI 10.2307/1971425 MR 1014929 12 +[21] E. +Schippers, +Distortion +theorems +for +higher +order +Schwarzian +derivat- +ives +of +univalent +functions, +Proc. +Amer. +Math. +Soc. +128 +(2000), +3241–3249. +DOI 10.1090/S0002-9939-00-05623-9 MR 1706981 21 +[22] H. Tamanoi, Higher Schwarzian operators and combinatorics of the Schwarzian de- +rivative, Math. Ann. 305 (1996), 127–151. DOI 10.1007/BF01444214 MR 1386108 5 +Faculty of Mathematics and Computer Science, UniDistance Suisse, Brig, +Switzerland +Email address: thomas.mettler@fernuni.ch +Faculty of Mathematics and Computer Science, UniDistance Suisse, Brig, +Switzerland +Email address: lukas.poerschke@fernuni.ch,lukpoerschke@gmail.com + diff --git a/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/load_file.txt b/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8dc61302c4832e4ea6ebcd37d8710376c35f58a --- /dev/null +++ b/e9FKT4oBgHgl3EQfAy1w/content/tmp_files/load_file.txt @@ -0,0 +1,696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf,len=695 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='11700v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='DG] 27 Jan 2023 The Schwarzian derivative and the degree of a classical minimal surface thomas mettler and lukas poerschke Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Using the Schwarzian derivative we construct a sequence {Pℓ}ℓ⩾2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The differentials {Pℓ}ℓ⩾2 are invariant under all deformations of the surface arising via the Weierstrass rep- resentation and depend on the induced metric and its derivatives only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A minimal surface is said to have degree n if its n-th differential is a polynomial expression in the differentials of lower degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We observe that several well-known minimal surfaces have small degree, including Enneper’s surface, the helicoid/catenoid and the Scherk – as well as the Schwarz family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Introduction In [2], a meromorphic quadratic differential P2 is introduced on every non- flat oriented minimal surface M in Euclidean 3-space E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The differential P2 arises as a conservation law for a certain curvature entropy functional and is hence called the entropy differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In this note we show that P2 is the first element in a geometrically natural sequence {Pℓ}ℓ⩾2 of meromorphic differen- tials on M, where Pℓ has degree ℓ, that is, Pℓ is a section of the ℓ-th tensorial power of the canonical bundle of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The entropy differentials {Pℓ}ℓ⩾2 arise as certain higher order Schwarzian derivatives of the stereographically pro- jected Gauss map G : M → C ∪ {∞};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' where we compute the Schwarzian derivative and its higher order descendants with respect to the Levi-Civita connection of the flat metric √ −Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Here g denotes the induced metric on M and K ⩽ 0 its Gauss curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The differentials {Pℓ}ℓ⩾2 satisfy the following properties: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let X : M → E3 be a non-flat minimal immersion of the oriented surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Denoting by {Pℓ}ℓ⩾2 the induced differentials, we have: (i) the differential Pℓ is holomorphic away from umbilic points and ex- tends meromorphically to all of M with a pole of order ℓ at the umbilic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' (ii) the differential Pℓ depends on the induced metric g and its derivatives up to order (ℓ+4) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In particular, the differentials {Pℓ}ℓ⩾2 are the same for all members of the associated family of minimal immersions of X : M → E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Date: 27th January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 1 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE (iii) the differentials {Pℓ}ℓ⩾2 are invariant under Goursat transforms of the minimal immersion X : M → E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' (iv) the differentials {Pℓ}ℓ⩾2 are invariant under (constant) rescaling of X : M → E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Having the infinite sequence {Pℓ}ℓ⩾2 at hand, we may say that a minimal immersion X : M → E3 has degree n if Pn vanishes identically or if Pn is a polynomial expression in the lower order differentials, that is, Pn = ψn(P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , Pn−1), where ψn is a weighted-homogeneous polynomial of degree n which we call the algebraic type of the immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We observe that several classical min- imal surfaces are surfaces of low degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Using [2], it follows that – up to Euclidean motion and the deformations listed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1 – open sub- sets of Enneper’s surface are the only surfaces of degree 2 and open subsets of the helicoid and catenoid are the only surfaces of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A complete description of degree four surfaces is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, we prove that if a degree four minimal surface admits an umbilic point, then it satisfies an equation of the form P4 + an(P2)2 = 0, where an = 12(n + 2)2 (3n2 + 4n), n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The equation P4 + an(P2)2 = 0 is satisfied by the Enneper surfaces with higher dihedral symmetry group Dn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, the sequence an con- verges to 4 as n goes to infinity and we show that at this limit coefficient there exists a 1-parameter family of immersed singly periodic degree four minimal deformations of the plane which keep the lines � z = 0, y = kπ 2 | k ∈ Z � fixed and so that the induced metric admits a Killing vector field that is the real part of a holomorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Likewise, a degree five minimal surface admitting an umbilic point satisfies P5 + 2anP3P2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We show that the smallest admissible coefficient 216/7 is realized by the Schwarz family (includ- ing the P- and D-surface and the Gyroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, the limit coefficient 8 is realized by the Scherk family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Also, at degree seven we encounter the k-noids of Jorge & Meeks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We also prove that locally and away from umbilic points a minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let X : M → E3 be a non-flat minimal immersion of an oriented surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then for every point p in the complement M′ of the umbilic locus there exists a neighbourhood Up and a sequence of minimal immersions Xn : Up → E3 with Xn having degree n so that limn→∞ ∥Xn − X∥gEucl = 0 locally uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Schwarzian derivative arises most naturally in the context of project- ive differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In Appendix A we discuss the relation between the definition of the Schwarzian derivative used in this note and the usual definition of projective differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' THE DEGREE OF A CLASSICAL MINIMAL SURFACE 3 Acknowledgements This article is based on the doctoral thesis of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='. The authors were par- tially supported by the priority programme SPP 2026 “Geometry at Infinity” of DFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The authors are grateful to Jacob Bernstein for several helpful dis- cussions regarding the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Weyl connections Let (M, [g]) be an oriented surface equipped with a conformal structure [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall that a Weyl connection for [g] (or [g]-conformal connection) is a torsion-free connection ∇ on TM preserving [g], that is, its parallel transport maps are angle preserving with respect to the conformal structure [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A torsion-free connection ∇ on TM preserves [g] if and only if for some – and hence any – Riemannian metric g ∈ [g], there exists a 1-form β ∈ Ω1(M), so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) ∇g = 2β ⊗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, it follows from Koszul’s identity that for every pair (g, β) there exists a unique affine torsion-free connection satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1), which is (g,β)∇ = g∇ + g ⊗ β# − β ⊗ Id − Id ⊗ β, where g∇ denotes the Levi-Civita connection of the metric g and β# denotes the g-dual vector field to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that for every smooth function u on M, the pair (e2ug, β + du) determines the same Weyl connection as the pair (g, β) (exp(2u)g,β+du)∇ = (g,β)∇, as can easily be verified using the identity (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='159]) e2ug∇ = g∇ − g ⊗ g∇u + du ⊗ Id + Id ⊗ du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We will use the notation [g]∇ to denote a Weyl connection for [g] and simply write ∇ when the conformal structure is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let J denote the complex structure on M induced by [g] and the ori- entation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Clearly, a torsion-free connection preserves [g] if and only if it preserves J and we may therefore also think of the Weyl connections for [g] as torsion-free J-linear connections, that is, torsion-free connections on TM whose parallel transports maps are J-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, a Weyl connec- tion ∇ induces connections on all tensorial powers of the canonical bundle L = T 1,0M∗ of (M, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By standard abuse of notation, we will denote these connections by ∇ as well, so that for all ℓ ∈ Z we have first order differential operators ∇ : Γ(Lℓ) → Ω1(M, Lℓ), sending smooth sections of Lℓ to Lℓ-valued 1-forms on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since Lℓ is a complex vector bundle, the Lℓ-valued 1-forms on M decompose Ω1(M, Lℓ) = Ω1,0(M, Lℓ) ⊕ Ω0,1(M, Lℓ) 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE into (1,0) and (0,1) parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that we may canonically identify Ω1,0(M, Lℓ) with Γ(Lℓ+1) so that the (1,0) part of ∇ may be thought of as a differential operator ∇1,0 : Γ(Lℓ) → Γ(Lℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For a smooth section σ ∈ Γ(Lℓ), we write σ′ = ∇1,0σ as well as σ′′ = (σ′)′ and likewise for higher order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For what follows it will be convenient to have local coordinate expres- sions for the differential operators ∇1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' To this end let z : U → C be a local holomorphic coordinate on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Extending ∇ complex-linearly to the complexified tangent bundle TM ⊗ C, it follows from the J-linearity of ∇ that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) ∇∂z∂¯z = 0 and furthermore that there exists a unique complex-valued function γ on U such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) ∇∂z∂z = γ ∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, a torsion-free connection satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) in every holo- morphic coordinate system is J-linear and hence a Weyl connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Suppose σ : U → Lℓ is a smooth section, we write σ = s dzℓ for some smooth complex-valued function s on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) ∇1,0s dzℓ = (∂zs − ℓγs) dzℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' If w : V → C is another local holomorphic coordinate with U ∩ V ̸= ∅, then writing σ = ˆsdwℓ and ∇∂w∂w = ˆγ ∂w for complex-valued functions ˆs, ˆγ on V , we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5) ˆs = (∂wz)ℓs, ˆγ = (∂wz) γ + (∂zw) ∂2 wwz, on U ∩ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5) it is easy to check that the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) does not depend on the chosen coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The reader less familiar with complex geometry may therefore take (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) as the definition of the operators ∇1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note in particular that ∇1,0 agrees with the usual “del-operator” ∂ on functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Higher order Schwarzian derivatives Let (M, [g]) be a Riemann surface and ∇ a Weyl connection for [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let f be a holomorphic function on M satisfying f ′ = ∂f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We define the Schwarzian derivative of f with respect to ∇ to be the quadratic differential S∇(f) = f ′′′ f ′ − 3 2 �f ′′ f ′ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In a local holomorphic coordinate system z : U → C with ∇∂z∂z = γ ∂z for some complex-valued function γ on U, we obtain with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6) S∇(f) = {f, z} dz2 + �1 2γ2 − ∂zγ � dz2, where {f, z} = ∂3 zzzf ∂zf − �∂2 zzf ∂zf �2 THE DEGREE OF A CLASSICAL MINIMAL SURFACE 5 denotes the classical Schwarzian derivative of f with respect to the coordinate z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Taking the Levi-Civita connection of the spherical metric g1 = � 2 1 + |z|2 �2 dz ◦ d¯z on the Riemann sphere ˆC gives g1∇∂z∂z = − 2¯z 1 + |z|2 ∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, we have γ = −2¯z/(1 + |z|2), so that 1 2γ2 − ∂zγ = 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6) simplifies to give the classical coordinate definition of the Schwar- zian derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall that the classical definition of the Schwarzian de- rivative extends to the set of locally injective meromorphic functions and consequently by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6), so does our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, a locally injective meromorphic function a defined on some domain Ω ⊂ ˆC satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) S g1∇(a) = 0 if and only if a is the restriction of a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A crucial property of the Schwarzian derivative is that it defines a cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' If f is a locally injective meromorphic function on M and a a local biholomorphism on ˆC so that a ◦ f is well defined, then one may easily verify that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='8) S∇(a ◦ f) = S∇(f) + f ∗ � S g1∇(a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' As a consequence of this identity and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we see that the Schwarzian de- rivative is invariant under post-composition by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Schwarzian derivative is the first in a sequence of differential operators enjoying invariance under post-composition by a Möbius transformation: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let (M, [g]) be a Riemann surface equipped with a Weyl connection ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For ℓ ⩾ 2 we define the ℓ-th Schwarzian derivative to be the (ℓ + 1)-th order differential operator defined by S∇ ℓ+1(f) = � S∇ ℓ (f) �′ where S∇ 2 (f) = f ′′′ f ′ − 3 2 �f ′′ f ′ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The ℓ-th Schwarzian derivative maps a locally injective meromorphic function f on M to a smooth section of Lℓ, that is, a smooth differential on M of degree ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By construction, the operators S∇ ℓ are invariant under post- composition by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, none of the higher order Schwarzian derivatives defines a cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Local coordinate definitions of higher order Möbius invariant Schwarzian differential operators previously appeared in [1] (see also [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Every non-vanishing holomorphic quadratic differential Q on a Riemann surface (M, [g]) gives rise to a flat [g]-conformal connection A∇ where A∇ denotes the Levi-Civita connection of the flat Lorentzian metric A = Re(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE Indeed, in a local holomorphic coordinate z : U → C on M so that Q = qdz2 for some holomorphic function q on U, we have A∇∂z∂¯z = 0 and A∇∂z∂z = γ ∂z, with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9) γ = ∂zq 2q , showing that A∇ is [g]-conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' If f is a locally injective meromorphic function on U, we obtain with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) S A∇ 2 (f) = � {f, z} + 5 8 �∂zq q �2 − 1 2 ∂2 zzq q � dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The entropy differentials and their invariance properties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The entropy differentials Let M be an oriented smooth surface and X = (Xi) : M → E3 a smooth immersion into Euclidean 3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let g and A denote the induced metric and second fundamental form on M g = dX · dX, A = −dN · dX, where N = (Ni) : M → S2 ⊂ E3 denotes the (orientation compatible) Gauss map of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Weingarten shape operator is the endomorphism S : TM → TM satisfying A(X, Y ) = g(S(X), Y ) = g(X, S(Y )) for all X, Y ∈ Γ(TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The operator S is g-symmetric and hence (pointwise) diagonalizable with real eigenvalues κ1, κ2, called the principal curvatures of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall that X is called minimal if its mean curvature H = 1 2 tr S = 1 2(κ1 + κ2) vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A point p ∈ M where κ1 = κ2 is an umbilic point, which – in the minimal case – amounts to κ1 = κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, the umbilic points are precisely those points where the Gauss curvature K = κ1κ2 = −(κ1)2 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The umbilic locus is the set U = {p ∈ M | κ1(p) = κ2(p) = 0} and we use M′ to denote its complement in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=', M′ = M \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The minimality of X is well-known to be equivalent to the meromorphicity of the stereographically projected Gauss-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' More precisely, we have the following lemma whose proof may be found in [11] or most standard texts on minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let X : M → E3 be a smooth immersion with stereographically projected Gauss map G = (N1+iN2)/(1−N3) : M → ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then X is minimal if and only if G is meromorphic with respect to the complex structure on M induced by g and the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Another key property of minimal surfaces is that the induced metric sat- isfies the so-called Ricci condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' THE DEGREE OF A CLASSICAL MINIMAL SURFACE 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The induced metric g of a minimal immersion X : M → E3 has non-positive Gauss curvature K and whenever K < 0, the metric g satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) ∆ log(−K) = 4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, if (M, g) is a simply connected Riemannian 2-manifold of strictly negative Gauss curvature satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1), then (M, g) can be immersed iso- metrically and minimally into E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Here ∆ denotes the Laplace-Beltrami operator with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For a proof the reader may consult [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' An immediate consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) is the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let g denote the induced metric of a minimal immersion X : M → E3 and K its Gauss curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then on M′ ⊂ M, the complement of the umbilic locus, g0 = √ −Kg defines a flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The proof is an immediate consequence of the well-known and easily- derived formula for the dependence of the Gauss curvature on a conformal change of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For u ∈ C∞(M), we have Ke2ug = e−2u (Kg − ∆u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, on M′ we obtain Kg0 = 1 √ −K � K − 1 2∆ log �√ −K �� = 1 √ −K � K − 1 4∆ log (−K) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ Clearly, the Levi-Civita connection g0∇ of g0 is a [g]-conformal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4 we immediately see that we obtain a sequence of differentials Pℓ on the complement M′ of the umbilic locus of a minimal immersion X : M → E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Indeed, since the Gauss map is a locally injective meromorphic function on M′, the differentials Pℓ = S g0∇ ℓ (G) are well defined on M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' On M′ we have another flat metric given by the second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' On M′ the Levi-Civita connection g0∇ of the flat Riemannian metric g0 and the Levi-Civita connection A∇ of the flat Lorentzian metric A agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Denote by J the complex structure on M induced by g and the orient- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then the Hopf differential Q = A + iAJ is a holomorphic quadratic differential [8], where we write AJ(X, Y ) = A(JX, Y ), for X, Y ∈ Γ(TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In a neighbourhood of every point p ∈ M′ we may find local holomorphic coordinates z = x + iy such that Q = dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We will henceforth call such coordinates Q-adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In such a coordinate system we have g = e2u dz ◦ d¯z = e2u � dx2 + dy2� , and A = Re(dz2) = dx2 − dy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE for some real-valued function u satisfying Liouville’s equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) 4∂2 z¯zu = e−2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It follows that g has Gauss curvature K = −e−4u and hence g0 = √ −Kg = dx2 + dy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, with respect to the coordinate z, the Christoffel symbols of both g0∇ and A∇ vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Covering M′ with local holomorphic Q- adapted coordinates implies that g0∇ = A∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ The differentials Pℓ have the property of being holomorphic away from umbilic points and extending meromorphically across umbilic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let X : M → E3 be a non-flat minimal immersion with in- duced differentials {Pℓ}ℓ⩾2 on the complement M′ of the umbilic locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then the differentials {Pℓ}ℓ⩾2 are holomorphic on M′ and extend meromorphically to all of M with Pℓ having a pole of order ℓ at the umbilic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, the residue of Pℓ at an umbilic point p ∈ M is Resp(Pℓ) = � −1 2 �ℓ+1 (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' (n + 2)ℓ−2(3n2 + 4n), where n denotes the order of vanishing of the Hopf differential at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Suppose that P is a meromorphic differential of degree ℓ ∈ N having a pole of order ℓ at some point p, so that there exists a local p-centred holomorphic coordinate z satisfying P(z) = f(z) zℓ dzℓ, where f is a holomorphic function near 0 with f(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then the residue of P at p is defined as Resp(P) = f(0), which is clearly well-defined, that is, independent of the choice of local p- centred holomorphic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let p ∈ M′ and let w : Up → C be a local holo- morphic coordinate defined in a neighbourhood of p so that Q = dw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' There- fore, in such a coordinate system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6) becomes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) P2 = {G, w} dw2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) Pℓ+1 = ∂wpℓdwℓ+1 where we write Pℓ = pℓdwℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since the Schwarzian derivative maps mero- morphic locally injective functions to holomorphic quadratic differentials, it follows that P2 and hence the differentials Pℓ on M′ are holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Now suppose z is any local holomorphic coordinate defined in some p- neighbourhood Vp so that Q = qdz2 for some holomorphic function q on Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that the umbilic points are isolated since they are precisely the points where the holomorphic quadratic differential Q vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Hence, at an THE DEGREE OF A CLASSICAL MINIMAL SURFACE 9 umbilic point p ∈ M we may choose a local p-centred holomorphic coordinate z so that X3 = Re(z) and G = 1 + azn+1 + bzn+2 + O(zn+2), where a ∈ C∗ and b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From this we compute that Q = −a(n + 1)zndz2 + O(zn), so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5) P2 = − ��3n2 + 4n 8 � z−2 + n(n + 2) (n + 1) b az−1 � dz2 + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Hence we may write P2 = f2,n(z) z2 dz2, where the complex-valued function f2,n is holomorphic near 0 and satisfies f2,n(0) = −3n2 + 4n 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5) together with a straightforward inductive argument we see that Pℓ = �ℓ−1 � k=0 cℓ,k zk−ℓ � b a �k� dzℓ + O(1) for some coefficients cℓ,k where cℓ,0 = � −1 2 �ℓ+1 (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' n(n + 2)ℓ−2(3n + 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It follows that we may write Pℓ = fℓ,n(z) zℓ dzℓ where the complex-valued function fℓ,n is holomorphic near 0 and satisfies Resp(Pℓ) = fℓ,n(0) = cℓ,0, thus completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In [2] it was shown that the real part of P2 is a conservation law for critical points of a certain curvature entropy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For this reason we call {Pℓ}ℓ⩾2 the entropy differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In fact, it was shown that if we write g = exp(2u)dz ◦ d¯z for some real-valued function u and some local Q-adapted holomorphic coordinate z so that Q = dz2, then P2 is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6) P2 = −2 � ∂2 zzu + (∂zu)2� dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Invariance properties The entropy sequence enjoys certain invariance properties which we will now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' First we observe (see also [2]): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The entropy sequence is intrinsic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Pℓ+2 depends on the induced metric and its derivatives up to order ℓ + 4 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In particular, Pℓ+2 is invariant under post-composing X : M → E3 with an ambient isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We work in a local Q-adapted local coordinate z so that Q = dz2 and g = e2udz ◦ d¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consider the metric ˆg = (−K)3/4g with Gauss curvature Kˆg = 1 2|Kg|1/4 where Kg denotes the Gauss curvature of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Now define T = ˆg˚∇2 log(Kˆg) = 1 4 ˆg˚∇2 log(−Kg) where ˆg˚∇2 denotes the trace-free Hessian of ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Clearly, the symmetric trace- less covariant 2-tensor field T depends on g and its derivatives up to order four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Using the identity u = − 1 4 log(−Kg), we compute T = −g0˚∇2u − du2 + 1 2g0(g0∇u, g0∇u)g0 = −2 Re �� ∂2 zzu + (∂zu)2� dz2� , which agrees with the real part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It follows that P2 depends on the induced metric only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since Pℓ+2 is just the ℓ-th derivative of P2 with respect to the Levi-Civita connection of the induced flat metric g0, we see that Pℓ+2 depends on the induced metric and its derivatives up to order ℓ+4 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ In order to discuss the further invariance properties we first recall the Weierstrass representation of minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We consider C3 and let φ denote the natural complex inner product φ(z, w) = z1w1 + z2w2 + z3w3, where z = (zi) and w = (wi) are elements of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Weierstrass observed that if (M, J) is a Riemann surface and ˜X : M → C3 is a holomorphic null immersion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' ˜X∗φ = 0, then X = Re ◦ ˜X : M → R3 is a conformal and minimal immersion of (M, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, every minimal immersion of a simply connected surface M arises in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Having a holomorphic null immersion ˜X : M → C3, the triple (M, G, η), where G is the stereographically projected Gauss map of the minimal immer- sion Re( ˜X) and η = d ˜X3 is called the Weierstrass data of the null immersion ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Standard computations give (see for instance [10]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) Q = − 1 GdG ◦ η, where Q denotes the Hopf differential of Re( ˜X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A (global) method of producing a holomorphic null immersion of a Riemann surface (M, J) into C3 from Weierstrass data was obtained by Osserman [17]: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let G : M → ˆC be a meromorphic function and η a holo- morphic 1-form on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Suppose that: (i) The zeroes of η coincide with the poles and zeros of G, with the same order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' (ii) For any closed curve γ ⊂ M, � γ Gη = � γ η G, Re � γ η = 0, where ¯z denotes complex conjugation of z ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' THE DEGREE OF A CLASSICAL MINIMAL SURFACE 11 then ˜X(p) − ˜X(p0) = � p p0 �1 2(G−1 − G), i 2(G−1 + G), 1 � η, yields a holomorphic null immersion ˜X : M → C3 so that Re( ˜X) has Weier- strass data (M, G, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The natural left action of the linear conformal group C∗ × SO(3, C) on C3 yields a left action on the space of holomorphic null immersions and consequently on the space W(M) of minimal immersions of M arising via the Weierstrass representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For an element X : M → E3 ∈ W(M) we call the deformations of X obtained by the C∗ × SO(3, C) action its Weier- strass deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Next, following [12], we study the space of Weierstrass deformations more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The action by the subgroup R+×SO(3, R) corresponds to similarity trans- formations of the minimal surface associated to the null immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It follows that the space of non-similar Weierstrass deformations is the homogeneous space (C∗ × SO(3, C)) / � R+ × SO(3, R) � ≃ S1 × (SO(3, C)/SO(3, R)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The first circle factor yields the well-known associated family (or Bonnet family) of minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The associated family of minimal surfaces has the properties of being locally isometric and sharing a common Gauss map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, the Hopf differential Q changes by a complex phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The latter factor gives rise to the so-called Goursat family of minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The group SO(3, C) ≃ PSL(2, C) acts on the Gauss-map by Möbius trans- formation and leaves the Hopf differential unchanged (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' [7, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1] or [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since under a Bonnet-transform the induced metric is unchanged, so is the induced flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It follows that the differentials {Pℓ}ℓ⩾2 are the same for the whole S1-family of associated minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, since the Hopf differential is unchanged under a Goursat transform, so is the Levi-Civita connection of the second fundamental form A∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5 implies that the Levi-Civita connection g0∇ of the flat metric is invariant under Goursat transform as well (this is noteworthy since the induced metric itself does change non-trivially under Goursat transforms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Hence it follows from the invariance of the Schwarzian derivative under post-composition by a Möbius transformation that P2 and hence all differentials {Pℓ}ℓ⩾2 are invariant under Goursat transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Concluding, we have: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The meromorphic differentials {Pℓ}ℓ⩾2 are invariant un- der Goursat – and Bonnet transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Finally, note that scaling the immersion X : M → E3 by a constant does neither change the Levi-Civita connection of the induced flat met- ric, nor the Gauss map and hence leaves the sequence {Pℓ}ℓ⩾2 unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' This fact together with the content of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='11 is summarized in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1 of the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The degree of a minimal surface The existence of an intrinsic sequence of meromorphic differentials on a min- imal surface motivates the following definition: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let M be an oriented smooth surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A non-flat minimal immersion X : M → E3 is said to have degree n ∈ N if Pn vanishes identically or if there exists a weighted-homogeneous polynomial ψn : Cn−2 → C of degree n with weights (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , n − 1) such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) Pn = ψn(P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Pn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, we call ψn the algebraic type of the immersion X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Clearly, if a non-flat minimal immersion has degree n, then it also has degree m for all m ⩾ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The degree will therefore always denote the smallest integer for which a relation of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall, polynomial ψn(z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , zn−1) which does not vanish identically is called weighted-homogeneous of degree n if there exist positive integers (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , wn−1), called the weights of the variables, such that for every λ ̸= 0 ψn(λw1z1, λw2z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , λwn−1zn−1) = λnψn(z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' , zn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A definition of degree for constant mean curvature surfaces without umbilic points was previously given by Pinkall and Sterling [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4 (Enneper’s surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Enneper’s surface is the minimal surface with Weierstrass data M = C, G(z) = z and η(z) = zdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we compute Q(z) = −dz2 and hence using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z) = {z, z} dz2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, Enneper’s surface has the lowest possible degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, it was shown in [2] that a minimal surface satisfying P2 = 0 is – up to Euc- lidean motion and scaling – an open subset of Enneper’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Thus En- neper’s surface and its Weierstrass deformations exhaust all degree 2 surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that Enneper’s Weierstrass deformations are just similarity transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A characterisation of Enneper’s surface in terms of so-called Chern–Ricci functions was given in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6 (The helicoid & catenoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The helicoid is the minimal surface with Weierstrass data M = C, G(z) = ez and η(z) = idz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we compute Q(z) = −idz2 and hence using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z) = {ez, z} dz2 = −1 2dz2, so that P3(z) = ∂z � −1 2 � dz3 = 0, THE DEGREE OF A CLASSICAL MINIMAL SURFACE 13 showing that the helicoid is a degree 3 surface (and consequently so is the catenoid, since it is a member of the helicoid’s associated family of minimal surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Conversely, it was proved in [2] that a minimal surface satisfying P2 = λQ for some non-zero complex constant λ is – up to Euclidean mo- tion, scaling and Goursat transform – an open subset of a member of the helicoid/catenoid family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since P2 = λQ precisely characterizes the degree 3 surfaces, it follows that the helicoid and its Weierstrass deformations exhaust all degree 3 surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Degree four surfaces A minimal immersion with entropy differentials {Pℓ}ℓ⩾2 is of degree 4 if and only if there exists a complex constant λ such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) P4 − 6λ(P2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, we have a (complex) one-dimensional space of algebraic types of degree 4 immersions and a complete classification is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, locally and away from umbilic points a somewhat explicit description is avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let therefore X : M → E3 be a degree 4 immersions and let z : U → C be a local holomorphic coordinate in which the entropy differential Q of X satisfies Q = dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Writing P2 = pdz2 for some holomorphic function p on U, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) becomes ∂2 zzp − 6λp2 = 0, so that for λ ̸= 0, we obtain p(z) = 1 λ℘(z + c1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 0, c2) for complex constants c1, c2 where ℘(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' g2, g3) denotes the Weierstrass elliptic function with invariants g2 and g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, the Schwarzian derivative of the Gauss map of a degree 4 immersion, computed with respect to the coordinate z, is a Weierstrass ℘-function or a polynomial of degree 1 in z (provided λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The space of algebraic types of degree four surfaces simplifies in the pres- ence of umbilic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Suppose X : M → E3 is a smooth minimal immersion of degree four whose Hopf differential vanishes to order n ∈ N at some umbilic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then the Hopf differential vanishes to order n at every umbilic point and the immersion X satisfies P4 + 12(n + 2)2 (3n2 + 4n)(P2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let p ∈ M denote the umbilic point at which the Hopf differential vanishes to order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6 implies that Resp(P2) = −3n2 + 4n 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' and hence Resp((P2)2) = (Resp(P2))2 = �3n2 + 4n 8 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE Since X has degree four there exists a complex-constant λ such that P4 − 6λ(P2)2 = 0 and hence using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='6 we must have Resp(P4) = − 3 16(n + 2)2(3n2 + 4n) = 6λ (Resp(P2))2 = 6λ �3n2 + 4n 8 �2 , which implies P4 + 12(n + 2)2 (3n2 + 4n)(P2)2 = 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Furthermore, note that the sequence an = 12(n+2)2 (3n2+4n) is injective which excludes the possibility of having two umbilic points at which the Hopf differential vanishes to different order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='8 (Enneper surfaces of higher dihedral symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The minimal surfaces arising from the Weierstrass data M = C, G(z) = zk and η(z) = zkdzk for some integer k ⩾ 2 are called Enneper surfaces of higher dihedral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) Q(z) = −kzk−1dz2, showing that these surfaces have an umbilic point at which the Hopf differ- ential vanishes to order k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7, if the Enneper surfaces of higher dihedral symmetry have degree four, then they will have to satisfy P4 + 12 (k + 1)2 (k − 1)(3k + 1)(P2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) we get γ = (k−1) 2z , hence with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z) = �� zk, z � − (k − 1)(k + 3) 8z2 � dz2 = − �(k − 1)(k + 1) 2z2 + (k − 1)(k + 3) 8z2 � dz2 = − 1 8z2 (k − 1)(3k + 1)dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) we have P3(z) = � −1 8(k − 1)(3k + 1)∂z � 1 z2 � + 1 4 (k − 1) 2z �(k − 1)(3k + 1) z2 �� dz3 = 1 8z3 (k − 1)(k + 1)(3k + 1)dz3, and likewise P4(z) = � − 3 8z4 (k − 1)(k + 1)(3k + 1) − 3 2z (k − 1) 1 8z3 (k − 1)(k + 1)(3k + 1) � dz4 = − 3 16z4 (k + 1)2(k − 1)(3k + 1)dz4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Hence we obtain P4 + 12 (k + 1)2 (k − 1)(3k + 1)(P2)2 = 0, THE DEGREE OF A CLASSICAL MINIMAL SURFACE 15 which agrees with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The sequence an = 12(n + 2)2 (3n2 + 4n) describing the admissible coefficients for degree four minimal immersions with umbilic points satisfies limn→∞ λn = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It is therefore natural to ask what (umbilic-free) minimal surfaces satisfy P4 + 4(P2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We will next study a 1-parameter family of such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9 (Limit degree four surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let M = C with Gt(z) = 1 t e−z and ηt(z) = 2tezdz where t > 0 is a real parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From this Weierstrass data we obtain a family of minimal immersions Xt : R2 → R3 given by (x, y) �→ � x − t2 2 e2x cos(2y), y + t2 2 e2x sin(2y), −2tex cos(y) � where z = x + iy denotes the standard coordinate on C ≃ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that the immersion X0 yields the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The induced metrics are gt = � 1 + t2e2x�2 � dx2 + dy2� , and the second fundamental forms are At = 2tex � cos(y) � dx2 − dy2� − 2 sin(y) (dx ◦ dy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The metrics gt have Gauss curvature Kt = − 4t2e2x (1 + t2e2x)4 Writing Σt = Xt(C), these surfaces are singly periodic with respect to the action Σt × Z → Σt (p, n) �→ p + \uf8eb \uf8ed 0 2nπ 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' for n ∈ Z and p ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The fundamental domain Xt(R×[−π, π]) is symmetric around y = 0 and contains two straight lines defined by y = ±(1/2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note moreover that ∂y is a Killing vector field for gt which is the imaginary part of a holomorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we obtain Q(z) = 2t ezdz2 and from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='9) we get γ = 1 2, hence with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z) = − 3 8dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Likewise we obtain P3(z) = 3 8dz3 and P4 = − 9 16dz4 so that P4 + 4(P2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A portion of the image of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Degree five surfaces A minimal immersion with entropy differentials {Pℓ}ℓ⩾2 is of degree 5 if and only if there exists a complex constant λ such that P5 + λP2P3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, again, we have a (complex) one-dimensional space of algebraic types of degree 5 immersions and a complete classification is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, in entirely similar fashion to the degree four case we can prove: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Suppose X : M → E3 is a smooth minimal immersion of degree five whose Hopf differential vanishes to order n ∈ N at some umbilic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then the Hopf differential vanishes to order n at every umbilic point and the immersion X satisfies P5 + 24 (n + 2)2 (3n + 4)nP3P2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The sequence an = 24 (n + 2)2 (3n + 4)n satisfies a1 = 216/7 and a∞ = limn→∞ an = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Both coefficients a1 and a∞ are realized by well-known surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='11 (Schwarz family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Schwarz primitive triply periodic sur- face is the minimal surface with Weierstrass data M = � (z, w) ∈ ˆC2 | w2 = z8 − 14z4 + 1 � and G(z, w) = z and η(z, w) = z wdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we compute Q(z, w) = − 1 wdz2 and hence using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z, w) = −42z2(z4 + 1)2 w4 dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Simple computations give P3(z, w) = 84z(z16 + 34z12 − 34z4 − 1) w6 dz3, THE DEGREE OF A CLASSICAL MINIMAL SURFACE 17 and P4(z, w) = − 84 w8 (z24 + 282z20 + 1887z16 − 884z12 + 1887z8 + 282z4 + 1)dz4, as well as P5(z, w) = 108864z3(z24 + 36z20 + 69z16 − 69z8 − 36z4 − 1) w10 dz5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Consequently, we have P5 + 216 7 P2P3 = 0, thus showing that the Schwarz family of minimal surfaces has degree 5 and realises the first possible coefficient 216 7 among degree 5 surfaces with umbilic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The limit a∞ = 8 is also realized by a well-known family of minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='12 (Scherk family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The singly-periodic Scherk surface is the minimal surface with Weierstrass data M = ˆC \\ {±1, ±i}, G(z) = z and η(z) = iz (z4−1)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) we compute Q(z) = − i z4 − 1dz2 and hence using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10) we obtain P2(z) = − 6z2 (z4 − 1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Simple computations give P3(z) = 12z(z4 + 1) (z4 − 1)3 , P4(z) = −12(z8 + 10z4 + 1) (z4 − 1)4 and P5(z) = 576z3(z4 + 1) (z4 − 1)5 , so that P5 + 8P2P3 = 0, showing that Scherk’s family of minimal surfaces has degree 5 and realizes the limit coefficient a∞ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Degree seven surfaces Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Taking M = ˆC \\ {z : zk = 1} and G(z) = zk−1 as well as η(z) = zk−1 (zk−1)2 dz gives the k-noids of Jorge & Meeks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' With the help of a computer algebra system one easily verifies that for k > 2 the k-noids satisfy 0 = P7 + akP5P2 + bkP4P3 + ckP3(P2)2 18 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE and hence are surfaces of degree 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The coefficients are ak = 16(3k − 2)(k − 2) 3k2 − 16k + 8 , bk = 8(27k4 − 144k3 − 56k2 + 128k − 32) (3k2 − 16k + 8)(k − 2)(3k − 2) , ck = 192k2 3k2 − 16k + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Approximating minimal surfaces It is natural to ask if a minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In this section we will show that locally and away from umbilic points this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let therefore M be an oriented surface and X : M → E3 a non-flat minimal immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For p ∈ M′ pick a simply connected neighbourhood Up so that Up does not contain any umbilic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' After possibly shrinking Up and applying an ambient rotation, we may assume that the stereographically projected Gauss map G of X does not attain the value ∞ on Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let Q denote the Hopf differential and P2 denote the first entropy differential of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We take a local p-centred holomorphic coordinate z so that Q = dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By analytic continuation we may extend z to all of Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We write P2 = ρ 2dz2 for some holomorphic function ρ on Up, where, by definition, we have ρ 2 = {G, z} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall the following classical fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let ρ be a holomorphic function on a simply connected do- main Ω in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then for every prescription of w(z0) and (∂zw)(z0) at some point z0 ∈ Ω, there exists a unique holomorphic function w on Ω satisfying Hill’s equation with potential ρ/4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) ∂zzw + ρ 4w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For a proof see for instance [14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By a straightforward power series coefficient comparison argument, we also obtain: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let D ⊂ C be the open unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For every holomorphic function ρ on D and for all w0, w′ 0 ∈ C, the sequence wn : D → C with wn being the unique solution of ∂2 zzwn + ρn 4 wn = 0, wn(0) = w0, (∂zwn)(0) = w′ 0, converges locally uniformly to the unique solution of ∂2 zzw + ρw = 0, w(0) = w0, (∂zw)(0) = w′ 0, where ρn denotes the Taylor series around 0 of ρ up to order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' THE DEGREE OF A CLASSICAL MINIMAL SURFACE 19 Hill’s equation is linear and by Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) its space of solutions is complex two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let therefore w1, w2 : Ω → C be a pair of linearly independent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Their Wronskian W(w1, w2) = w1∂zw2 − w2∂zw1 is a non-zero constant W by Abel’s identity, and w1 and w2 cannot vanish simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Writing ˆG = w1 w2, we obtain ∂z ˆG = w2∂zw1 − w1∂zw2 (w2)2 = − W (w2)2 so that ˆG is a locally injective meromorphic map except possibly on the zero locus of w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Computing ∂z(1/ ˆG) we see that ˆG is locally injective away from there zero locus of w1, and thus everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A simple computation now gives { ˆG, z} = −2∂2 zzw2 w2 = ρ 2, where we have used the fact that w2 solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' From the cocycle prop- erty (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='8) of the Schwarzian derivative we see that ˆG differs from the stereo- graphically projected Gauss map by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In particular, on Up, there exists a pair of solutions w1, w2 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) – unique up to res- caling by a common constant – so that G = w1/w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since G does never attain the value ∞ on Up the function w2 is non vanishing on Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Fix such a pair (w1, w2) and let w1,n and w2,n be the unique solutions on Up to Hill’s equation with potential ρn/4 which satisfy wi,n(p) = wi(p), and (∂zwi,n)(p) = (∂zwi)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Here ρn denotes the power series of ρn around p of order (n−3) with respect to the coordinate z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Note that it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='7) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10 that we may recover X : Up → E3 by taking the Weierstrass data (G, −G/(∂zG)dz) on Up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, we may also take (Gn, −Gn/(∂zGn)dz) as Weierstrass data on Up where Gn are the locally injective meromorphic functions on Up defined by Gn = w1,n w2,n After possibly shrinking Up again, we can assume that the functions Gn are holomorphic for sufficiently large n, and hence by applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='10 again we obtain a sequence of minimal immersions � Xn : Up → E3� n⩾n0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By construction, Xn has degree n, since ρn is just a polynomial of order (n − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2, the functions wi,n converge locally uniformly to wi as n tends to infinity and since w2 is non-vanishing, Gn converges locally uniformly to G as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Since Xn arises by integration from the Weierstrass data (Gn, −Gn/(∂zGn)dz), it follows that ∥Xn − X∥gEucl converges to zero locally uniformly as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 20 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Higher order Schwarzian derivatives For background about the Schwarzian derivative we refer to reader to [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We have defined the Schwarzian derivative with respect to a conformal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' However, less geometric structure is required on a Riemann surface (M, [g]) to define the Schwarzian derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' All one needs is a Möbius structure [4], which is a generalization of the classical notion of a complex projective structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A Möbius structure is a linear second order differential operator which can be written as the sum of the symmetrized trace-free Hessian of a [g]-conformal connection ∇ and the real part of a quadratic differential Q H = Sym0(Hess(∇)) + Re(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The differential operator H acts on densities of weight −1/2 on M and takes values in the symmetric and [g]-traceless covariant 2-tensor fields of weight −1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' H : Γ � Λ2(T ∗M)−1/2� → Γ � S2 0(T ∗M) ⊗ Λ2(T ∗M)−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Schwarzian derivative of a local biholomorphism ϕ : (M, [g], H) → (M′, [g]′, H′) between Riemann surfaces equipped with Möbius structures is then defined to be ϕ∗H′−H, where the pullback is defined in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The classical definition of the Schwarzian derivative is recovered by taking both (M, [g]) and (M′, [g]′) to be ˆC and H the Möbius structure associated to the Levi-Civita connection of the spherical metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A Möbius structure H on (M, [g]) is called flat if in a neighbourhood of every point of M there exists a local holomorphic coordinate z so that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1) H = Re(∂2 zz) with respect to the canonical local trivialisations of the vector bundles Λ2(T ∗M)−1/2 and S2(T ∗M) ⊗ Λ2(T ∗M)−1/2 induced by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We call such a coordinate sys- tem adapted to the flat Möbius structure H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' It is straightforward to check that any two overlapping adapted holomorphic coordinates are related by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall that a complex projective structure on a Riemann surface (M, [g]) is a (maximal) atlas of charts mapping open sets into CP1 so that trans- ition functions are (restrictions) of Möbius transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Therefore, a flat Möbius structure induces a complex projective structure and conversely every complex projective structure induces a flat Möbius structure by defin- ing H as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We refer the reader to [4] for additional details on Möbius structures and to [6] for complex projective structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' A crucial difference between having a conformal connection at hand versus a Möbius structure only, is when it comes to defining higher order Schwarzian derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The operators SH ℓ from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1 below are well defined on a Riemann surface (M, [g]) equipped with a (flat) Möbius structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Al- though they arise as derivatives of the Schwarzian derivative – and hence may be thought of as higher order Schwarzian derivatives – for ℓ ⩾ 3 they all do lack the invariance under post-composition by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let (M, [g]) be a Riemann surface equipped with a flat Möbius structure H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Then for ℓ ⩾ 2 there exists a differential operator SH ℓ THE DEGREE OF A CLASSICAL MINIMAL SURFACE 21 of order ℓ + 1 mapping locally injective meromorphic functions on M to holomorphic differentials of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In a local H-adapted coordinate z on M the operators are given by SH ℓ+1(f) = � ∂z (sℓ(f)) − ℓsℓ(f)∂2 zzf ∂zf � dzℓ+1 with SH 2 (f) = � ∂3 zzzf ∂zf − �∂2 zzf ∂zf �2� dz2 where we write SH ℓ (f) = sℓ(f)dzℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' These operators can also be defined in the case where H is not flat, for the sake of brevity we will however not show this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' In local coordinates, they appeared in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For instance, in a local H-adapted coordinate z we have SH 3 (f) = � ∂4 zzzzf ∂zf − 6 (∂3 zzzf)(∂2 zzf) (∂zf)2 + 6 �∂2 zzf ∂zf �3� dz3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The reader may easily verify that SH 3 (f) is not invariant under post-compo- sition by a Möbius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' For comparison, we mention that if ∇ is a [g]-conformal connection admitting a local holomorphic coordinate z in which 1 2γ2 − ∂zγ = 0, then we have S∇ 3 (f) = � ∂4 zzzzf ∂zf − 4 (∂3 zzzf)(∂2 zzf) (∂zf)2 + 3 �∂2 zzf ∂zf �3� dz3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We only need to show that the definition of the operators SH ℓ is invariant under coordinate changes of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2) w = az + b cz + d, with � a b c d � ∈ SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We have dw = τdz and ∂w = τ −1∂z with τ = (ad − bc) (cz + d)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The proof is by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Recall the (well-known) fact that the Schwarzian SH 2 as defined above is invariant under coordinate changes of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Let us therefore assume that SH ℓ is invariant under coordinate changes of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' We write SH ℓ (f) = ˆsℓ(f)dwℓ = sℓ(f)dzℓ so that ˆsℓ(f) = τ −ℓsℓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Now using ∂zτ = − 2c (cz+d)τ, we obtain (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) ∂w (ˆsℓ(f)) dwℓ+1 = � τ −(ℓ+1)∂z(sℓ(f)) + τ −1sℓ(f)∂zτ −l� dwℓ+1 = ∂z(sℓ(f))dzℓ+1 + sℓ(f) 2c (cz + d)ℓdzℓ+1, 22 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' METTLER AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' POERSCHKE as well as (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) ℓˆsℓ(f)∂w (∂wf) ∂wf dwℓ+1 = ℓτ −ℓsℓ(f)∂z � τ −1∂zf � ∂zf τ ℓ+1dzℓ+1 = ℓsℓ(f)∂2 zz(f) ∂zf dzℓ+1 + ℓτsℓ(f)τ −1 2c (cz + d)dzℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='4) proves the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' The Schwarzian derivative can be generalized to higher di- mensions from the conformal viewpoint [16] as well as from the (complex) projective viewpoint [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Aharonov, A necessary and sufficient condition for univalence of a meromorphic function, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 36 (1969), 599–604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' MR 0249622 5 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Bernstein, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Mettler, Characterizing classical minimal surfaces via the en- tropy differential, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' 27 (2017), 2235–2268.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='2307/1971425 MR 1014929 12 [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Schippers, Distortion theorems for higher order Schwarzian derivat- ives of univalent functions, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' Soc.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='1007/BF01444214 MR 1386108 5 Faculty of Mathematics and Computer Science, UniDistance Suisse, Brig, Switzerland Email address: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='mettler@fernuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='ch Faculty of Mathematics and Computer Science, UniDistance Suisse, Brig, Switzerland Email address: lukas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='poerschke@fernuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='ch,lukpoerschke@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FKT4oBgHgl3EQfAy1w/content/2301.11700v1.pdf'} diff --git a/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/2301.02752v1.pdf.txt b/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/2301.02752v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9a1a93a8fa9dfa8c242c66e48d15a68de981b33 --- /dev/null +++ b/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/2301.02752v1.pdf.txt @@ -0,0 +1,694 @@ +arXiv:2301.02752v1 [math.GR] 7 Jan 2023 +finite normal subgroups of strongly verbally closed groups +Filipp D. Denissov +Faculty of Mathematics and Mechanics of Moscow State University +Moscow 119991, Leninskie gory, MSU. +Moscow center for Fundamental and Applied Mathematics. +denissov.filipp@gmail.com +In the recent paper by A. A. Klyachko, V. Yu. Miroshnichenko, and A. Yu. Olshanskii, it is proven +that the center of any finite strongly verbally closed group is its direct factor. One of the results of the +current paper is the generalization of this nontrivial fact to the case of finite normal subgroups of any +strongly verbally closed groups. It follows from this generalization that finitely generated nilpotent +groups with nonabelian torsion subgroups are not strongly verbally closed. +1. Introduction +A subgroup H of a group G is called verbally closed [MR14] if any equation of the form +w(x1, x2, . . . , xn) = h, where w is an element of the free group F(x1, . . . , xn) and h ∈ H, +having solutions in G has a solution in H. If each system of equations with coefficients from H +{w1(x1, . . . ) = 1, . . . , wm(x1, . . . ) = 1}, where wi ∈ H ∗ F(x1, . . . , xn) (and ∗ means the free product), +having solutions in G has a solution in H, then the subgroup H is called algebraically closed in G. Note +that if the subgroup H is algebraically closed in the group G, then it is verbally closed in G. +A group G is called strongly verbally closed if it is algebraically closed in any group containing G as a +verbally closed subgroup. Thus, the verbal closedness (as well as the algebraic closedness) is a property +of a subgroup, while the strong verbal closedness is a property of an abstract group. The class of strongly +verbally closed groups is fairly wide. For example, it includes +— all abelian groups [Maz18], +— all free groups [KM18], +— all virtually free groups containing no nontrivial finite normal subgroups [KM18], [KMM18], +— all groups decomposing nontrivially into a free product [Maz19], +— fundamental groups of all connected surfaces except the Klein bottle [Maz18], [Kly21], +— all finite groups with nonabelian monolith [KMO21], +— inifinite dihedral group [KMM18] and any finite dihedral group whose order is not divisible by 8 +[KMO21], +— all acylindrically hyperbolic groups with no nontrivial finite normal subgroups [Bog22]. +The class of non-strongly-verbally-closed groups is fairly wide too. Among such groups are the following: +— the already mentioned fundamental group of the Klein bottle [Kly21], +— the discrete Heisenberg group [KMO21], +— any finite group, whose center is not its direct factor (in particular, any finite nonabelian nilpotent +group) [KMO21], [RKK17], [KM18]. +Proving the strong verbal closedness (as well as its absence) of a group is not easy. In [KMO21], for +example, a question is raised: +Question 1. Does there exist a finitely generated nilpotent nonabelian strongly verbally closed group? +A negative answer to this question would yield a broad generalization of the last two examples of +non-strongly-verbally-closed groups mentioned above. So far, we managed to give a partial answer to +this question. More precisely, we proved the absence of strong verbal closedness of finitely generated +nilpotent groups with nonabelian torsion subgroups and of some finitely generated nilpotent nonabelian +groups with abelian torsion subgroups. +1 + +A property that is stronger than strong verbal closedness is the property of being a strong retract +[KMO21]. A group H is called a strong retract if it is a retract of any group G ⩾ H from the variety +var H. +Let us recall some terminology [Neu67]: +— the variety generated by a class of groups K is the class of all groups satisfying all identities that +hold in all groups from K, +— the variety generated by a group G is designated by var G. +This gives rise to the following question from [KMO21]: +Question 2. What is an arbitrary finite strong retract? +In [KMO21] some examples of strong retracts are provided. In the next section, we describe all the +nilpotent strong retracts. +Below we provide a brief list of notation we use. +If x, y are elements of some group, then the symbol [x, y] denotes their commutator x−1y−1xy. The +symbol ord(x) denotes the order of an element x of a group G. The center of a group G is denoted +by Z(G), and its commutator subgroup is denoted by G′. Centralizer of a subset X of a group G is +denoted by C(X). The symbol ⟨⟨X⟩⟩ stands for the normal closure of a subset X of a group G (that is +the intersection of all normal subgroups of G containing X). The free group with a basis X is denoted as +F(X) or Fn in case X has n ∈ N elements. Identical mapping from X to itself is denoted by id. We use +the symbol H ∼= G to express the fact that groups H and G are isomorphic. Finally, the symbol H ⩽ G +denotes the fact that a group H is a subgroup of G. The symbol H ⊴ G denotes the fact that H is a +normal subgroup of G. +The author is grateful to his supervisor Anton Alexandrovich Klyachko for formulation of the problem +and for valuable remarks during the work. +2. Nilpotent strong retracts +Note that in case when G is an abelian group, H ⩽ G is its retract if and only if H is a direct summand +of G. It means that the property of being a strong retract for the abelian group G is equivalent to the +property of G being a direct summand of any group H ∈ var G containing G. For the further discussion, +we need the description of all varieties of abelian groups (see [Fuc70], paragpaph 18, exercise 7): +Varieties of abelian groups are precisely the following classes of groups: 1) the class of all +abelian groups; 2) the class of all abelian groups of exponent n ∈ N. +To begin with, consider the case, when G is not a group of bounded period. Then, according to the +description, var G is the class of all abelian groups. The following is true of divisible abelian groups (see, +for example, [Kur60]): +If G is a divisible abelian group, and H is an abelian group such that G ⩽ H, then G is a +direct summand of H. +Let us remind that a group G is called divisible if for any g ∈ G and n ∈ N, the equation xn = g has a +solution in G. +Proposition 1. An abelian group G of unbounded period is a strong retract if and only if it is divisible. +Proof. Sufficiency follows from the fact provided above. Let G be an abelian group of unbounded period. +Then, as it was noted earlier, var G is the class of all abelian groups. In particular, var G contains a +divisible group H containing G [Kur60]. Though, if G is not divisible itself, it is not a direct summand +of H (as direct summands of a divisible group are divisible themselves [Kur60]), so G is not a strong +retract. +Let us move on to abelian groups of bounded period. The first Pr¨ufer theorem provides a complete +description of these groups [Kur60]: +2 + +An abelian group G of bounded period d is a direct sum of primary cyclic groups, i.e. G ∼= +� +i∈I Zpki +i , where pi are prime numbers and ki are natural numbers such that pki +i |d, i ∈ I (I +is an index set). +We need the following variation of the Zorn’s lemma [Fuc70]: +Let M ̸= ∅ be a partially ordered set. Suppose that every chain in M (a totally ordered +subset of M) has an upper bound. Then M contains a maximal element. +Now, we are ready to proceed with our description: +Proposition 2. An abelian group G of bounded period is a strong retract if and only if in its decom- +position into the direct sum of primary cyclic groups, orders of any distinct direct summands are either +equal or coprime: +G ∼= +m +� +i=1 +Cpki +i (ni), where Cpki +i (ni) is equal to the direct sum of ni copies of the group Zpki +i , +where all prime numbers pi are distinct, m, ki ∈ N, and ni are some cardinal numbers. +Proof. Suppose that G cannot be decomposed into such a direct sum. We may assume that +G = +m +� +i=1 +� +j∈Ii +Zp +kj +i +, +(1) +where m ∈ N, |Ii| = ni and among kj, j ∈ Ii there are only finitely many different ones (because G is a +group of bounded period) but there exists i ∈ {1, . . . , m} such that for some j1, j2 ∈ Ii, kj1 ̸= kj2. +Consider the group: H = +m +� +i=1 +Cpsi +i (ni), where +si = max{kj | Zp +kj +i +is a direct summand in the decomposition (1)}, i = 1, 2, . . . , m. +Since both G and H are of the same exponent: exp(G) = �m +i=1 si = exp(H), it follows from the +description of abelian varieties that H ∈ var G . +Consider the injection f : G → H, which works on each direct summand from (1) as follows: let +i ∈ {1, . . . , m}, j ∈ Ii, f : Zp +kj +i +֒→ Zpsi +i , where Zpsi +i +is the jth summand from the decomposition of +Cpsi +i (ni) into the direct sum. Every direct summand from (1) is mapped into the corresponding direct +summand of the decomposition of H, so that the restriction of f to Zp +kj +i +is a natural injection: if kj = si, +then it is the identical map; otherwise it is a mapping to the subgroup of Zpsi +i +of order pkj +i . From the +uniqueness of the decomposition of an abelian group of bounded period into the direct sum of primary +cyclic groups [Fuc70], it follows that f(G) is not a direct summand of H. Thus, G is not a strong retract. +Now, suppose that G has the decomposition from the statement of the theorem. Let H ∈ var G and +let f : G ֒→ H be a monomorphism. As any monomorphism preserves the order of an element, the pith +component of G is mapped into the pith component of H under f, so it suffices to prove the theorem +only for the case G = Cpk(n), where p is prime, k ∈ N, and n is some cardinal number. +Let us show that there exists such X ⩽ H that H = f(G) ⊕ X. In Zorn’s lemma, take set of all +subgroups of H having trivial intersection with f(G) as M: +M = {Y ⩽ H | Y ∩ f(G) = {0}}. +Order on M is introduced as follows: for X, Y ∈ M, X ⩽ Y if X is a subgroup of Y . It can be verified +directly that this is an order on M. Set M is nonempty: {0} ∈ M. Any chain {Yα} ⊆ M of subgroups +having trivial intersection with f(G) is bounded by an element Y ∈ M, where Y = ∪αYα. Consequently, +Zorn’s lemma is applicable, and M contains a maximal element X: X ⩽ H, X ∩ f(G) = {0}, and X +3 + +is not a subgroup of any bigger (relatively to the order we introduced earlier) subgroup satisfying this +property. +From X ∩f(G) = {0} it follows that f(G)+X = f(G)⊕X. It remains to prove that H = f(G)+X. +Let h ∈ H. There exists such k ∈ N that kh ∈ f(G) + X. Indeed, otherwise ⟨h⟩ ∩ (f(G) + X) = {0}, +which means that (⟨h⟩ + X) ∩ f(G) = {0}, which leads to a contradiction with the maximality of X. +Let s be the least of such numbers k. Without loss of generality, assume that s is prime or that s = 1 +(otherwise, take a power of h instead of h). Two cases are possible: +1) s = p. Then, ph = f(g) + x for some g ∈ G, x ∈ X. If g = pg1, g1 ∈ G (g1 may be equal to +zero), then ph − f(pg1) = x. However, from h − f(g1) ̸∈ X (as h ̸∈ f(G) + X) it can be obtained that +(X + ⟨h − f(g1)⟩) ∩ f(G) = {0}, which leads to a contradiction with the maximality of X. Consequently, +g ̸= pg1 for any g1 ∈ G. +As g ̸= 0, ord(g) = pk. +Though, ord(ph) = pr < pk, so pr(ph) = 0 = +pr(f(g)) + prx. As the sum f(G) + X is direct, prf(g) = prx = 0, which means that prg = 0, which is +impossible. +2) s ̸= p. For abelian groups of period p, the mapping g �→ sg is an automorphism, so, as sh = f(g)+x +for some g ∈ G, x ∈ X, there exist such g1 ∈ G, x1 ∈ X that g = sg1, x = sx1. Thus, s(h−f(g1)−x1) = 0. +No nontrivial element of H has the order of s, so h = f(g1) + x1. +As a result, H = f(G) ⊕ X, and G is a strong retract. +The next simple theorem shows that consideration of nilpotent groups does not yield any new strong +retracts: +Proposition 3. Nilpotent strong retract is an abelian group. +Proof. Let G be a nilpotent strong retract. Then (see [KM79]), Z(G) ̸= {1}. Put N = G′ ∩ Z(G). +Consider the central product of G with its copy �G with joined subgroup N: +K = G +× +N= � +N +�G = (G × �G)/{(g, g−1)|g ∈ N}, +where � +N = �G′ ∩ Z( �G). The group K is contained in var G as it is a quotient group of the direct product +of groups G, �G ∈ var G. Let ρ be a hypothetical retraction from the group K to its subgroup G. From +the fact that in the group K, the group G commutes with the group �G, we obtain ρ( �G) ⩽ Z(G). Image +of the commutator subgroup under a homomorphism into an abelian group is trivial, so ρ( � +N) = {1}. +Finally, from the definition of retraction, we obtain N = {1}, so G′ = {1}. +As a result, we proved the following theorem: +Nilpotent-strong-retract theorem. Nilpotent strong retracts are precisely divisible abelian groups +and abelian groups of bounded exponent in whose decomposition into the direct sum of primary cyclic +groups, orders of any distinct direct summands are either equal or coprime. +In the next paragraph we show that many nilpotent groups are not even strongly verbally closed. +3. Finite normal subgroups of strongly verbally closed groups +We say that a group presentation ⟨X | R⟩ is finitely presented over a group presentation ⟨Y | S⟩, if there +exist such finite sets A and B that ⟨X | R⟩ ∼= ⟨X′ | R′⟩, where X′ = Y ∪ A, R′ = S ∪ B. +The following lemma reveals that this definition is, in fact, a group property (which means it does not +depend on the choice of a group presentation), so it makes sense to speak about the finite presentability +of one group over the other group: +Lemma 1. Suppose that a group presentation ⟨X | R⟩ is finitely presented over a group presentation +⟨Y | S⟩ and ⟨Y | S⟩ ∼= ⟨Y ′ | S′⟩. Then ⟨X | R⟩ is finitely presented over ⟨Y ′ | S′⟩. +Proof. We may assume that X = Y ∪ A and R = S ∪ B for some finite sets A and B. It is known (see, +for example, [LS15]) that groups defined by group presentations ⟨Y | S⟩ and ⟨Y ′ | S′⟩ are isomorphic if +and only if presentation ⟨Y ′ | S′⟩ is obtained from presentation ⟨Y | S⟩ by applying a finite number of +Tietze transformations: +— adding to the set S an arbitrary set T ⊆ ⟨⟨S⟩⟩ ⊴ F(Y ) of its consequences, +4 + +— adding to the set Y an arbitrary set �Y while ading to S a set {�y = w�y | �y ∈ �Y , w�y ∈ F(Y )}, +and their inverses. +It is sufficient to prove the lemma only for the case, when ⟨Y ′ | S′⟩ is obtained +from ⟨Y | S⟩ by applying one Tietze transformation. +One can easily verify that in case of the first +transformation, X′ = X and R′ = R ∪ T, while in case of the second transformation, X′ = X ∪ �Y and +R′ = R ∪ {�y = w�y | �y ∈ �Y , w�y ∈ F(Y )} provide the desired group presentation. +By virtue of Lemma 1, the following definition may be introduced: +A group G is finitely presented over a group H, if there exists such a presentation of G that it is +finitely presented over any presentation of H. +Lemma 2. Suppose that G contains a subgroup H and a finite normal subgroup N such that G/N is +finitely presented over H/(H ∩ N). Then G is finitely presented over H. +Proof (with minor changes) replicates the proof of the Hall theorem [Hal54] about preservation of finite +presentability of a group under extensions (see also [Rob82]). +Let G be a group, H = ⟨X | R⟩ ⩽ G, and N = ⟨Y | S⟩ ⊴ G be its finite subgroup, where Y and S are +finite sets. By condition of the lemma, the group G/N is finitely presented over H/(H∩N) = ⟨X | R∪C⟩, +where ⟨⟨C⟩⟩ = H ∩ N and the set C is finite. Consequently +G/N ∼= ⟨X ∪ A | R ∪ C ∪ B⟩, +where sets A and B are finite. +Let us construct a presentation of the group G. As the set of generators, take X ∪ A ∪ Y , where +sets X, A, Y are in one-to-one correspondence with sets X, A, Y respectively. The sets R, S, C, and +B are in correspondence with the sets R, S, C, and B respectively. As the set of defining relations, +take the union of the following sets: R, S, C1 = {cw−1 +c +| c ∈ C, wc ∈ F(Y )}, B1 = {bw−1 +b +| b ∈ +B, wb ∈ F(Y )} (c ∈ C and b ∈ B are considered as words from F(X) and from F(X ∪ A) respectively), +T = {a−1yaw−1 +a,y, aya−1v−1 +a,y | a ∈ A, y ∈ Y , wa,y, va,y ∈ F(Y )}: +�G = ⟨X ∪ A ∪ Y | R ∪ S ∪ C1 ∪ B1 ∪ T⟩. +Consider a surjective homomorphism θ : �G → G, defined with the following bijections X → X, A → A, +Y → Y on the generators (defining relations are mapped into true identities under such a map on +generators, so such a homomorphism exists). The restriction θ|K : K → N on the subgroup K = ⟨Y ⟩ ⩽ �G +is an isomorphism as all the relations in the alphabet Y in �G are consequences of the defining relations +S. Besides, K ⊴ �G. +Homomorphism �θ : �G/K → G/N generated by θ, is an isomorphism too. Now, let g ∈ ker θ. Then +gK ∈ ker �θ, but �θ is an isomorphism, so g ∈ K. Finally, θ|K is an isomorphism, so g = 1. +The following lemma provides a criterion for algebraic closedness of a subgroup H of a group G in case, +when G is finitely presented over H (for similar propositions, refer to [MR14]): +Lemma 3. Suppose that H = ⟨X | R⟩ is a subgroup of G and G is finitely presented over H. The +subgroup H is algebraically closed in G if and only if H is a retract of G. +Proof. Suppose H is algebraically closed in G and A = {a1, . . . , am}, B = {s1, . . . , sn} are the sets from +the definition of finite presentability of G over H. The relations si(a1, . . . , am, X) = 1, i = 1, . . . , n are +corresponded to a system of equations with coefficients from H: + + + + + +s1(t1, . . . , tm, X) = 1 +. . . +sn(t1, . . . , tm, X) = 1 +which, by condition, has a solution t1 = a1, . . . , tm = am. By virtue of algebraic closedness of H in G, +this system has a solution t1 = h1, . . . , tm = hm in H. Mapping X ⊔ {a1, . . . , am} → H, x ∈ X �→ x, +ai �→ hi extends to a surjective homomorphism ϕ : G → H, as defining relations of G are mapped into +true identities under such a mapping of generators (note that R is the set of words in the alphabet X). +5 + +This homomorphism is a desired retraction: let h ∈ H, h = v(x1, . . . , xr), xi ∈ X. Applying to this +word the homomorphism ϕ, we get: ϕ(h) = v(ϕ(x1), . . . , ϕ(xr)) = h. +Algebraic closedness of a subgroup H of a group G follows from retractness of H in G for every group +G [MR14]. +Approximation lemma. Let C be a finite elementary abelian p-group (where p is a prime number). +For any k ∈ N, there exists t ⩾ k such that the direct product P = ×t +i=1Ci of copies Ci of C contains a +subgroup R invariant with respect to the diagonal action on P of the endomorphism algebra End C with +the following properties: +1) R ⊆ � ker ρj, where ρj : P → Cj, j = 1, . . . , t are the natural projections, +2) But R · ×j̸∈JCj = P for any subset J ⊆ {1, . . . , t} of cardinality |J| = k, +3) Moreover, each such J is contained in a set J′ ⊇ J such that P = R × (×j̸∈J′Cj); and there +exist integers nij ∈ Z such that the projection π : P → ×j̸∈J′Cj with the kernel R acts as: +Ci ∋ ci �→ � +j̸∈J′ cnij +j +, where cj ∈ Cj is the element corresponding to ci under the isomorphism +Ci ∼= C ∼= Cj. +The following theorem provides a generalization of the result from [KMO21] about the center of a finite +strongly verbally closed group. +The proof is also analogical to the proof of that theorem, with the +exception of some nuances. +Finite-normal-subgroup theorem. Let H be a strongly verbally closed group. For any finite normal +subgroup T of H, for any abelian subgroup A of T, normal in H, it is true that Z(CT (A)) is a direct +factor of CT (A), and some complement is normal in H. Here CT (X) = C(X) ∩ T. +Proof. Let H be such a group, and let L = CT (A). It suffices, for each prime p, to find a homomorphism +ψp : L → Z(L) commuting with the action H ↷ L by conjugations (this action is well-defined as L ⊴ H) +and injective on the p-component of the center Zp(L) of L. Then the homomorphism ψ : L → Z(L), +x �→ +� +p +πp(ψp(x)), where πp : Z(L) → Zp(L) is the projection on the p-component, is injective on Z(L), +so its kernel is the desired complement D (normality of D in H follows from the fact that ψ commutes +with H ↷ L). +Suppose that there are no such homomorphisms for some prime p, i.e. every homomorphism f : +L → Z(L) commuting with the action H ↷ L is not injective on Zp(L). Then it is not injective on the +maximal elementary abelian p-subgroup C ⩽ Zp(L) (it is finite as L is finite). Indeed, if x ̸= 1 ∈ Zp(L) is +an element such that f(x) = 1, then, raising it to the appropriate power d, we get f(xd) = 1 and xd ∈ C, +xd ̸= 1. +Choose t by the approximation lemma applied to C (for some k to be specified later) and consider +the fibered product of t copies of the group H: +Q = {(h1, . . . , ht) | h1L = · · · = htL} ⩽ Ht. +First of all, let us show that the subgroup R ⩽ Ct ⩽ Q from the approximation lemma is normal in Q. +Subgroup R is invariant under the diagonal action of automorphisms Aut C ⩽ End C. It remains to show +that Q acts by conjugations on P = Ct diagonally. It follows from the lemma: +Lemma 4. Let G be a group, and N ⊴ G. If xC(N) = yC(N) for some x, y ∈ G, then x and y act on +N (by conjugations) identically. +Proof. From xC(N) = yC(N) it follows that for some c ∈ C(N), x = yc. Then for n ∈ N, we have: +x−1nx = c−1(y−1ny)c = y−1ny. +The last identity is true, as (due to normality) y−1ny ∈ N and c ∈ C(N). +Let q = (q1, . . . , qt) ∈ Q, p = (p1, . . . , pt) ∈ P. As q1L = q2L = · · · = qtL, then (according to Lemma +4) q−1pq = �q−1p�q, where �q = (q1, . . . , q1). It means that the conjugation action of Q on P is diagonal. +On the other hand, diagonal action by conjugations induces an endomorphism of Ct (due to normality +6 + +of C ⊴ H), and R is invariant with respect to the diagonal action of such endomorphisms, leading to +normality of R in Q. +Put G = Q/R. First, let us show that H embeds into G. The group H embeds into Q diagonally: +h �→ (h, . . . , h), h ∈ H. This homomorphism serves as embedding into G as well, as all projections of any +nontrivial diagonal element of Q are nontrivial (and R is contained in the union of the kernels of these +projections). +Now, let us prove verbal closedness of this diagonal subgroup (denote it as H too) in G. Consider an +equation +w(x1, . . . , xn) = (h, . . . , h) +having a solution in G and let �x1, . . . , �xn be a preimage (in Q) of a solution x1, . . . , xn. Then (in Q): +w(�x1, . . . , �xn) = (hc1, . . . , hct), +where (c1, . . . , ct) ∈ R. By the property 1) of the approximation lemma, ci = 1 for some i. It means that +in H (the group itself) w(�xi +1, . . . , �xi +n) = h, where �xi +j is the ith coordinate of the vector �xj, j = 1, . . . , n. +Let us take yj = (�xi +j, . . . , �xi +j), j = 1, . . . , n. Then in H ⩽ G the following is true: +w(y1, . . . , yn) = (h, . . . , h), +which proves verbal closedness of H in G. +Let U ⩽ L. We use the following denotion: +Ui := {(1, . . . , 1, u, 1, . . . , 1) | u ∈ U} ⩽ Q, i = 1, . . . , t (coordinate u stands on the ith place). +It remains to prove that H is not algebraically closed in G. +Lemma 5. The group Q is finitely presented over its subgroup H. +Proof. According to Lemma 2, it is sufficient to show that Q/(L1 × · · · × Lt) is finitely presented over +H/�L, where �L = {(l, . . . , l) | l ∈ L}. However, Q = H · (L1 × · · · × Lt), so the statement we prove follows +from this fact (see [KM79], theorem 4.2.4): +Suppose that G is a group, F is its subgroup, and K is its normal subgroup. Then (K·F)/K ∼= +F/(F ∩ K). +Thus, the group Q/(L1 × · · · × Lt) is not just finitely presented over H/�L but is isomorphic to it. +From Lemma 3 and Lemma 5, it follows that it suffices to show that H is not a retract of G. Let +ρ : G → H be a hypothetical retraction, and let ˆρ : Q → H be its composition with the natural +epimorphism Q → Q/R = G. Henceforth, all subgroups and centralizers we refer to relate to Q. +Let us verify that ˆρ(Li) ⩽ CT (CT (L)) ⩽ L for every i. First, prove the left inclusion. Let h ∈ CT (L). +Then, h commutes with every element from L; consequently, h, as an element of Q, commutes with Li. +Applying the retraction ˆρ to this identity, we get that ˆρ(h) (= h) commutes with the subgroup ˆρ(Li), +which (by definition of the centralizer) proves the inclusion. The second inclusion follows from the fact +that L = CT (A) = C(A) ∩ T, which means that +CT (CT (L)) ⩽ CT (A ∩ T) = CT (A) = L. +The first inclusion here is true as C(L) ⩾ A. The following equality is true as A ⩽ T. +On the other hand, for i ̸= j, the mutual commutator subgroup [Li, Lj] is trivial (as in case i and +j are different, Li and Lj are contained in different components of the fibered product). +It means +that the image of this mutual commutator subgroup is trivial too: [ˆρ(Li), ˆρ(Lj)] = {1}. Consequently, +[Li, � +j̸=i Lj] = {1} and [ˆρ(Li), � +j̸=i ˆρ(Lj)] = {1}. If ˆρ(Li) = ˆρ(Ll) for some i ̸= l, then (by the virtue +of well-known commutator identities) [ˆρ(Li), � +j ˆρ(Lj)] = {1}, which means that ˆρ(Li) ⩽ CT (L) (as +L = ˆρ(L) ⩽ � +j ˆρ(Lj)). +Thereby, if for some different i and j, ˆρ(Li) = ˆρ(Lj), then ˆρ(Li) ⩽ CT (L). From here and from the +inclusion we proved earlier, we get ˆρ(Li) ⩽ L ∩ CT (L) = Z(L). +7 + +Let us take k in the approximation lemma to be the number of all subgroups of T, and let J be the +set of all exclusive numbers i, namely such that for any l ̸= i, ˆρ(Li) ̸= ˆρ(Ll). Since among ˆρ(Li) ⩽ T +there are no more than k different subgroups, |J| ⩽ k. Thus, from the property 3) of the approximation +lemma, we have a decomposition: +×t +i=1Ci = R × (×i∈ICi), +where I ⊆ {1, . . . , t}\J is some set of non-exclusive elements. Again, according to the property 3) of the +approximation lemma, the projection π : ×t +i=1Ci → ×i∈ICi onto the second factor of this decomposition +is defined by an integer matrix (nij), namely, for ci ∈ Ci, π : ci �→ � +j∈I cnij +j +, where cj are elements +corresponding to ci under the isomorphism Ci ∼= C ∼= Cj. +This means that the restriction of π to C = {(c, . . . , c) | c ∈ C ⩽ H} is defined by formula: +ˆπ : (c, . . . , c) �→ +� +j∈I +cmj +j , mj = +� +i +nij. +Here cj are elements corresponding to c under the isomorphism C ∼= Cj. +Then (as i ∈ I are non-exclusive, we have ˆρ(Li) ⩽ Z(L)), consider the composition: +Ψ : C ⩽ Q → Z(L), c π�→ +� +j∈I +cmj +j +ˆρ�→ +� +j∈I +ˆρ(cmj +j ). +It extends to a homomorphism Φ : L → Z(L) defined by the similar formula: +Φ : g �→ +� +j∈I +ˆρ(gmj +j +), +where g ∈ L and gj ∈ Lj are elements corresponding to g ∈ L. Obviously, it is an extension of Ψ and +a homomorphism, as for j ∈ I, ˆρ(Lj) ⩽ Z(L) and the group Z(L) is abelian. This homomorphism +commutes with the conjugation action of H on L. Indeed, let g ∈ H and let g be the action of g on L +by conjugation, namely, for x ∈ L, g(x) = g−1xg. Let us show that Φ ◦ g = g ◦ Φ. Let h ∈ L. Then +Φ(g(h)) = � +j∈I ˆρ(g−1hmj +j g) = � +j∈I g−1ˆρ(hmj +j )g = g(Φ(h)). Penult identity is true, as ˆρ is a retraction +on H, so it acts identically on H itself. By assumption we made in the beginning, the kernel of this +homomorphism has nontrivial intersection with C: ker Φ ∩ C ̸= {1}, so the restriction Ψ = Φ|C has a +nontrivial kernel too. +On the other hand, Ψ is the identical mapping, since Ψ = ˆρ|C ◦ π|C = ˆρ|C ◦ ˆπ = ˆρ|C (the last identity +is true as ˆπ is a projection «forgetting» the R coordinate, and ˆρ(R) = {1} is a composition of the natural +homomorphism to the quotient group and of the retraction to H) and ˆρ|C = id, as ˆρ is the retraction +from Q to H, so it acts trivially on C. The obtained contradiction completes the proof. +Let us provide some corollaries of this theorem: +Corollary 1. Finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly +verbally closed. +Proof. Let us take the torsion subgroup of such group as T from the theorem, and the center of this +torsion subgroup as A ̸= {1}. Since T is nilpotent and nonabelian, every nontrivial normal subgroup of +T has a nontrivial intersection with A [KM79], so A is not a direct factor of T. +Corollary 2 [KMO21]. A finite group, whose center is not a direct factor is not strongly verbally +closed. +This theorem does not cover the case of finitely generated nilpotent nonabelian groups with abelian +torsion subgroups, and it is still unknown whether there are strongly verbally closed groups among such +groups. So far, we can provide only a partial answer to this question (see the first proposition of the +following paragraph). +8 + +4. Nilpotent non-strongly-verbally-closed groups +Let us remind that the discrete Heisenberg group is the free nilpotent group of nilpotency class two with +two free generators. It can be easily verified that this group admits a faithful representation in the group +of upper triangular matrices of size 3 by 3. +Proposition 4. Let H be the discrete Heisenberg group with a and b being its free generators and N +being its subgroup: +N = ⟨⟨aα, [a, b]n⟩⟩, α, n ⩾ 0. +The group G = H/N is strongly verbally closed if and only if gcd(α, n) = 1. +Proof. Let T(G) be the torsion subgroup of G. If T(G) = {1}, then G is the discrete Heisenberg group, +whose non-strong-verbal-closedness was proved in [KMO21]. If gcd(α, n) = 1, then G is abelian, since +[a, b]α = [aα, b] ∈ N; consequently, it is strongly verbally closed, as it follows from the theorem from +[Maz18]. +Consider the case, when gcd(α, n) = d ̸= 1. Without loss of generality, we may assume that α and n +are the least of non-negative numbers such that aα ∈ N, [a, b]n ∈ N. Consider the central product of G +with its copy �G with joined commutator subgroup: +K = G +× +G′= � +G′ +�G = (G × �G)/{(c, c−1)|c ∈ G′}. +The group G is not algebraically closed in K, since G is not a retract of K. Indeed, let ρ be a hypothetical +retraction. The group G commutes with �G in K, so ρ( �G) ⩽ Z(G) and ρ( �G′) = {1}, which leads to a +contradiction with the definition of retraction. However, G is verbally closed in K. Let w ∈ F(t1, . . . , ts) +be some word and +w((h1N, h′ +1N), . . . , (hsN, h′ +sN)) = (hN, N) +for some hN, hiN ∈ G, h′ +iN ∈ �G. Then, for some cN ∈ G′, the following holds: +� +w(h′ +1, . . . , h′ +s)N = cN +w(h1, . . . , hs)N = hc−1N +By an automorphism of the free group, the word w can be reduced to a normal form [KMO21]: +w(t1, . . . , ts) = tm +1 w′(t1, . . . , ts), where m ∈ N, w′ ∈ F ′ +s. From the first equation, we get cN ∈ G′ ∩ ϕ(Gs), +where ϕ : Gs → G, (g1, . . . , gs) �→ w(g1, . . . , gs) is a verbal mapping. +This means that for some +w1, w2 ∈ N, in H it is true that: +� +w(h′ +1, . . . , h′ +s) = cw1 +w(h1, . . . , hs) = hc−1w2 +Let us show that in G the identity (aN)x = [aN, bN]z doesn’t hold for x ̸∈ αZ. Converse would mean +that in the discrete Heisenberg group the following holds: +ax[a, b]−z = b−ka−laαt[a, b]nsalbk +for some k, l, t, s ∈ Z. After some reductions, we get: ax−αt = [a, b]ns+z+αtk. In H it is possible only +if x = αt ∈ αZ. We obtained a contradiction. Thus, h′ +1 = [a, b]γ for some γ ∈ Z, and, consequently, +cw1 ∈ H′. Since for any verbal mapping ϕ in the discrete Heisenberg group (see [KMO21]) +for any g ∈ ϕ(Hs), it is true that g(ϕ(Hs) ∩ H′) ⊆ ϕ(Hs), +for some g1, . . . , gs ∈ H, w(g1, . . . , gs) = w(h1, . . . , hs)c. It means that: +w(g1, . . . , gs) = hw3 +for some w3 ∈ W, and in G: +w(g1, . . . , gs)N = hN, +which proves verbal closedness of G in K. +9 + +At last, let us prove that higher dimensional Heisenberg groups over any field are not strongly verbally +closed: +The Heisenberg group of dimension 2n+1 over a field K, where n ∈ N is the group of upper triangular +matrices of the kind +Hn(K) = +� +T(¯a,¯b, c) = + + +1 +¯a +c +0 +In +¯b +0 +0 +1 + + +�����¯a, (¯b)⊺ ∈ Kn, c ∈ K +� +, +where In is the identity matrix of size n. +Proposition 5. The group Hn(K) is not strongly verbally closed. +Proof. Consider the central product of Hn(K) with its copy �Hn(K) with joined commutator subgroup: +G = Hn(K) +× +Hn(K)′= � +Hn(K)′ +�Hn(K). +Denote with the symbol H the first factor of this central product. Let us show that H is not algebraically +closed in G. The group H is linear, and, consequently, it is equationally noetherian [BMR99], so it is +algebraically closed in G if and only if it is a retract of every finitely generated over H subgroup of G +[KMM18]. In particular, of such a group: +¯H = ⟨H, (1, h1), . . . , (1, hn), (1, g1), . . . , (1, gn)⟩, where hi = + + +1 +¯ai +0 +0 +In +0 +0 +0 +1 + + , gi = + + +1 +0 +0 +0 +In +¯bi +0 +0 +1 + + , +where ¯ai = (0, . . . , 1, . . . , 0) = (¯bi)⊺, (unit is on the ith place). Thus, N = ⟨h1, . . . , hn, g1, . . . , gn⟩ is +a subgroup of �Hn(K), isomorphic to the discrete Heisenberg group of dimension (2n + 1). Let ρ be a +hypothetical retraction. Since in G the group H commutes with N, we get that ρ(N ′) = {1}, which leads +to a contradiction with the definition of retraction. +Nevertheless, subgroup H is verbally closed in G: let w ∈ Fs be some word (without loss of generality, +this word is in the normal form we established earlier), and let ϕ : Hs → H be the verbal mapping +associated with this word. Suppose that for some hi, h ∈ H, h′ +i ∈ �H, c ∈ H′: +� +w(h′ +1, . . . , h′ +s) = c +w(h1, . . . , hs) = hc−1 +In general, on matrices gi = T(¯ai,¯bi, ci), i = 1, . . . , s, the mapping ϕ acts like that: +ϕ(g1, . . . , gs) = + + +1 +m¯a1 +mc1 + f(¯a1, . . . , ¯as;¯b1, . . . ,¯bs) +0 +In +m¯b1 +0 +0 +1 + + , +where f : (Kn)s × (Kn)s → K is some function linear in every argument. The image of f is either trivial +or is equal to K, which leads to: +ϕ(Hn(K)s) = + + + + + +{1}, +if m = 0 and the image of f is trivial +(Hn(K))′, +if m = 0 and the image of f equals K +Hn(K), +if m ̸= 0 +Then ϕ(Hn(K)s) ∩ (Hn(K))′ ⩽ Hn(K) and for every element h ∈ ϕ(Hn(K)s) it is true that +h(ϕ(Hn(K)s) ∩ (Hn(K))′) ⊆ ϕ(Hn(K)s), +whence verbal closedness follows. +10 + +REFERENCES +[BMR99] +G. Baumslag, A. Myasnikov, and V. Remeslennikov. Algebraic geometry over groups I. Al- +gebraic sets and ideal theory. J. Algebra, 219:1, 1999, 16–79. +[Bog22] +O. Bogopolski. Equations in acylindrically hyperbolic groups and verbal closedness. Groups, +Geom. Dyn., 16:2, 2022, 613–682. +[Fuc70] +L. Fuchs. Infinite Abelian Groups, v. 1. Academic Press, 1970. +[Hal54] +P. Hall. Finiteness conditions for soluble groups. Proc. London Math. Soc., 4:1, 1954, 419– +436. +[Kly21] +A. A. Klyachko. The Klein bottle group is not strongly verbally closed, though awfully close +to being so. Canadian Mathematical Bulletin, 64:2, 2021, 491–497. +[KM18] +A. A. Klyachko and A. M. Mazhuga. Verbally closed virtually free subgroups. Sb. Math., +209:6, 2018, 850–856. +[KM79] +M. I. Kargapolov and Ju. I. Merzljakov. Fundamentals of the theory of groups. Graduate +Texts in Mathematics, 62, Springer, 1979. +[KMM18] +A. A. Klyachko, A. M. Mazhuga, and V. Yu. Miroshnichenko. Virtually free finite-normal- +subgroup-free groups are strongly verbally closed. J. Algebra, 510, 2018, 319–330. +[KMO21] +A. A. Klyachko, V. Yu. Miroshnichenko, and A. Yu. Olshanskii. Finite and nilpotent strongly +verbally closed groups. J. Algebra Its Appl. (to appear), 2021. +[Kur60] +A. G. Kurosh. The theory of groups. Chelsea Publishing Company, 1960. +[LS15] +R. Lyndon and P. Schupp. Combinatorial group theory. Springer, 2015. +[Maz18] +A. M. Mazhuga. Strongly verbally closed groups. J. Algebra, 493, 2018, 171–184. +[Maz19] +A. M. Mazhuga. Free products of groups are strongly verbally closed. Sb. Math., 210:10, 2019, +1456–1492. +[MR14] +A. Myasnikov and V. Roman’kov. Verbally closed subgroups of free groups. J. Group Theory, +17:1, 2014, 29–40. +[Neu67] +H. Neumann. Varieties of groups. Springer-Verlag, Berlin-Heidelberg-New York, 1967. +[RKK17] +V. A. Roman’kov, N. G. Khisamiev, and A. A. Konyrkhanova. Algebraically and verbally +closed subgroups and retracts of finitely generated nilpotent groups. Siberian Math. J., 58:3, +2017, 536–545. +[Rob82] +D. J. S. Robinson. A course in the theory of groups. Springer-Verlag, 1982. +11 + diff --git a/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/load_file.txt b/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6a4ca50a451ddb5cd620758ba6b88b677630696 --- /dev/null +++ b/eNE0T4oBgHgl3EQf5gKT/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf,len=673 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='02752v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='GR] 7 Jan 2023 finite normal subgroups of strongly verbally closed groups Filipp D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Denissov Faculty of Mathematics and Mechanics of Moscow State University Moscow 119991, Leninskie gory, MSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Moscow center for Fundamental and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' denissov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='filipp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='com In the recent paper by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Klyachko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Miroshnichenko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Olshanskii, it is proven that the center of any finite strongly verbally closed group is its direct factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' One of the results of the current paper is the generalization of this nontrivial fact to the case of finite normal subgroups of any strongly verbally closed groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It follows from this generalization that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Introduction A subgroup H of a group G is called verbally closed [MR14] if any equation of the form w(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xn) = h, where w is an element of the free group F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xn) and h ∈ H, having solutions in G has a solution in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If each system of equations with coefficients from H {w1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' ) = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , wm(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' ) = 1}, where wi ∈ H ∗ F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xn) (and ∗ means the free product), having solutions in G has a solution in H, then the subgroup H is called algebraically closed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Note that if the subgroup H is algebraically closed in the group G, then it is verbally closed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' A group G is called strongly verbally closed if it is algebraically closed in any group containing G as a verbally closed subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, the verbal closedness (as well as the algebraic closedness) is a property of a subgroup, while the strong verbal closedness is a property of an abstract group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The class of strongly verbally closed groups is fairly wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' it includes — all abelian groups [Maz18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — all free groups [KM18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — all virtually free groups containing no nontrivial finite normal subgroups [KM18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' [KMM18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — all groups decomposing nontrivially into a free product [Maz19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — fundamental groups of all connected surfaces except the Klein bottle [Maz18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' [Kly21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — all finite groups with nonabelian monolith [KMO21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — inifinite dihedral group [KMM18] and any finite dihedral group whose order is not divisible by 8 [KMO21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' — all acylindrically hyperbolic groups with no nontrivial finite normal subgroups [Bog22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The class of non-strongly-verbally-closed groups is fairly wide too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Among such groups are the following: — the already mentioned fundamental group of the Klein bottle [Kly21], — the discrete Heisenberg group [KMO21], — any finite group, whose center is not its direct factor (in particular, any finite nonabelian nilpotent group) [KMO21], [RKK17], [KM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proving the strong verbal closedness (as well as its absence) of a group is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In [KMO21], for example, a question is raised: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Does there exist a finitely generated nilpotent nonabelian strongly verbally closed group?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' A negative answer to this question would yield a broad generalization of the last two examples of non-strongly-verbally-closed groups mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' So far, we managed to give a partial answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' More precisely, we proved the absence of strong verbal closedness of finitely generated nilpotent groups with nonabelian torsion subgroups and of some finitely generated nilpotent nonabelian groups with abelian torsion subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 1 A property that is stronger than strong verbal closedness is the property of being a strong retract [KMO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' A group H is called a strong retract if it is a retract of any group G ⩾ H from the variety var H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us recall some terminology [Neu67]: — the variety generated by a class of groups K is the class of all groups satisfying all identities that hold in all groups from K, — the variety generated by a group G is designated by var G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This gives rise to the following question from [KMO21]: Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' What is an arbitrary finite strong retract?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In [KMO21] some examples of strong retracts are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In the next section, we describe all the nilpotent strong retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Below we provide a brief list of notation we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If x, y are elements of some group, then the symbol [x, y] denotes their commutator x−1y−1xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The symbol ord(x) denotes the order of an element x of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The center of a group G is denoted by Z(G), and its commutator subgroup is denoted by G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Centralizer of a subset X of a group G is denoted by C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The symbol ⟨⟨X⟩⟩ stands for the normal closure of a subset X of a group G (that is the intersection of all normal subgroups of G containing X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The free group with a basis X is denoted as F(X) or Fn in case X has n ∈ N elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Identical mapping from X to itself is denoted by id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We use the symbol H ∼= G to express the fact that groups H and G are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finally, the symbol H ⩽ G denotes the fact that a group H is a subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The symbol H ⊴ G denotes the fact that H is a normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The author is grateful to his supervisor Anton Alexandrovich Klyachko for formulation of the problem and for valuable remarks during the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Nilpotent strong retracts Note that in case when G is an abelian group, H ⩽ G is its retract if and only if H is a direct summand of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It means that the property of being a strong retract for the abelian group G is equivalent to the property of G being a direct summand of any group H ∈ var G containing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' For the further discussion, we need the description of all varieties of abelian groups (see [Fuc70], paragpaph 18, exercise 7): Varieties of abelian groups are precisely the following classes of groups: 1) the class of all abelian groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 2) the class of all abelian groups of exponent n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' To begin with, consider the case, when G is not a group of bounded period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then, according to the description, var G is the class of all abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The following is true of divisible abelian groups (see, for example, [Kur60]): If G is a divisible abelian group, and H is an abelian group such that G ⩽ H, then G is a direct summand of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us remind that a group G is called divisible if for any g ∈ G and n ∈ N, the equation xn = g has a solution in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' An abelian group G of unbounded period is a strong retract if and only if it is divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Sufficiency follows from the fact provided above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let G be an abelian group of unbounded period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then, as it was noted earlier, var G is the class of all abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In particular, var G contains a divisible group H containing G [Kur60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Though, if G is not divisible itself, it is not a direct summand of H (as direct summands of a divisible group are divisible themselves [Kur60]), so G is not a strong retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us move on to abelian groups of bounded period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The first Pr¨ufer theorem provides a complete description of these groups [Kur60]: 2 An abelian group G of bounded period d is a direct sum of primary cyclic groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' G ∼= � i∈I Zpki i , where pi are prime numbers and ki are natural numbers such that pki i |d, i ∈ I (I is an index set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We need the following variation of the Zorn’s lemma [Fuc70]: Let M ̸= ∅ be a partially ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that every chain in M (a totally ordered subset of M) has an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then M contains a maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Now, we are ready to proceed with our description: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' An abelian group G of bounded period is a strong retract if and only if in its decom- position into the direct sum of primary cyclic groups, orders of any distinct direct summands are either equal or coprime: G ∼= m � i=1 Cpki i (ni), where Cpki i (ni) is equal to the direct sum of ni copies of the group Zpki i , where all prime numbers pi are distinct, m, ki ∈ N, and ni are some cardinal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that G cannot be decomposed into such a direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We may assume that G = m � i=1 � j∈Ii Zp kj i , (1) where m ∈ N, |Ii| = ni and among kj, j ∈ Ii there are only finitely many different ones (because G is a group of bounded period) but there exists i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , m} such that for some j1, j2 ∈ Ii, kj1 ̸= kj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the group: H = m � i=1 Cpsi i (ni), where si = max{kj | Zp kj i is a direct summand in the decomposition (1)}, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Since both G and H are of the same exponent: exp(G) = �m i=1 si = exp(H), it follows from the description of abelian varieties that H ∈ var G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the injection f : G → H, which works on each direct summand from (1) as follows: let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , m}, j ∈ Ii, f : Zp kj i ֒→ Zpsi i , where Zpsi i is the jth summand from the decomposition of Cpsi i (ni) into the direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Every direct summand from (1) is mapped into the corresponding direct summand of the decomposition of H, so that the restriction of f to Zp kj i is a natural injection: if kj = si, then it is the identical map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' otherwise it is a mapping to the subgroup of Zpsi i of order pkj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From the uniqueness of the decomposition of an abelian group of bounded period into the direct sum of primary cyclic groups [Fuc70], it follows that f(G) is not a direct summand of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, G is not a strong retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Now, suppose that G has the decomposition from the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let H ∈ var G and let f : G ֒→ H be a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As any monomorphism preserves the order of an element, the pith component of G is mapped into the pith component of H under f, so it suffices to prove the theorem only for the case G = Cpk(n), where p is prime, k ∈ N, and n is some cardinal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us show that there exists such X ⩽ H that H = f(G) ⊕ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In Zorn’s lemma, take set of all subgroups of H having trivial intersection with f(G) as M: M = {Y ⩽ H | Y ∩ f(G) = {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Order on M is introduced as follows: for X, Y ∈ M, X ⩽ Y if X is a subgroup of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It can be verified directly that this is an order on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Set M is nonempty: {0} ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Any chain {Yα} ⊆ M of subgroups having trivial intersection with f(G) is bounded by an element Y ∈ M, where Y = ∪αYα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consequently, Zorn’s lemma is applicable, and M contains a maximal element X: X ⩽ H, X ∩ f(G) = {0}, and X 3 is not a subgroup of any bigger (relatively to the order we introduced earlier) subgroup satisfying this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From X ∩f(G) = {0} it follows that f(G)+X = f(G)⊕X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It remains to prove that H = f(G)+X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' There exists such k ∈ N that kh ∈ f(G) + X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Indeed, otherwise ⟨h⟩ ∩ (f(G) + X) = {0}, which means that (⟨h⟩ + X) ∩ f(G) = {0}, which leads to a contradiction with the maximality of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let s be the least of such numbers k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Without loss of generality, assume that s is prime or that s = 1 (otherwise, take a power of h instead of h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Two cases are possible: 1) s = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then, ph = f(g) + x for some g ∈ G, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If g = pg1, g1 ∈ G (g1 may be equal to zero), then ph − f(pg1) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' However, from h − f(g1) ̸∈ X (as h ̸∈ f(G) + X) it can be obtained that (X + ⟨h − f(g1)⟩) ∩ f(G) = {0}, which leads to a contradiction with the maximality of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consequently, g ̸= pg1 for any g1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As g ̸= 0, ord(g) = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Though, ord(ph) = pr < pk, so pr(ph) = 0 = pr(f(g)) + prx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As the sum f(G) + X is direct, prf(g) = prx = 0, which means that prg = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 2) s ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' For abelian groups of period p, the mapping g �→ sg is an automorphism, so, as sh = f(g)+x for some g ∈ G, x ∈ X, there exist such g1 ∈ G, x1 ∈ X that g = sg1, x = sx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, s(h−f(g1)−x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' No nontrivial element of H has the order of s, so h = f(g1) + x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As a result, H = f(G) ⊕ X, and G is a strong retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The next simple theorem shows that consideration of nilpotent groups does not yield any new strong retracts: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Nilpotent strong retract is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let G be a nilpotent strong retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then (see [KM79]), Z(G) ̸= {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Put N = G′ ∩ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the central product of G with its copy �G with joined subgroup N: K = G × N= � N �G = (G × �G)/{(g, g−1)|g ∈ N}, where � N = �G′ ∩ Z( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group K is contained in var G as it is a quotient group of the direct product of groups G, �G ∈ var G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let ρ be a hypothetical retraction from the group K to its subgroup G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From the fact that in the group K, the group G commutes with the group �G, we obtain ρ( �G) ⩽ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Image of the commutator subgroup under a homomorphism into an abelian group is trivial, so ρ( � N) = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finally, from the definition of retraction, we obtain N = {1}, so G′ = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As a result, we proved the following theorem: Nilpotent-strong-retract theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Nilpotent strong retracts are precisely divisible abelian groups and abelian groups of bounded exponent in whose decomposition into the direct sum of primary cyclic groups, orders of any distinct direct summands are either equal or coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In the next paragraph we show that many nilpotent groups are not even strongly verbally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finite normal subgroups of strongly verbally closed groups We say that a group presentation ⟨X | R⟩ is finitely presented over a group presentation ⟨Y | S⟩, if there exist such finite sets A and B that ⟨X | R⟩ ∼= ⟨X′ | R′⟩, where X′ = Y ∪ A, R′ = S ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The following lemma reveals that this definition is, in fact, a group property (which means it does not depend on the choice of a group presentation), so it makes sense to speak about the finite presentability of one group over the other group: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that a group presentation ⟨X | R⟩ is finitely presented over a group presentation ⟨Y | S⟩ and ⟨Y | S⟩ ∼= ⟨Y ′ | S′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then ⟨X | R⟩ is finitely presented over ⟨Y ′ | S′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We may assume that X = Y ∪ A and R = S ∪ B for some finite sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It is known (see, for example, [LS15]) that groups defined by group presentations ⟨Y | S⟩ and ⟨Y ′ | S′⟩ are isomorphic if and only if presentation ⟨Y ′ | S′⟩ is obtained from presentation ⟨Y | S⟩ by applying a finite number of Tietze transformations: — adding to the set S an arbitrary set T ⊆ ⟨⟨S⟩⟩ ⊴ F(Y ) of its consequences, 4 — adding to the set Y an arbitrary set �Y while ading to S a set {�y = w�y | �y ∈ �Y , w�y ∈ F(Y )}, and their inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It is sufficient to prove the lemma only for the case, when ⟨Y ′ | S′⟩ is obtained from ⟨Y | S⟩ by applying one Tietze transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' One can easily verify that in case of the first transformation, X′ = X and R′ = R ∪ T, while in case of the second transformation, X′ = X ∪ �Y and R′ = R ∪ {�y = w�y | �y ∈ �Y , w�y ∈ F(Y )} provide the desired group presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' By virtue of Lemma 1, the following definition may be introduced: A group G is finitely presented over a group H, if there exists such a presentation of G that it is finitely presented over any presentation of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that G contains a subgroup H and a finite normal subgroup N such that G/N is finitely presented over H/(H ∩ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then G is finitely presented over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof (with minor changes) replicates the proof of the Hall theorem [Hal54] about preservation of finite presentability of a group under extensions (see also [Rob82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let G be a group, H = ⟨X | R⟩ ⩽ G, and N = ⟨Y | S⟩ ⊴ G be its finite subgroup, where Y and S are finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' By condition of the lemma, the group G/N is finitely presented over H/(H∩N) = ⟨X | R∪C⟩, where ⟨⟨C⟩⟩ = H ∩ N and the set C is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consequently G/N ∼= ⟨X ∪ A | R ∪ C ∪ B⟩, where sets A and B are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us construct a presentation of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As the set of generators, take X ∪ A ∪ Y , where sets X, A, Y are in one-to-one correspondence with sets X, A, Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The sets R, S, C, and B are in correspondence with the sets R, S, C, and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As the set of defining relations, take the union of the following sets: R, S, C1 = {cw−1 c | c ∈ C, wc ∈ F(Y )}, B1 = {bw−1 b | b ∈ B, wb ∈ F(Y )} (c ∈ C and b ∈ B are considered as words from F(X) and from F(X ∪ A) respectively), T = {a−1yaw−1 a,y, aya−1v−1 a,y | a ∈ A, y ∈ Y , wa,y, va,y ∈ F(Y )}: �G = ⟨X ∪ A ∪ Y | R ∪ S ∪ C1 ∪ B1 ∪ T⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider a surjective homomorphism θ : �G → G, defined with the following bijections X → X, A → A, Y → Y on the generators (defining relations are mapped into true identities under such a map on generators, so such a homomorphism exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The restriction θ|K : K → N on the subgroup K = ⟨Y ⟩ ⩽ �G is an isomorphism as all the relations in the alphabet Y in �G are consequences of the defining relations S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Besides, K ⊴ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Homomorphism �θ : �G/K → G/N generated by θ, is an isomorphism too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Now, let g ∈ ker θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then gK ∈ ker �θ, but �θ is an isomorphism, so g ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finally, θ|K is an isomorphism, so g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The following lemma provides a criterion for algebraic closedness of a subgroup H of a group G in case, when G is finitely presented over H (for similar propositions, refer to [MR14]): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that H = ⟨X | R⟩ is a subgroup of G and G is finitely presented over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The subgroup H is algebraically closed in G if and only if H is a retract of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose H is algebraically closed in G and A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , am}, B = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , sn} are the sets from the definition of finite presentability of G over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The relations si(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , am, X) = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , n are corresponded to a system of equations with coefficients from H: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 s1(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , tm, X) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' sn(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , tm, X) = 1 which, by condition, has a solution t1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , tm = am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' By virtue of algebraic closedness of H in G, this system has a solution t1 = h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , tm = hm in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Mapping X ⊔ {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , am} → H, x ∈ X �→ x, ai �→ hi extends to a surjective homomorphism ϕ : G → H, as defining relations of G are mapped into true identities under such a mapping of generators (note that R is the set of words in the alphabet X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 5 This homomorphism is a desired retraction: let h ∈ H, h = v(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xr), xi ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Applying to this word the homomorphism ϕ, we get: ϕ(h) = v(ϕ(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ϕ(xr)) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Algebraic closedness of a subgroup H of a group G follows from retractness of H in G for every group G [MR14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Approximation lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let C be a finite elementary abelian p-group (where p is a prime number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' For any k ∈ N, there exists t ⩾ k such that the direct product P = ×t i=1Ci of copies Ci of C contains a subgroup R invariant with respect to the diagonal action on P of the endomorphism algebra End C with the following properties: 1) R ⊆ � ker ρj, where ρj : P → Cj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , t are the natural projections, 2) But R · ×j̸∈JCj = P for any subset J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , t} of cardinality |J| = k, 3) Moreover, each such J is contained in a set J′ ⊇ J such that P = R × (×j̸∈J′Cj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' and there exist integers nij ∈ Z such that the projection π : P → ×j̸∈J′Cj with the kernel R acts as: Ci ∋ ci �→ � j̸∈J′ cnij j , where cj ∈ Cj is the element corresponding to ci under the isomorphism Ci ∼= C ∼= Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The following theorem provides a generalization of the result from [KMO21] about the center of a finite strongly verbally closed group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The proof is also analogical to the proof of that theorem, with the exception of some nuances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finite-normal-subgroup theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let H be a strongly verbally closed group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' For any finite normal subgroup T of H, for any abelian subgroup A of T, normal in H, it is true that Z(CT (A)) is a direct factor of CT (A), and some complement is normal in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Here CT (X) = C(X) ∩ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let H be such a group, and let L = CT (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It suffices, for each prime p, to find a homomorphism ψp : L → Z(L) commuting with the action H ↷ L by conjugations (this action is well-defined as L ⊴ H) and injective on the p-component of the center Zp(L) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then the homomorphism ψ : L → Z(L), x �→ � p πp(ψp(x)), where πp : Z(L) → Zp(L) is the projection on the p-component, is injective on Z(L), so its kernel is the desired complement D (normality of D in H follows from the fact that ψ commutes with H ↷ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that there are no such homomorphisms for some prime p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' every homomorphism f : L → Z(L) commuting with the action H ↷ L is not injective on Zp(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then it is not injective on the maximal elementary abelian p-subgroup C ⩽ Zp(L) (it is finite as L is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Indeed, if x ̸= 1 ∈ Zp(L) is an element such that f(x) = 1, then, raising it to the appropriate power d, we get f(xd) = 1 and xd ∈ C, xd ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Choose t by the approximation lemma applied to C (for some k to be specified later) and consider the fibered product of t copies of the group H: Q = {(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ht) | h1L = · · · = htL} ⩽ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' First of all, let us show that the subgroup R ⩽ Ct ⩽ Q from the approximation lemma is normal in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Subgroup R is invariant under the diagonal action of automorphisms Aut C ⩽ End C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It remains to show that Q acts by conjugations on P = Ct diagonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It follows from the lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let G be a group, and N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If xC(N) = yC(N) for some x, y ∈ G, then x and y act on N (by conjugations) identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From xC(N) = yC(N) it follows that for some c ∈ C(N), x = yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then for n ∈ N, we have: x−1nx = c−1(y−1ny)c = y−1ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The last identity is true, as (due to normality) y−1ny ∈ N and c ∈ C(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , qt) ∈ Q, p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , pt) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' As q1L = q2L = · · · = qtL, then (according to Lemma 4) q−1pq = �q−1p�q, where �q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It means that the conjugation action of Q on P is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' On the other hand, diagonal action by conjugations induces an endomorphism of Ct (due to normality 6 of C ⊴ H), and R is invariant with respect to the diagonal action of such endomorphisms, leading to normality of R in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Put G = Q/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' First, let us show that H embeds into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group H embeds into Q diagonally: h �→ (h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h), h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This homomorphism serves as embedding into G as well, as all projections of any nontrivial diagonal element of Q are nontrivial (and R is contained in the union of the kernels of these projections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Now, let us prove verbal closedness of this diagonal subgroup (denote it as H too) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider an equation w(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xn) = (h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h) having a solution in G and let �x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , �xn be a preimage (in Q) of a solution x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then (in Q): w(�x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , �xn) = (hc1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hct), where (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ct) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' By the property 1) of the approximation lemma, ci = 1 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It means that in H (the group itself) w(�xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , �xi n) = h, where �xi j is the ith coordinate of the vector �xj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us take yj = (�xi j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , �xi j), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then in H ⩽ G the following is true: w(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , yn) = (h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h), which proves verbal closedness of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let U ⩽ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We use the following denotion: Ui := {(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , 1, u, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , 1) | u ∈ U} ⩽ Q, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , t (coordinate u stands on the ith place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It remains to prove that H is not algebraically closed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group Q is finitely presented over its subgroup H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' According to Lemma 2, it is sufficient to show that Q/(L1 × · · · × Lt) is finitely presented over H/�L, where �L = {(l, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , l) | l ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' However, Q = H · (L1 × · · · × Lt), so the statement we prove follows from this fact (see [KM79], theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='4): Suppose that G is a group, F is its subgroup, and K is its normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then (K·F)/K ∼= F/(F ∩ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, the group Q/(L1 × · · · × Lt) is not just finitely presented over H/�L but is isomorphic to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From Lemma 3 and Lemma 5, it follows that it suffices to show that H is not a retract of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let ρ : G → H be a hypothetical retraction, and let ˆρ : Q → H be its composition with the natural epimorphism Q → Q/R = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Henceforth, all subgroups and centralizers we refer to relate to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us verify that ˆρ(Li) ⩽ CT (CT (L)) ⩽ L for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' First, prove the left inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let h ∈ CT (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then, h commutes with every element from L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' consequently, h, as an element of Q, commutes with Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Applying the retraction ˆρ to this identity, we get that ˆρ(h) (= h) commutes with the subgroup ˆρ(Li), which (by definition of the centralizer) proves the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The second inclusion follows from the fact that L = CT (A) = C(A) ∩ T, which means that CT (CT (L)) ⩽ CT (A ∩ T) = CT (A) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The first inclusion here is true as C(L) ⩾ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The following equality is true as A ⩽ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' On the other hand, for i ̸= j, the mutual commutator subgroup [Li, Lj] is trivial (as in case i and j are different, Li and Lj are contained in different components of the fibered product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It means that the image of this mutual commutator subgroup is trivial too: [ˆρ(Li), ˆρ(Lj)] = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consequently, [Li, � j̸=i Lj] = {1} and [ˆρ(Li), � j̸=i ˆρ(Lj)] = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If ˆρ(Li) = ˆρ(Ll) for some i ̸= l, then (by the virtue of well-known commutator identities) [ˆρ(Li), � j ˆρ(Lj)] = {1}, which means that ˆρ(Li) ⩽ CT (L) (as L = ˆρ(L) ⩽ � j ˆρ(Lj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thereby, if for some different i and j, ˆρ(Li) = ˆρ(Lj), then ˆρ(Li) ⩽ CT (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From here and from the inclusion we proved earlier, we get ˆρ(Li) ⩽ L ∩ CT (L) = Z(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 7 Let us take k in the approximation lemma to be the number of all subgroups of T, and let J be the set of all exclusive numbers i, namely such that for any l ̸= i, ˆρ(Li) ̸= ˆρ(Ll).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Since among ˆρ(Li) ⩽ T there are no more than k different subgroups, |J| ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, from the property 3) of the approximation lemma, we have a decomposition: ×t i=1Ci = R × (×i∈ICi), where I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , t}\\J is some set of non-exclusive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Again, according to the property 3) of the approximation lemma, the projection π : ×t i=1Ci → ×i∈ICi onto the second factor of this decomposition is defined by an integer matrix (nij), namely, for ci ∈ Ci, π : ci �→ � j∈I cnij j , where cj are elements corresponding to ci under the isomorphism Ci ∼= C ∼= Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This means that the restriction of π to C = {(c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , c) | c ∈ C ⩽ H} is defined by formula: ˆπ : (c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , c) �→ � j∈I cmj j , mj = � i nij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Here cj are elements corresponding to c under the isomorphism C ∼= Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then (as i ∈ I are non-exclusive, we have ˆρ(Li) ⩽ Z(L)), consider the composition: Ψ : C ⩽ Q → Z(L), c π�→ � j∈I cmj j ˆρ�→ � j∈I ˆρ(cmj j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It extends to a homomorphism Φ : L → Z(L) defined by the similar formula: Φ : g �→ � j∈I ˆρ(gmj j ), where g ∈ L and gj ∈ Lj are elements corresponding to g ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Obviously, it is an extension of Ψ and a homomorphism, as for j ∈ I, ˆρ(Lj) ⩽ Z(L) and the group Z(L) is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This homomorphism commutes with the conjugation action of H on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Indeed, let g ∈ H and let g be the action of g on L by conjugation, namely, for x ∈ L, g(x) = g−1xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us show that Φ ◦ g = g ◦ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let h ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then Φ(g(h)) = � j∈I ˆρ(g−1hmj j g) = � j∈I g−1ˆρ(hmj j )g = g(Φ(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Penult identity is true, as ˆρ is a retraction on H, so it acts identically on H itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' By assumption we made in the beginning, the kernel of this homomorphism has nontrivial intersection with C: ker Φ ∩ C ̸= {1}, so the restriction Ψ = Φ|C has a nontrivial kernel too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' On the other hand, Ψ is the identical mapping, since Ψ = ˆρ|C ◦ π|C = ˆρ|C ◦ ˆπ = ˆρ|C (the last identity is true as ˆπ is a projection «forgetting» the R coordinate, and ˆρ(R) = {1} is a composition of the natural homomorphism to the quotient group and of the retraction to H) and ˆρ|C = id, as ˆρ is the retraction from Q to H, so it acts trivially on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The obtained contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us provide some corollaries of this theorem: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us take the torsion subgroup of such group as T from the theorem, and the center of this torsion subgroup as A ̸= {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Since T is nilpotent and nonabelian, every nontrivial normal subgroup of T has a nontrivial intersection with A [KM79], so A is not a direct factor of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Corollary 2 [KMO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' A finite group, whose center is not a direct factor is not strongly verbally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This theorem does not cover the case of finitely generated nilpotent nonabelian groups with abelian torsion subgroups, and it is still unknown whether there are strongly verbally closed groups among such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' So far, we can provide only a partial answer to this question (see the first proposition of the following paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Nilpotent non-strongly-verbally-closed groups Let us remind that the discrete Heisenberg group is the free nilpotent group of nilpotency class two with two free generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It can be easily verified that this group admits a faithful representation in the group of upper triangular matrices of size 3 by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let H be the discrete Heisenberg group with a and b being its free generators and N being its subgroup: N = ⟨⟨aα, [a, b]n⟩⟩, α, n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group G = H/N is strongly verbally closed if and only if gcd(α, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let T(G) be the torsion subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If T(G) = {1}, then G is the discrete Heisenberg group, whose non-strong-verbal-closedness was proved in [KMO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' If gcd(α, n) = 1, then G is abelian, since [a, b]α = [aα, b] ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' consequently, it is strongly verbally closed, as it follows from the theorem from [Maz18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the case, when gcd(α, n) = d ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Without loss of generality, we may assume that α and n are the least of non-negative numbers such that aα ∈ N, [a, b]n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the central product of G with its copy �G with joined commutator subgroup: K = G × G′= � G′ �G = (G × �G)/{(c, c−1)|c ∈ G′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group G is not algebraically closed in K, since G is not a retract of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Indeed, let ρ be a hypothetical retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group G commutes with �G in K, so ρ( �G) ⩽ Z(G) and ρ( �G′) = {1}, which leads to a contradiction with the definition of retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' However, G is verbally closed in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let w ∈ F(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ts) be some word and w((h1N, h′ 1N), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , (hsN, h′ sN)) = (hN, N) for some hN, hiN ∈ G, h′ iN ∈ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Then, for some cN ∈ G′, the following holds: � w(h′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h′ s)N = cN w(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hs)N = hc−1N By an automorphism of the free group, the word w can be reduced to a normal form [KMO21]: w(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ts) = tm 1 w′(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ts), where m ∈ N, w′ ∈ F ′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' From the first equation, we get cN ∈ G′ ∩ ϕ(Gs), where ϕ : Gs → G, (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs) �→ w(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs) is a verbal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' This means that for some w1, w2 ∈ N, in H it is true that: � w(h′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h′ s) = cw1 w(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hs) = hc−1w2 Let us show that in G the identity (aN)x = [aN, bN]z doesn’t hold for x ̸∈ αZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Converse would mean that in the discrete Heisenberg group the following holds: ax[a, b]−z = b−ka−laαt[a, b]nsalbk for some k, l, t, s ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' After some reductions, we get: ax−αt = [a, b]ns+z+αtk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In H it is possible only if x = αt ∈ αZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' We obtained a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, h′ 1 = [a, b]γ for some γ ∈ Z, and, consequently, cw1 ∈ H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Since for any verbal mapping ϕ in the discrete Heisenberg group (see [KMO21]) for any g ∈ ϕ(Hs), it is true that g(ϕ(Hs) ∩ H′) ⊆ ϕ(Hs), for some g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs ∈ H, w(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs) = w(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hs)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' It means that: w(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs) = hw3 for some w3 ∈ W, and in G: w(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs)N = hN, which proves verbal closedness of G in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 9 At last, let us prove that higher dimensional Heisenberg groups over any field are not strongly verbally closed: The Heisenberg group of dimension 2n+1 over a field K, where n ∈ N is the group of upper triangular matrices of the kind Hn(K) = � T(¯a,¯b, c) = \uf8eb \uf8ed 1 ¯a c 0 In ¯b 0 0 1 \uf8f6 \uf8f8 �����¯a, (¯b)⊺ ∈ Kn, c ∈ K � , where In is the identity matrix of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group Hn(K) is not strongly verbally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Consider the central product of Hn(K) with its copy �Hn(K) with joined commutator subgroup: G = Hn(K) × Hn(K)′= � Hn(K)′ �Hn(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Denote with the symbol H the first factor of this central product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let us show that H is not algebraically closed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The group H is linear, and, consequently, it is equationally noetherian [BMR99], so it is algebraically closed in G if and only if it is a retract of every finitely generated over H subgroup of G [KMM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' In particular, of such a group: ¯H = ⟨H, (1, h1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , (1, hn), (1, g1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , (1, gn)⟩, where hi = \uf8eb \uf8ed 1 ¯ai 0 0 In 0 0 0 1 \uf8f6 \uf8f8 , gi = \uf8eb \uf8ed 1 0 0 0 In ¯bi 0 0 1 \uf8f6 \uf8f8 , where ¯ai = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , 0) = (¯bi)⊺, (unit is on the ith place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Thus, N = ⟨h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hn, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gn⟩ is a subgroup of �Hn(K), isomorphic to the discrete Heisenberg group of dimension (2n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Let ρ be a hypothetical retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Since in G the group H commutes with N, we get that ρ(N ′) = {1}, which leads to a contradiction with the definition of retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Nevertheless, subgroup H is verbally closed in G: let w ∈ Fs be some word (without loss of generality, this word is in the normal form we established earlier), and let ϕ : Hs → H be the verbal mapping associated with this word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Suppose that for some hi, h ∈ H, h′ i ∈ �H, c ∈ H′: � w(h′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , h′ s) = c w(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , hs) = hc−1 In general, on matrices gi = T(¯ai,¯bi, ci), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , s, the mapping ϕ acts like that: ϕ(g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , gs) = \uf8eb \uf8ed 1 m¯a1 mc1 + f(¯a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' , ¯as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content='¯b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' ,¯bs) 0 In m¯b1 0 0 1 \uf8f6 \uf8f8 , where f : (Kn)s × (Kn)s → K is some function linear in every argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' The image of f is either trivial or is equal to K, which leads to: ϕ(Hn(K)s) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 {1}, if m = 0 and the image of f is trivial (Hn(K))′, if m = 0 and the image of f equals K Hn(K), if m ̸= 0 Then ϕ(Hn(K)s) ∩ (Hn(K))′ ⩽ Hn(K) and for every element h ∈ ϕ(Hn(K)s) it is true that h(ϕ(Hn(K)s) ∩ (Hn(K))′) ⊆ ϕ(Hn(K)s), whence verbal closedness follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' 10 REFERENCES [BMR99] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Baumslag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Myasnikov, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Remeslennikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Algebraic geometry over groups I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Al- gebraic sets and ideal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Algebra, 219:1, 1999, 16–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' [Bog22] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Bogopolski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Equations in acylindrically hyperbolic groups and verbal closedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Groups, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE0T4oBgHgl3EQf5gKT/content/2301.02752v1.pdf'} +page_content=', 16:2, 2022, 613–682.' metadata={'source': 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Garc´ıa1, Shu-Wei Chou-Chen2, Luis A. Barboza4, Maria L. +Daza–Torres1, J. Cricelio Montesinos-L´opez1, Paola V´asquez3, Juan G. +Calvo4, Miriam Nu˜no1, and Fabio Sanchez4 +1Department of Public Health Sciences, University of California Davis, CA, USA. +2Centro de Investigaci´on en Matem´atica Pura y Aplicada - Escuela de Estad´ıstica, Universidad de Costa Rica, +San Jos´e, Costa Rica. +3Centro de Investigaci´on en Matem´atica Pura y Aplicada, Universidad de Costa Rica, San Jos´e, Costa Rica. +4Centro de Investigaci´on en Matem´atica Pura y Aplicada - Escuela de Matem´atica, Universidad de Costa Rica, +San Jos´e, Costa Rica +Abstract +Throughout history, prevention and control of dengue transmission have chal- +lenged public health authorities worldwide. In the last decades, the interaction of +multiple factors, such as environmental and climate variability, has influenced incre- +ments in incidence and geographical spread of the virus. In Costa Rica, a country +characterized by multiple microclimates separated by short distances, dengue has +been endemic since its introduction in 1993. Understanding the role of climatic and +environmental factors in the seasonal and inter-annual variability of disease spread is +essential to develop effective surveillance and control efforts. In this study, we con- +ducted a wavelet time series analysis of weekly climate, local environmental variables, +and dengue cases (2001-2019) from 32 cantons in Costa Rica to identify significant pe- +riods (e.g., annual, biannual) in which climate and environmental variables co-varied +with dengue cases. Wavelet coherence analysis was used to characterize seasonality, +multi-year outbreaks, and relative delays between the time series. Results show that +dengue outbreaks occurring every 3 years in cantons located in the country’s Central, +North, and South Pacific regions were highly coherent with the Oceanic Ni˜no 3.4 +and the Tropical North Caribbean Index (TNA). Dengue cases were in phase with El +Ni˜no 3.4 and TNA, with El Ni˜no 3.4 ahead of dengue cases by roughly nine months +and TNA ahead by less than three months. Annual dengue outbreaks were coherent +with local environmental variables (NDWI, EVI, Evapotranspiration, and Precipita- +tion) in most cantons except those located in the Central, South Pacific, and South +Caribbean regions of the country. The local environmental variables were in phase +with dengue cases and were ahead by around three months. +1 +arXiv:2301.02286v1 [q-bio.PE] 3 Jan 2023 + +Introduction +Dengue is a mosquito-borne viral infection caused by four antigenically distinct dengue +virus serotypes (DENV1-4). The transmission to humans occurs by the bite of an infected +female mosquito. The Aedes aegypti is the primary vector, more common in rural areas, +and Aedes albopictus is a secondary vector, considered one of the 100 worst invasive species +in the world [1], currently present on all continents except Antarctica [2, 3]. Dengue is +a flu-like illness that affects individuals of all ages, causing significant health, economic, +and social burdens on a population [4]. The clinical profile of patients can range from +asymptomatic infection to severe cases. +In recent years, the complex interaction of biological, socioeconomic, environmental, +and climatic factors has facilitated the rapid emergence of this viral infection throughout +the world, becoming endemic and a relevant public health problem in more than 100 coun- +tries [5]. In the last decades, the number of dengue cases reported to the World Health +Organization (WHO) has increased from 505,430 cases in 2000 to more than 4.2 million in +2019 [6, 5]. +Seasonal case patterns and vector abundance suggest that dengue transmission is sen- +sitive to climatic and environmental factors [7, 8]. Climatic conditions can alter spatial +and temporal dynamics of vector ecology, potentially increasing vector ranges, lengthening +the duration of vector activity, and increasing the mosquito’s infectious period [7]. Pre- +cipitation provides habitats for the aquatic stages of the mosquito life cycle and strongly +influences vector distribution [8, 9]. On the other hand, water temperature plays a signif- +icant role in mosquito reproduction since it directly affects survival at all stages of its life +cycle [10]. Temperature increases are associated with a faster rate of viral replication within +the vector and a shorter extrinsic incubation period [7]. Furthermore, higher humidity is +associated with an increase in Ae. aegypti feeding activity, which enhances the spread of +the diseases. +The complexity of dengue transmission has driven many studies to assess its correlation +with meteorological and ecological variables [11, 12, 13, 14, 15, 16, 17]. Most of these works +evaluated the effects and correlation between dengue cases and climate variables [18, 19] +and seasonal vegetation dynamics, which may also influence the biology of the vector +populations at relatively local scales [20, 21, 22]. Barrera et al. [23] suggested that dense +vegetation can promote Ae. aegypti pupal productivity by contributing organic material +to the habitat and influencing water temperature and evaporation by creating shades. +This study focuses on Costa Rica, where, in 1961, successfully eradicated the mosquito +Ae. aegypti after intense prevention and control campaigns. In 1970, the lack of continuity +in active surveillance caused the mosquito to be found again on the Pacific coast. +By +September 1993, health officials reported the first dengue case. The patterns and periodicity +of transmission are different across the country. Despite the coastal regions being the most +affected areas, trends observed over the years show variations in transmission peaks in +all affected areas, challenging public health authorities to allocate and optimize available +resources. +This work aims to describe the incidence patterns of dengue in 32 different cantons +(of interest to the Ministry of Health) of Costa Rica and its correlation with climatic and +local environmental variables using wavelet coherence and wavelet cluster analysis. Wavelet +2 + +transform decomposes a time series in both the time and frequency domains, revealing how +different periods change over time into non-stationary signals [25, 26]. Furthermore, it +allows conclusions to be drawn about the synchronicity of the series in specific periods [16]. +Wavelets have been used to study time series with different purposes: to evaluate the +main characteristics of non-stationary time series [27, 16, 28, 15, 29], to analyze spatial +patterns [16, 14], to study the relationship between population and environmental time +series; finding the phase and/or synchrony patterns [15, 16], and to study multiple time +series [30, 31]. +In this study, we describe and compare the relationships between El Ni˜no-Southern +Oscillation (ENSO), Tropical North Caribbean Index (TNA), Normalized Difference Water +Index (NDWI), Enhanced Vegetation Index (EVI), Evapotranspiration (ET), precipitation, +and dengue cases in 32 cantons in Costa Rica using a wavelet approach. +Materials and methods +Study Area +119 +201 +205 +209 +213 +305 +410 +501 +502 +503 +505 +506 +510 +601 +602 +604 +605 +606 +607 +609 +610 +611 +701 +702 +703 +704 +705 +706 +101 +103 +109 +110 +1. San José +5. Guanacaste +7. Limón +101 San José +501 Liberia +701 Limón +103 Desamparados +502 Nicoya +702 Pococí +109 Santa Ana +503 Santa Cruz +703 Siquirres +110 Alajuelita +505 Carrillo +704 Talamanca +119 Pérez Zeledón +506 Cañas +705 Ma�na +510 La Cruz +706 Guácimo +2. Alajuela +201 Alajuela +6. Puntarenas +205 Atenas +601 Puntarenas +209 Oro�na +602 Esparza +213 Upala +604 Montes de Oro +605 Osa +3. Cartago +606 Quepos +305 Turrialba +607 Golfito +609 Parrita +4. Heredia +610 Corredores +410 Sarapiqui +611 Garabito +Figure 1: Costa Rica Study Area. The colored cantons correspond to the 32 cantons +highlighted as being of interest to the Minister of Health of Costa Rica due to the prevalence +of dengue. +Costa Rica is a Central American country bordering Nicaragua to the North, the +Caribbean Sea to the northeast, Panama to the southeast, and the Pacific Ocean to the +Southwest. It has a population of around five million in a land area of 51,060 km2. Almost +half of the population is concentrated in the Great Metropolitan Area, which includes the +capital, San Jos´e. Costa Rica is divided into seven provinces: San Jos´e, Alajuela, Heredia, +Cartago, Guanacaste, Puntarenas, and Lim´on. These provinces are further divided into +3 + +N +W +E +sdifferent cantons (83 in total). Due to the high number of dengue cases, we focused this +analysis on 32 cantons reported by the local public health authorities as areas of interest +(Fig 1). +The country has a tropical climate with various microclimates. On the Pacific coast and +the Central region, there are a dry season from December to April and a rainy season from +May to November, during which rainfall is abundant. In contrast, in the eastern plains and +coasts (but also in the southern part of the Pacific coast), the climate is equatorial, with +abundant rainfall throughout the year. +In Costa Rica, the success in the fight against Ae. aegypti in the 1950s led to the +declaration of the country as free of the vector by 1961. However, the lack of continuity in +active surveillance caused, around 1971, the detection of positive places for the presence of +the mosquito. After that time, a new eradication campaign was established. However, by +1992, the vector was already in almost all national territories. By September 1993, the first +dengue cases were reported on the Pacific coast. Since then, the disease has exhibited endo- +epidemic transmission with the circulation of the four dengue virus serotypes. According +to data from the Ministry of Health, from 1993 to 2021, the country has notified a total of +398.546 cases. During these years, transmission has been characterized by epidemic peaks +occurring every 2 to 5 years, with 2013 being the year with the highest number of cases +(49,993 cases) reported by health centers around the country, followed by 2005 (37,798 +cases) and 2010 (31,484 cases) [68]. The year 2007 reported the highest number of severe +dengue cases for 318 patients, which represented 1.19% of the total cases reported for that +period. Despite the coastal regions being the most affected areas, trends observed over the +years show variations in transmission peaks in all affected areas (Fig S1). +Data +Dengue cases +The Ministry of Health of Costa Rica provided weekly dengue case records for all can- +tons. For this analysis, the data were aggregated monthly, square-root transformed, and +standardized due to their asymmetry [16]. +Climate variables +Anomalies in the Caribbean and Pacific Ocean were considered throughout the Tropical +North Caribbean Index (TNA) [32] and El Ni˜no-Southern Oscillation (ENSO) for region +3.4 [33], respectively [34, 32, 35]. +The TNA SST anomaly index indicates the surface temperatures in the eastern tropical +North Caribbean Ocean. El Ni˜no (positive phase of ENSO) occurs, on average, every 3-7 +years with episodes typically lasting 9-12 months [36], and is characterized by sea surface +temperature (SST) above the mean in region 3.4 of the equatorial Pacific [36]. +ENSO +is the Earth’s strongest inter-annual climate cycle and is the leading cause of climatic +variability in Northeastern South America [37]. ENSO data was obtained from the Climate +Prediction Center (CPC) of the United States National Oceanographic and Atmospheric +Administration (NOAA) [33]. +4 + +Environmental variables +We considered four environmental variables that provide local information on vegetation, +precipitation, and evapotranspiration. The Normalized Difference Water Index (NDWI) +describes changes in the liquid water content of leaves, providing a proxy for water stress +in vegetation [38, 39, 40]. The Enhanced Vegetation Index (EVI) provides spatial and tem- +poral information about vegetation. It can be used to quantify vegetation greenness [41]. +Evapotranspiration (ET) can be used to calculate regional water and energy balance and soil +water status; hence, it provides key information for water resource management [42]. Fi- +nally, precipitation index corresponds to rainfall estimates and was obtained from CHIRPS +(Climate Hazards Group InfraRed Precipitation with Station) [43]. All data was stan- +dardized to provide consistency and allow for comparisons across different time series and +corresponding results. +Wavelets analysis +To understand the time-frequency variability of dengue, climate, and environmental vari- +ables, we conducted a wavelet time series analysis over a time period of nineteen years +(2001-2019). Wavelet time series analyses are ideal for noisy, non-stationary data, such +as dengue case data, demonstrating strong seasonality and inter-annual variability (yearly +changes) [44, 25]. In the following, we present the critical details of our analyses. For +further depth, Torrence and Compo [45] provide a comprehensive presentation of wavelet +time series analysis, and Cazellez et al. [26, 25, 46] offer a perspective of the use of these +techniques in ecological and epidemic applications. +Wavelet power spectra +The wavelet analysis is based on a wavelet transform defined as: +Wx(s, τ) = 1 +√s +� ∞ +−∞ +x(t)Ψ∗ +�t − τ +s +� +dt = +� ∞ +−∞ +x(t)Ψ∗ +s,τ(t)dt, +(1) +where ∗ denotes the complex conjugate form and Ψs,τ(t) represent a family of functions +derived from a single function called the “mother wavelet”. The signal is decomposed in +these functions which can be expressed in terms of two parameters, one for the time position +τ, and the other for the scale of the wavelets s, given by +Ψs,τ(t) = 1 +√sΨ +�t − τ +s +� +. +(2) +For this analysis, we use the R-packages WaveletComp [47] that analyzes the frequency +structure of uni- and bivariate time series using the Morlet mother wavelet [25, 26] +Ψ(t) = π +−1 +4 eiωte +−t2 +2 . +(3) +This leads to a continuous complex-valued wavelet transform that can be separated +into its real and imaginary parts, providing information on both local amplitude and in- +stantaneous phase of any periodic process across time — a prerequisite for investigating +coherency between two-time series [47]. +5 + +Analyzing two time series +Cross-wavelet and wavelet coherence allowed us to compare two time series, such as climate +and dengue, and to identify synchronous periods or signals. The cross-wavelet transform of +two-time series (xt) and (yt), with respective wavelet transforms Wx, and Wy decomposes +the Fourier co- and quadrature-spectra in the time-frequency (or time-scale) domain. The +concepts of cross-wavelet analysis provide a tool for (i) comparing the frequency of two-time +series, (ii) concluding the series’ synchronicity at specific periods and across certain ranges +of time [47]. The cross-wavelet transform of two time series (xt) and (yt) is given by +Wx,y(τ, s) = 1 +sWx(τ, s)W ∗ +y (τ, s) +Its modulus can be interpreted as cross-wavelet power. +Power.xy(τ, s) = |Wx,y(τ, s)| +In a geometric sense, the cross-wavelet transform is the analog of the covariance. How- +ever, it lends itself to certain limitations for interpretation concerning the degree of associ- +ation of the two series that can be remedied by coherence. +Wavelet coherence +Wavelet Coherence is a normalized measure of dependence for which it is possible to con- +struct confidence intervals, commonly considered more interpretable than the cross-wavelet +transform [47]. Fourier coherency measures the cross-correlation between two time series +as a function of frequency; an analogous concept in wavelet theory is the notion of wavelet +coherency. In a geometric sense, coherency is the analog of classical correlation. Conse- +quently, in analogy with the notion of Fourier coherence and the coefficient of determination +in statistics, wavelet coherence is given by squared coherence. +Rx,y(s, τ) = +|⟨Wx,y(s, τ)⟩|2 +|⟨Wx(s, τ)⟩|2|⟨Wy(s, τ)⟩|2 +(4) +The angle brackets indicate smoothing in both time and frequency, Wx(s, τ) and Wy(s, τ) +are the wavelet transform of the series xt and yt, respectively. The value of Rx,y(s, τ) ranges +between 0 and 1, where 1 represents a perfect linear relationship between the time series +xt and yt. +The phase difference associated to the two signals, provides information about series +synchronization (i.e., in phase or out of phase). The Morlet wavelet is a complex wavelet, +so the phase difference can be computed in terms of the real (R) and the imaginary (I) +part, as follows +Φx,y(s, τ) = I(⟨Wx,y(s, τ)⟩) +R(⟨Wx,y(s, τ)⟩) +(5) +The instantaneous time lag between the time series xt and yt is also computed [16]. +6 + +Wavelet clusters +Clustering is partitioning a set of objects into groups (clusters) so that objects within a +group are more similar to each other than objects in different groups according to some +variables. Most of the clustering algorithms depend on some assumptions in order to define +the subgroups present in a data set. Consequently, the resulting clustering scheme requires +some evaluation regarding its validity. +Since dengue and climatic data are time-dependent and non-stationary, this paper fo- +cuses on performing clustering analysis based on vector wavelet coherence, proposed by +Oygur and Unal [48]. This methodology allows us to measure the synchronicity and co- +movements between vectors of different climatic time series and dengue to perform cluster +analysis based on their synchronicity for 32 cantons. +The cluster analysis based on vector wavelet coherence is performed using Wald’s +method [49]. Since the clustering is done by using a dissimilarity matrix constructed by +the vector wavelets coherence, we were able to compute Silhouette [50], Frey [51], McClain +[52], Cindex [53] and Dunn [54] indices, by setting minimum and maximum clusters as 2 +and 15 respectively, in order to determine the optimal number of clusters. However, all +these indices have differing optimal numbers of clusters. We decided to combine the Cindex +criterion with the expert criteria so that the results could be interpretable. +Initially, an analysis was carried out in which all the time series were considered. How- +ever, the results could have been more readily interpretable. The cluster algorithms could +only establish relationships between two or three cantons, so most groups were made up of +a single item. Thus, considering the nature of the data, the analysis was divided into two +moments. One in which the coherence between dengue cases and climatic variables was an- +alyzed, and the second considers the coherence between four different local environmental +variables and dengue cases. +Computing environment +All analyses were performed using the statistical package R version 2.4 [55]. To perform the +analysis, we first compute the multiple wavelet coherence using the vwc function in the R- +packages vectorwavelet [48]. Then, the function wclust in the R-package biwavelet [56] +is used to compute the dissimilarity matrices, and NbClust [57] is used to create the clusters +and indices calculation. Finally, the package WaveletComp version 1.1 [47] is considered +to obtain more interpretable results on the coherence between the incidence of dengue and +each climatic and environmental variable. Wavelet coherence, power average, and phase +difference are plotted and grouped according to the clusters obtained with the biowavelet +package. +7 + +Results +Climate variables and dengue cases +The coherence between climatic variables (TNA and El Ni˜no 3.4) and dengue cases changes +over time and geographically. A description of particular characteristics observed in the +different clusters is presented below. However, some general features are common to the +places where coherence was observed: 1) The 3-y dengue outbreaks were highly coherent +with climate time series after 2008 in cantons located in the central and those close to the +Pacific ocean (See Fig 2). Dengue and the climate time series were synchronized, with El +Ni˜no 3.4 leading by around nine months and TNA ahead by less than three months. 2) +Associations between climate and dengue cases are also observed in the 1, 1.5, and 2 yr +periods, with dengue time series leading. 3) There was no significant multiyear coherence +between climate variables and dengue cases in cantons mainly located in the Caribbean, and +some others, such as Parrita and Quepos, in the central Pacific. 4) The cantons grouped in +the clusters share behavior patterns regarding the periods in which the highest significant +areas appear and the time series synchronization. +(a) +(b) +Figure 2: Cantons where dengue cases co-varied with El Ni˜no and TNA. Fig +(a) represents the cantons (red) where dengue outbreaks every 3 years showed significant +coherence with El Ni˜no and TNA. Fig (b) corresponds to those cantons (red) where the an- +nual dengue outbreaks showed significant coherence with TNA, with TNA variable leading +only by short periods. Data for cantons in white where no available for this analysis. +8 + +Characteristics of clusters that include climate variables (CV) +Cluster 1: +Alajuela, Orotina, Perez Zeled´on, San Jos´e, Santa Ana, +Sarapiqu´ı +Wavelet coherence shows an area of joint significance in the 3-yr +period after 2010 for both TNA and El Ni˜no 3.4 with dengue +cases. Except for Puntarenas and Alajuela, the coherence with +TNA is mainly in the period of 1-yr and needs to be identifiable +between El Ni˜no 3.4 and Puntarenas. The horizontal arrows in +period 3 pointing to the right indicate that the two series TNA +and dengue cases are in phase with vanishing phase differences +from 2011 to 2017. For El Ni˜no 3.4 and dengue cases, the time +series are also in phase after 2010, with El Ni˜no 3.4 leading by +roughly 9 months. +Cluster 2: +Alajuelita, Ca˜nas, Esparza, La Cruz, Talamanca, Upala +Dengue incidence showed significant coherence with TNA in +the 1-yr period, mainly after 2010 for all cantons except in +Alajuelita, where the coherence is around the 3-yr period from +2010-2018. The time series are in phase, but the leading time +series changes. +In Alajuelita, Ca˜nas, and La Cruz, the TNA +time series is the leading one by less than 2 months, contrary +to Esparza, Talamanca, and Upala. The wavelet coherence for +El Ni˜no 3.4 and dengue cases show a significant coherence in +the period of 3-yr in Upala and Alajuelita, with El Ni˜no time +series ahead by roughly nine months after 2007. Small areas of +significance are observed for the remaining cantons in the 1 and +2-yr periods with dengue time series ahead. +9 + +Cluster 3: +Atenas, Desamparados, Liberia, Parrita, Pococi, Santa +Cruz. +Wavelet coherence shows an area of joint significance between +TNA and dengue cases in the 1-yr period, mainly between 2012 +and 2017, in all cantons. +However, TNA is the leading time +series only in Liberia and Santa Cruz. There is also an area of +joint significance in the 3-yr period between TNA and dengue +cases in Atenas, Desamparados, and Santa Cruz, with the +time series in-phase with vanishing phase differences between +2010-2016, except in Desamparados, where TNA was leading +by a couple of weeks. The wavelet coherence for El Ni˜no 3.4 +and dengue cases show an area of joint significance in the 3-yr +period for Atenas, Liberia, Santa Cruz, and Desamparados. +The time series are in phase after 2010 with El Ni˜no 3.4 leading +by around nine months. For Parrita and Pococ´ı, there are no +identifiable areas of significance. +Cluster 4: +Carrillo, Guacimo, Lim´on, Nicoya, Siquirres +There are no identifiable common patterns in all cantons. Co- +herence for dengue cases with both TNA and El Ni˜no 3.4 shows +an area of joint significance in the 3-yr period in Carrillo and +Nicoya. TNA and dengue cases have vanished phase differences +between 2008 and 2016 approximately in both cantons, and +El Ni˜no 3.4 and dengue cases are in phase with El Ni˜no 3.4, +leading by a lag of 9 months from 2008-2014. TNA and dengue +cases also have an area of joint significance in the 1-yr period. +The time series are in-phase after 2012, with TNA leading in +Carrillo and Nicoya. +For the remaining cantons, the wavelet coherence shows coher- +ence between El Ni˜no 3.4 and dengue cases in 1 and 2-yr periods, +with the dengue time series ahead. +Cluster 5 +Corredores, Golfito, Osa +There is an area of joint significance between TNA and +dengue cases in all cantons in the 1-yr and 3-yr periods where +the leading time series is dengue cases. +The 3-yr dengue +outbreaks were highly coherent with El Ni˜no 3.4 after 2007, +with El Ni˜no 3.4 leading by a lag ranging between 0 to 9 months. +10 + +Cluster 6 +Garabito, Matina, Montes de Oro, Quepos, Turrialba +There is only an area of joint significance between climate vari- +ables and dengue cases in Garabito and Montes de Oro in the +period of 3-yr after 2010. TNA and dengue time series are in- +phase with vanishing phase differences from 2011 to 2016, and +El Ni˜no 3.4 has an advantage of approximately 9 months over +dengue cases. For the remaining cantons, small areas of signifi- +cance are observed in 1 and 2-yr periods with dengue time series +ahead. +Local environmental variables and dengue cases +Using coherence analysis to compare these time series in the frequency domain, we found +that the annual dengue outbreaks were highly coherent with all environmental variables. +In all cases, the phase difference shows an increase in dengue incidence after an increase in +the local environment variables with a lag ranging between 0 to 3 months. The association +was nonstationary, with disruption from around 2007 to 2010. Dengue incidence showed +significant coherence with the four variables EVI, NDWI, ET, and precipitation except +in Lim´on, Matina, Talamanca, Corredores, Golfito, and Osa, located in the South Pacific +and South Caribbean of the country, and Parrita, Garabito, and Quepos in the central +Pacific (see Fig 3). The wavelet coherence for the last cantons showed areas of high signif- +icance between dengue, mainly with ET and precipitation in short periods. This finding +is expected due to the regular seasonality observed on the Pacific coast and the country’s +Central region. The main differences observed between the clusters are the periods in which +the synchronicity of the dengue cases and local environmental time series is observed. +Figure 3: Cantons where dengue cases co-varied with the environmental vari- +ables. Red cantons are those in which the annual dengue outbreaks was highly coherent +with all environmental variables. +11 + +Characteristics of clusters that include local environmental variables (LEV) +Cluster 1 +Alajuela, Montes de Oro, Orotina, Parrita, Pococci, +Santa Ana, Turrialba +Annual dengue outbreaks showed significant coherence with +the four environmental variables, except in Parrita and Pococ´ı. +In all cases, the leading time series were the environmental +variables with 0 to 3 months ahead of dengue cases. Wavelet +coherence in Parrita shows an area of joint significance only with +precipitation before 2009 with the time series in phase. Wavelet +coherence in Pococ´ı has a continuous area of joint significance +with all variables. While for the rest of the cantons, a disruption +is observed between the areas of significance around 2007 to 2010. +Cluster 2 +Alajuelita, Lim´on +Wavelet coherence in Alajuela shows two areas of joint signifi- +cance with all variables in the 1-yr period with disruption around +2007 to 2010. The leading time series were the environmental +variables with 0 to 3 months ahead of dengue cases. Wavelet +coherence in Lim´on shows an area of high significance only with +ET in two short periods with an interruption between 2009-2010. +The coherence with the other variables is not conclusive. +Cluster 3 +Ca˜nas, Matina, Osa, Puntarenas +Wavelet coherence in Ca˜nas and Puntarenas show a correlation +with the four vegetation variables in the 1 yr period (in Ca˜nas, +there is a continuous area of joint significance over time, while +in Puntarenas is interrupted between 2008-2010). +Smaller +areas of significance are observed between dengue cases in Osa +and the vegetation variables, mainly after 2013. +In all cases, +environmental variables are the leading time series with a lag +of fewer than 3 months. Wavelet coherence in Matina shows an +area of high significance only with ET and EVI in 1-yr. The +environmental time series go ahead by less than 3 months. +12 + +Cluster 4 +Carrillo, +Desamparados, +Esparza, +La +Cruz, +Perez +Zeled´on, San Jose, Santa Cruz, Sarapiqu´ı, Siquirres, Ta- +lamanca +Wavelet coherence shows areas of joint significance between +dengue and the four vegetation variables in the 1-yr period +with an interruption around 2009-2010, except for San Jos´e +and Talamanca. In all cases, the leading time series were the +environmental variable by approximately 3 months ahead of +dengue cases. +There is no observed coherence between ET +and dengue cases in San Jos´e, and dengue cases in Talamanca +showed significant coherence only with ET in a short period +(around 2011-2015). +The area of high significance is smaller +in Perez Zeled´on, Sarapiqu´ı, and Siquirres than in the other +cantons in the cluster. Wavelet coherence in Talamanca shows +an area of joint significance only with ET in a short period +(2011-2015 approx) +Cluster 5 +Corredores, Golfito, Liberia, Nicoya +Wavelet coherence shows areas of joint significance between dengue +cases in Liberia and Nicoya with the four vegetation variables in +the 1-yr period with minor interruptions over time. Dengue cases in +Corredores and Golfito showed significant coherence only with ET +and precipitation in the 1-yr period. Local environmental variables +are the leading time series in all cases by approximately 3 months +ahead of dengue cases. +Cluster 6 +Garabito, Gu´acimo +Dengue time series in Gu´acimo showed significant coherence +with all variables after 2009 in the 1-yr period, with the en- +vironmental variables ahead of dengue cases by approximately +3 months. +Dengue time series in Garabito showed significant +coherence only with ET and precipitation before 2012 in the +1-yr period. +When creating the clusters, Atenas, Upala, and Quepos stayed isolated as the unique +canton in their groups. Quepos is a touristic canton located in the central Pacific where +the rain is abundant. The wavelet coherence shows an area of joint significance only between +dengue cases and ET and precipitation for a short period, with the dengue time series the +leading one. Upala is located in the North in a subregion where the climate is classified as +rainy with monsoon influence (in essence, a tropical monsoon climate tends to have more +13 + +precipitation and less pronounced dry seasons). The wavelet coherence shows a region of +high joint significance between 2012 and 2015 (the worst dengue outbreak in Upala was +registered in 2013) with all variables except EVI. The leading time series is dengue except in +the coherence with ET, where the time series were in phase with a vanish phase difference. +Atenas is located in the Central of the country. The wavelet coherence shows two areas +of joint significance in the 1-yr period, with a disruption between 2007-2009. The time +series are in-phase with environmental variables leading, except between 2009-2013, when +the dengue time series started to be the leading time series. The lag change between 0-3 +months. +Discussion +Climate variables +Our results show that the 3-year dengue outbreaks were highly coherent with El Ni˜no 3.4 +and TNA indicators, mainly in cantons located in the North, South Pacific coastal, and +Center region of Costa Rica. When analyzing phase differences between the time series in +these regions, synchrony between dengue cases and climatic variables was observed. The +time series of El Ni˜no and TNA used to be ahead of dengue cases by nine and less than +three months, respectively. In some cantons, TNA and dengue cases are almost in perfect +sync. +The synchronization of the time series began to be observed mainly after 2008. +The periods in which areas of high significance become evident and the year in which the +time series starts to be synchronized change between cantons and over time. Those timing +periods and the series leading are the main differences between the clusters. +Our results also showed that annual dengue outbreaks were highly coherent with TNA in +cantons located in the North Pacific and the Central region of the country, with the dengue +and TNA time series synchronized for short periods with a vanishing phase difference. In +some cantons, associations between climate and dengue cases were also observed in the 1, +1.5, and 2-year periods, with dengue time series leading. However, dengue cases influencing +climate variables are a biologically unlikely relationship. It is more plausible that changes +in El Ni˜no 3.4 and TNA increase subsequent dengue transmission [59], given that the +anomalies in the ocean temperatures affect local temperature and precipitation, which are +directly associated with the virus ecology and transmission [8, 7, 9]. +Significant coherence between climate variables and dengue cases was observed mainly in +the cantons in which there are periods of drought. There is no evidence of a contemporary +statistical association between the SST anomalies in the two oceans. But, it has been +found that seasonal precipitation is dependent upon the simultaneous interaction of both +anomalies. The draught, which typifies the Pacific slope during years of warm phase ENSO, +is less severe when the Caribbean is warmer, but the reduction in rainfalls may spread over +the entire country [60]. +The North Pacific belongs to the Pacific precipitation regime, +known for a well-defined dry and rainy period [61]. The geographical characteristics in the +central and South Pacific generate a climate where the dry period is very favorable and +short and the rainy intense [61]. The years of intense droughts in those regions, except +for 2001, have coincided with El Ni˜no years [62]. There has been observed that during El +Ni˜no, there is a greater probability that the entire Pacific slope and the Central region of +14 + +Costa Rica will experience dry to extremely dry conditions. At the same time, there is a +greater probability of extreme rainy scenarios in the Caribbean [62]. +Environmental variables +The annual dengue cycle was highly coherent with the four local environmental variables in +all cantons except those located in the central Pacific, South Pacific, and South Caribbean. +For those, dengue incidence showed a significant coherence with ET and precipitation over +short periods. In those regions, the rain is abundant, and there is not a well-defined dry +season [60] and the time series for the vegetation indices (EVI, NDWI) does not show an +annual seasonality like that observed in the North Pacific and the Central valley of the +country. Regarding synchronization, dengue and environmental time series were in phase +with the environmental time series ahead of dengue cases by approximately 3 months. +Climate and environmental variables showed significant coherence with dengue epi- +demics across the geographically diverse regions of Costa Rica. However, spatial hetero- +geneity in their effects exists. Even when the association was established, it was not possible +to provide those variables’ causal effects on the dengue cases’ dynamic. Epidemic data is +typically noisy, complex, and non-stationary. Changes in the periodicity over time may also +be to external factors or inherent characteristics of the disease. Alternative explanations for +the multi-annual outbreaks of dengue epidemics have been proposed, such as partial cross- +immunity among the four serotypes of the dengue virus [63]. Moreover, sociological factors +such as population structure, unplanned urbanization, and international transportation of +infected people and mosquitoes may also affect dengue transmission [64]. +The complex interaction of biological, socioeconomic, environmental, and climatic fac- +tors creates a substantial spatiotemporal heterogeneity in dengue outbreak intensity. Un- +derstanding this heterogeneity in dengue transmission could improve epidemic prediction +and disease control. There are some previous works in which predictive models have been +proposed for the country using climatic and vegetation variables. Fuller et al. [65] pro- +posed a simple structural model incorporating lagged SST and MODIS vegetation indices, +explaining 83% of the variance in the total weekly cases of DF/FHD in Costa Rica. How- +ever, the cantonal-level analysis conducted in this study highlights the spatial heterogeneity +of the effect of climate and environmental factors on dengue incidence, which reveals that +the effect of those variables on dengue transmission on a local scale might differ from global +expectations. Thus, for the design of interventions and resource allocation, more localized +predictions may be helpful. Other studies show that when incorporating the same climatic +variables in all cantons to predict relative risk areas of dengue outbreak, the models pro- +jections may fail in some places [66, 67], which reinforces the importance of understanding +the local correlations between dengue cases and external factor such as climate and socioe- +conomic drivers to improve local dengue predictions. +Many studies have been conducted in tropical and subtropical regions to elucidate the +complex interactions between climate variables and dengue transmission [11, 12, 13, 14, +15, 16, 17]. However, this study is unique in the localized analysis that takes into account +variables such as TNA, EVI, NDWI, evapotranspiration, and precipitation. Wavelet time +serires analysis allows a retrospective study to characterize outbreaks over time, which +provides important guidelines for future modeling approaches in which explicit mechanisms +15 + +can be incorporated. However, as climate factors are not the only predictors influencing +the rise in dengue infection, future studies are needed to include other factors unique to +this area, such as the predominant circulating dengue viruses, anthropogenic factors, and +herd immunity, to name a few. +Acknowledgments +This research was supported by a Seed Grant for International Activities from Global +Affairs and the School of Medicine at the University of California, Davis. +Additional information +Codes: All the codes and files necessary to reproduce the results presented in this doc- +ument will be available in the GitHub repository, https://github.com/yurygarcia26/ +Wavelets_Costa_Rica.git, after publication. +References +[1] Outammassine A, Zouhair S, Loqman S. Global potential distribution of three under- +appreciated arboviruses vectors (Aedes japonicus, Aedes vexans and Aedes vittatus) un- +der current and future climate conditions. 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[cited +15 November 2022] +Available from https://www.ministeriodesalud.go.cr/ +index.php/biblioteca-de-archivos-left/documentos-ministerio-de-salud/ +vigilancia-de-la-salud/analisis-de-situacion-salud. +21 + diff --git a/etE0T4oBgHgl3EQfXAB2/content/tmp_files/load_file.txt b/etE0T4oBgHgl3EQfXAB2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec9553ca6bd6829e9a9cc6249d86acd0bc6cf11d --- /dev/null +++ b/etE0T4oBgHgl3EQfXAB2/content/tmp_files/load_file.txt @@ -0,0 +1,830 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf,len=829 +page_content='Common patterns between dengue cases, climate, and local environmental variables in Costa Rica: A Wavelet Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Yury E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Garc´ıa1, Shu-Wei Chou-Chen2, Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Barboza4, Maria L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Daza–Torres1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cricelio Montesinos-L´opez1, Paola V´asquez3, Juan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Calvo4, Miriam Nu˜no1, and Fabio Sanchez4 1Department of Public Health Sciences, University of California Davis, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 2Centro de Investigaci´on en Matem´atica Pura y Aplicada - Escuela de Estad´ıstica, Universidad de Costa Rica, San Jos´e, Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 3Centro de Investigaci´on en Matem´atica Pura y Aplicada, Universidad de Costa Rica, San Jos´e, Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 4Centro de Investigaci´on en Matem´atica Pura y Aplicada - Escuela de Matem´atica, Universidad de Costa Rica, San Jos´e, Costa Rica Abstract Throughout history, prevention and control of dengue transmission have chal- lenged public health authorities worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In the last decades, the interaction of multiple factors, such as environmental and climate variability, has influenced incre- ments in incidence and geographical spread of the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In Costa Rica, a country characterized by multiple microclimates separated by short distances, dengue has been endemic since its introduction in 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Understanding the role of climatic and environmental factors in the seasonal and inter-annual variability of disease spread is essential to develop effective surveillance and control efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In this study, we con- ducted a wavelet time series analysis of weekly climate, local environmental variables, and dengue cases (2001-2019) from 32 cantons in Costa Rica to identify significant pe- riods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=', annual, biannual) in which climate and environmental variables co-varied with dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence analysis was used to characterize seasonality, multi-year outbreaks, and relative delays between the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Results show that dengue outbreaks occurring every 3 years in cantons located in the country’s Central, North, and South Pacific regions were highly coherent with the Oceanic Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and the Tropical North Caribbean Index (TNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue cases were in phase with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and TNA, with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 ahead of dengue cases by roughly nine months and TNA ahead by less than three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Annual dengue outbreaks were coherent with local environmental variables (NDWI, EVI, Evapotranspiration, and Precipita- tion) in most cantons except those located in the Central, South Pacific, and South Caribbean regions of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The local environmental variables were in phase with dengue cases and were ahead by around three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='02286v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='PE] 3 Jan 2023 Introduction Dengue is a mosquito-borne viral infection caused by four antigenically distinct dengue virus serotypes (DENV1-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The transmission to humans occurs by the bite of an infected female mosquito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The Aedes aegypti is the primary vector, more common in rural areas, and Aedes albopictus is a secondary vector, considered one of the 100 worst invasive species in the world [1], currently present on all continents except Antarctica [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue is a flu-like illness that affects individuals of all ages, causing significant health, economic, and social burdens on a population [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The clinical profile of patients can range from asymptomatic infection to severe cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In recent years, the complex interaction of biological, socioeconomic, environmental, and climatic factors has facilitated the rapid emergence of this viral infection throughout the world, becoming endemic and a relevant public health problem in more than 100 coun- tries [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In the last decades, the number of dengue cases reported to the World Health Organization (WHO) has increased from 505,430 cases in 2000 to more than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='2 million in 2019 [6, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Seasonal case patterns and vector abundance suggest that dengue transmission is sen- sitive to climatic and environmental factors [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Climatic conditions can alter spatial and temporal dynamics of vector ecology, potentially increasing vector ranges, lengthening the duration of vector activity, and increasing the mosquito’s infectious period [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Pre- cipitation provides habitats for the aquatic stages of the mosquito life cycle and strongly influences vector distribution [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' On the other hand, water temperature plays a signif- icant role in mosquito reproduction since it directly affects survival at all stages of its life cycle [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Temperature increases are associated with a faster rate of viral replication within the vector and a shorter extrinsic incubation period [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Furthermore, higher humidity is associated with an increase in Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' aegypti feeding activity, which enhances the spread of the diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The complexity of dengue transmission has driven many studies to assess its correlation with meteorological and ecological variables [11, 12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Most of these works evaluated the effects and correlation between dengue cases and climate variables [18, 19] and seasonal vegetation dynamics, which may also influence the biology of the vector populations at relatively local scales [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Barrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [23] suggested that dense vegetation can promote Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' aegypti pupal productivity by contributing organic material to the habitat and influencing water temperature and evaporation by creating shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' This study focuses on Costa Rica, where, in 1961, successfully eradicated the mosquito Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' aegypti after intense prevention and control campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In 1970, the lack of continuity in active surveillance caused the mosquito to be found again on the Pacific coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' By September 1993, health officials reported the first dengue case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The patterns and periodicity of transmission are different across the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Despite the coastal regions being the most affected areas, trends observed over the years show variations in transmission peaks in all affected areas, challenging public health authorities to allocate and optimize available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' This work aims to describe the incidence patterns of dengue in 32 different cantons (of interest to the Ministry of Health) of Costa Rica and its correlation with climatic and local environmental variables using wavelet coherence and wavelet cluster analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet 2 transform decomposes a time series in both the time and frequency domains, revealing how different periods change over time into non-stationary signals [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Furthermore, it allows conclusions to be drawn about the synchronicity of the series in specific periods [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelets have been used to study time series with different purposes: to evaluate the main characteristics of non-stationary time series [27, 16, 28, 15, 29], to analyze spatial patterns [16, 14], to study the relationship between population and environmental time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' finding the phase and/or synchrony patterns [15, 16], and to study multiple time series [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In this study, we describe and compare the relationships between El Ni˜no-Southern Oscillation (ENSO), Tropical North Caribbean Index (TNA), Normalized Difference Water Index (NDWI), Enhanced Vegetation Index (EVI), Evapotranspiration (ET), precipitation, and dengue cases in 32 cantons in Costa Rica using a wavelet approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Materials and methods Study Area 119 201 205 209 213 305 410 501 502 503 505 506 510 601 602 604 605 606 607 609 610 611 701 702 703 704 705 706 101 103 109 110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' San José 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Guanacaste 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Limón 101 San José 501 Liberia 701 Limón 103 Desamparados 502 Nicoya 702 Pococí 109 Santa Ana 503 Santa Cruz 703 Siquirres 110 Alajuelita 505 Carrillo 704 Talamanca 119 Pérez Zeledón 506 Cañas 705 Ma�na 510 La Cruz 706 Guácimo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Alajuela 201 Alajuela 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Puntarenas 205 Atenas 601 Puntarenas 209 Oro�na 602 Esparza 213 Upala 604 Montes de Oro 605 Osa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cartago 606 Quepos 305 Turrialba 607 Golfito 609 Parrita 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Heredia 610 Corredores 410 Sarapiqui 611 Garabito Figure 1: Costa Rica Study Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The colored cantons correspond to the 32 cantons highlighted as being of interest to the Minister of Health of Costa Rica due to the prevalence of dengue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Costa Rica is a Central American country bordering Nicaragua to the North, the Caribbean Sea to the northeast, Panama to the southeast, and the Pacific Ocean to the Southwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' It has a population of around five million in a land area of 51,060 km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Almost half of the population is concentrated in the Great Metropolitan Area, which includes the capital, San Jos´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Costa Rica is divided into seven provinces: San Jos´e, Alajuela, Heredia, Cartago, Guanacaste, Puntarenas, and Lim´on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' These provinces are further divided into 3 N W E sdifferent cantons (83 in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Due to the high number of dengue cases, we focused this analysis on 32 cantons reported by the local public health authorities as areas of interest (Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The country has a tropical climate with various microclimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' On the Pacific coast and the Central region, there are a dry season from December to April and a rainy season from May to November, during which rainfall is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In contrast, in the eastern plains and coasts (but also in the southern part of the Pacific coast), the climate is equatorial, with abundant rainfall throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In Costa Rica, the success in the fight against Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' aegypti in the 1950s led to the declaration of the country as free of the vector by 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, the lack of continuity in active surveillance caused, around 1971, the detection of positive places for the presence of the mosquito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' After that time, a new eradication campaign was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, by 1992, the vector was already in almost all national territories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' By September 1993, the first dengue cases were reported on the Pacific coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Since then, the disease has exhibited endo- epidemic transmission with the circulation of the four dengue virus serotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' According to data from the Ministry of Health, from 1993 to 2021, the country has notified a total of 398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='546 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' During these years, transmission has been characterized by epidemic peaks occurring every 2 to 5 years, with 2013 being the year with the highest number of cases (49,993 cases) reported by health centers around the country, followed by 2005 (37,798 cases) and 2010 (31,484 cases) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The year 2007 reported the highest number of severe dengue cases for 318 patients, which represented 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='19% of the total cases reported for that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Despite the coastal regions being the most affected areas, trends observed over the years show variations in transmission peaks in all affected areas (Fig S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Data Dengue cases The Ministry of Health of Costa Rica provided weekly dengue case records for all can- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For this analysis, the data were aggregated monthly, square-root transformed, and standardized due to their asymmetry [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Climate variables Anomalies in the Caribbean and Pacific Ocean were considered throughout the Tropical North Caribbean Index (TNA) [32] and El Ni˜no-Southern Oscillation (ENSO) for region 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 [33], respectively [34, 32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The TNA SST anomaly index indicates the surface temperatures in the eastern tropical North Caribbean Ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' El Ni˜no (positive phase of ENSO) occurs, on average, every 3-7 years with episodes typically lasting 9-12 months [36], and is characterized by sea surface temperature (SST) above the mean in region 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 of the equatorial Pacific [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' ENSO is the Earth’s strongest inter-annual climate cycle and is the leading cause of climatic variability in Northeastern South America [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' ENSO data was obtained from the Climate Prediction Center (CPC) of the United States National Oceanographic and Atmospheric Administration (NOAA) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 4 Environmental variables We considered four environmental variables that provide local information on vegetation, precipitation, and evapotranspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The Normalized Difference Water Index (NDWI) describes changes in the liquid water content of leaves, providing a proxy for water stress in vegetation [38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The Enhanced Vegetation Index (EVI) provides spatial and tem- poral information about vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' It can be used to quantify vegetation greenness [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Evapotranspiration (ET) can be used to calculate regional water and energy balance and soil water status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' hence, it provides key information for water resource management [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Fi- nally, precipitation index corresponds to rainfall estimates and was obtained from CHIRPS (Climate Hazards Group InfraRed Precipitation with Station) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' All data was stan- dardized to provide consistency and allow for comparisons across different time series and corresponding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelets analysis To understand the time-frequency variability of dengue, climate, and environmental vari- ables, we conducted a wavelet time series analysis over a time period of nineteen years (2001-2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet time series analyses are ideal for noisy, non-stationary data, such as dengue case data, demonstrating strong seasonality and inter-annual variability (yearly changes) [44, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In the following, we present the critical details of our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For further depth, Torrence and Compo [45] provide a comprehensive presentation of wavelet time series analysis, and Cazellez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [26, 25, 46] offer a perspective of the use of these techniques in ecological and epidemic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet power spectra The wavelet analysis is based on a wavelet transform defined as: Wx(s, τ) = 1 √s � ∞ −∞ x(t)Ψ∗ �t − τ s � dt = � ∞ −∞ x(t)Ψ∗ s,τ(t)dt, (1) where ∗ denotes the complex conjugate form and Ψs,τ(t) represent a family of functions derived from a single function called the “mother wavelet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The signal is decomposed in these functions which can be expressed in terms of two parameters, one for the time position τ, and the other for the scale of the wavelets s, given by Ψs,τ(t) = 1 √sΨ �t − τ s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' (2) For this analysis, we use the R-packages WaveletComp [47] that analyzes the frequency structure of uni- and bivariate time series using the Morlet mother wavelet [25, 26] Ψ(t) = π −1 4 eiωte −t2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' (3) This leads to a continuous complex-valued wavelet transform that can be separated into its real and imaginary parts, providing information on both local amplitude and in- stantaneous phase of any periodic process across time — a prerequisite for investigating coherency between two-time series [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 5 Analyzing two time series Cross-wavelet and wavelet coherence allowed us to compare two time series, such as climate and dengue, and to identify synchronous periods or signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The cross-wavelet transform of two-time series (xt) and (yt), with respective wavelet transforms Wx, and Wy decomposes the Fourier co- and quadrature-spectra in the time-frequency (or time-scale) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The concepts of cross-wavelet analysis provide a tool for (i) comparing the frequency of two-time series, (ii) concluding the series’ synchronicity at specific periods and across certain ranges of time [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The cross-wavelet transform of two time series (xt) and (yt) is given by Wx,y(τ, s) = 1 sWx(τ, s)W ∗ y (τ, s) Its modulus can be interpreted as cross-wavelet power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='xy(τ, s) = |Wx,y(τ, s)| In a geometric sense, the cross-wavelet transform is the analog of the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' How- ever, it lends itself to certain limitations for interpretation concerning the degree of associ- ation of the two series that can be remedied by coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence Wavelet Coherence is a normalized measure of dependence for which it is possible to con- struct confidence intervals, commonly considered more interpretable than the cross-wavelet transform [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Fourier coherency measures the cross-correlation between two time series as a function of frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' an analogous concept in wavelet theory is the notion of wavelet coherency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In a geometric sense, coherency is the analog of classical correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Conse- quently, in analogy with the notion of Fourier coherence and the coefficient of determination in statistics, wavelet coherence is given by squared coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Rx,y(s, τ) = |⟨Wx,y(s, τ)⟩|2 |⟨Wx(s, τ)⟩|2|⟨Wy(s, τ)⟩|2 (4) The angle brackets indicate smoothing in both time and frequency, Wx(s, τ) and Wy(s, τ) are the wavelet transform of the series xt and yt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The value of Rx,y(s, τ) ranges between 0 and 1, where 1 represents a perfect linear relationship between the time series xt and yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The phase difference associated to the two signals, provides information about series synchronization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=', in phase or out of phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The Morlet wavelet is a complex wavelet, so the phase difference can be computed in terms of the real (R) and the imaginary (I) part, as follows Φx,y(s, τ) = I(⟨Wx,y(s, τ)⟩) R(⟨Wx,y(s, τ)⟩) (5) The instantaneous time lag between the time series xt and yt is also computed [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 6 Wavelet clusters Clustering is partitioning a set of objects into groups (clusters) so that objects within a group are more similar to each other than objects in different groups according to some variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Most of the clustering algorithms depend on some assumptions in order to define the subgroups present in a data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Consequently, the resulting clustering scheme requires some evaluation regarding its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Since dengue and climatic data are time-dependent and non-stationary, this paper fo- cuses on performing clustering analysis based on vector wavelet coherence, proposed by Oygur and Unal [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' This methodology allows us to measure the synchronicity and co- movements between vectors of different climatic time series and dengue to perform cluster analysis based on their synchronicity for 32 cantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The cluster analysis based on vector wavelet coherence is performed using Wald’s method [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Since the clustering is done by using a dissimilarity matrix constructed by the vector wavelets coherence, we were able to compute Silhouette [50], Frey [51], McClain [52], Cindex [53] and Dunn [54] indices, by setting minimum and maximum clusters as 2 and 15 respectively, in order to determine the optimal number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, all these indices have differing optimal numbers of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' We decided to combine the Cindex criterion with the expert criteria so that the results could be interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Initially, an analysis was carried out in which all the time series were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' How- ever, the results could have been more readily interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The cluster algorithms could only establish relationships between two or three cantons, so most groups were made up of a single item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Thus, considering the nature of the data, the analysis was divided into two moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' One in which the coherence between dengue cases and climatic variables was an- alyzed, and the second considers the coherence between four different local environmental variables and dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Computing environment All analyses were performed using the statistical package R version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' To perform the analysis, we first compute the multiple wavelet coherence using the vwc function in the R- packages vectorwavelet [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Then, the function wclust in the R-package biwavelet [56] is used to compute the dissimilarity matrices, and NbClust [57] is used to create the clusters and indices calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Finally, the package WaveletComp version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='1 [47] is considered to obtain more interpretable results on the coherence between the incidence of dengue and each climatic and environmental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence, power average, and phase difference are plotted and grouped according to the clusters obtained with the biowavelet package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 7 Results Climate variables and dengue cases The coherence between climatic variables (TNA and El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4) and dengue cases changes over time and geographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' A description of particular characteristics observed in the different clusters is presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, some general features are common to the places where coherence was observed: 1) The 3-y dengue outbreaks were highly coherent with climate time series after 2008 in cantons located in the central and those close to the Pacific ocean (See Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue and the climate time series were synchronized, with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 leading by around nine months and TNA ahead by less than three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 2) Associations between climate and dengue cases are also observed in the 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='5, and 2 yr periods, with dengue time series leading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 3) There was no significant multiyear coherence between climate variables and dengue cases in cantons mainly located in the Caribbean, and some others, such as Parrita and Quepos, in the central Pacific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 4) The cantons grouped in the clusters share behavior patterns regarding the periods in which the highest significant areas appear and the time series synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' (a) (b) Figure 2: Cantons where dengue cases co-varied with El Ni˜no and TNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Fig (a) represents the cantons (red) where dengue outbreaks every 3 years showed significant coherence with El Ni˜no and TNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Fig (b) corresponds to those cantons (red) where the an- nual dengue outbreaks showed significant coherence with TNA, with TNA variable leading only by short periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Data for cantons in white where no available for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 8 Characteristics of clusters that include climate variables (CV) Cluster 1: Alajuela, Orotina, Perez Zeled´on, San Jos´e, Santa Ana, Sarapiqu´ı Wavelet coherence shows an area of joint significance in the 3-yr period after 2010 for both TNA and El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 with dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Except for Puntarenas and Alajuela, the coherence with TNA is mainly in the period of 1-yr and needs to be identifiable between El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and Puntarenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The horizontal arrows in period 3 pointing to the right indicate that the two series TNA and dengue cases are in phase with vanishing phase differences from 2011 to 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and dengue cases, the time series are also in phase after 2010, with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 leading by roughly 9 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 2: Alajuelita, Ca˜nas, Esparza, La Cruz, Talamanca, Upala Dengue incidence showed significant coherence with TNA in the 1-yr period, mainly after 2010 for all cantons except in Alajuelita, where the coherence is around the 3-yr period from 2010-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The time series are in phase, but the leading time series changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In Alajuelita, Ca˜nas, and La Cruz, the TNA time series is the leading one by less than 2 months, contrary to Esparza, Talamanca, and Upala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence for El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and dengue cases show a significant coherence in the period of 3-yr in Upala and Alajuelita, with El Ni˜no time series ahead by roughly nine months after 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Small areas of significance are observed for the remaining cantons in the 1 and 2-yr periods with dengue time series ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 9 Cluster 3: Atenas, Desamparados, Liberia, Parrita, Pococi, Santa Cruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence shows an area of joint significance between TNA and dengue cases in the 1-yr period, mainly between 2012 and 2017, in all cantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, TNA is the leading time series only in Liberia and Santa Cruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' There is also an area of joint significance in the 3-yr period between TNA and dengue cases in Atenas, Desamparados, and Santa Cruz, with the time series in-phase with vanishing phase differences between 2010-2016, except in Desamparados, where TNA was leading by a couple of weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence for El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and dengue cases show an area of joint significance in the 3-yr period for Atenas, Liberia, Santa Cruz, and Desamparados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The time series are in phase after 2010 with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 leading by around nine months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For Parrita and Pococ´ı, there are no identifiable areas of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 4: Carrillo, Guacimo, Lim´on, Nicoya, Siquirres There are no identifiable common patterns in all cantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Co- herence for dengue cases with both TNA and El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 shows an area of joint significance in the 3-yr period in Carrillo and Nicoya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' TNA and dengue cases have vanished phase differences between 2008 and 2016 approximately in both cantons, and El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and dengue cases are in phase with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4, leading by a lag of 9 months from 2008-2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' TNA and dengue cases also have an area of joint significance in the 1-yr period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The time series are in-phase after 2012, with TNA leading in Carrillo and Nicoya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For the remaining cantons, the wavelet coherence shows coher- ence between El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and dengue cases in 1 and 2-yr periods, with the dengue time series ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 5 Corredores, Golfito, Osa There is an area of joint significance between TNA and dengue cases in all cantons in the 1-yr and 3-yr periods where the leading time series is dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The 3-yr dengue outbreaks were highly coherent with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 after 2007, with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 leading by a lag ranging between 0 to 9 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 10 Cluster 6 Garabito, Matina, Montes de Oro, Quepos, Turrialba There is only an area of joint significance between climate vari- ables and dengue cases in Garabito and Montes de Oro in the period of 3-yr after 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' TNA and dengue time series are in- phase with vanishing phase differences from 2011 to 2016, and El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 has an advantage of approximately 9 months over dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For the remaining cantons, small areas of signifi- cance are observed in 1 and 2-yr periods with dengue time series ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Local environmental variables and dengue cases Using coherence analysis to compare these time series in the frequency domain, we found that the annual dengue outbreaks were highly coherent with all environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In all cases, the phase difference shows an increase in dengue incidence after an increase in the local environment variables with a lag ranging between 0 to 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The association was nonstationary, with disruption from around 2007 to 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue incidence showed significant coherence with the four variables EVI, NDWI, ET, and precipitation except in Lim´on, Matina, Talamanca, Corredores, Golfito, and Osa, located in the South Pacific and South Caribbean of the country, and Parrita, Garabito, and Quepos in the central Pacific (see Fig 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence for the last cantons showed areas of high signif- icance between dengue, mainly with ET and precipitation in short periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' This finding is expected due to the regular seasonality observed on the Pacific coast and the country’s Central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The main differences observed between the clusters are the periods in which the synchronicity of the dengue cases and local environmental time series is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Figure 3: Cantons where dengue cases co-varied with the environmental vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Red cantons are those in which the annual dengue outbreaks was highly coherent with all environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 11 Characteristics of clusters that include local environmental variables (LEV) Cluster 1 Alajuela, Montes de Oro, Orotina, Parrita, Pococci, Santa Ana, Turrialba Annual dengue outbreaks showed significant coherence with the four environmental variables, except in Parrita and Pococ´ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In all cases, the leading time series were the environmental variables with 0 to 3 months ahead of dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence in Parrita shows an area of joint significance only with precipitation before 2009 with the time series in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence in Pococ´ı has a continuous area of joint significance with all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' While for the rest of the cantons, a disruption is observed between the areas of significance around 2007 to 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 2 Alajuelita, Lim´on Wavelet coherence in Alajuela shows two areas of joint signifi- cance with all variables in the 1-yr period with disruption around 2007 to 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The leading time series were the environmental variables with 0 to 3 months ahead of dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence in Lim´on shows an area of high significance only with ET in two short periods with an interruption between 2009-2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The coherence with the other variables is not conclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 3 Ca˜nas, Matina, Osa, Puntarenas Wavelet coherence in Ca˜nas and Puntarenas show a correlation with the four vegetation variables in the 1 yr period (in Ca˜nas, there is a continuous area of joint significance over time, while in Puntarenas is interrupted between 2008-2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Smaller areas of significance are observed between dengue cases in Osa and the vegetation variables, mainly after 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In all cases, environmental variables are the leading time series with a lag of fewer than 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence in Matina shows an area of high significance only with ET and EVI in 1-yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The environmental time series go ahead by less than 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 12 Cluster 4 Carrillo, Desamparados, Esparza, La Cruz, Perez Zeled´on, San Jose, Santa Cruz, Sarapiqu´ı, Siquirres, Ta- lamanca Wavelet coherence shows areas of joint significance between dengue and the four vegetation variables in the 1-yr period with an interruption around 2009-2010, except for San Jos´e and Talamanca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In all cases, the leading time series were the environmental variable by approximately 3 months ahead of dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' There is no observed coherence between ET and dengue cases in San Jos´e, and dengue cases in Talamanca showed significant coherence only with ET in a short period (around 2011-2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The area of high significance is smaller in Perez Zeled´on, Sarapiqu´ı, and Siquirres than in the other cantons in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet coherence in Talamanca shows an area of joint significance only with ET in a short period (2011-2015 approx) Cluster 5 Corredores, Golfito, Liberia, Nicoya Wavelet coherence shows areas of joint significance between dengue cases in Liberia and Nicoya with the four vegetation variables in the 1-yr period with minor interruptions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue cases in Corredores and Golfito showed significant coherence only with ET and precipitation in the 1-yr period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Local environmental variables are the leading time series in all cases by approximately 3 months ahead of dengue cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Cluster 6 Garabito, Gu´acimo Dengue time series in Gu´acimo showed significant coherence with all variables after 2009 in the 1-yr period, with the en- vironmental variables ahead of dengue cases by approximately 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Dengue time series in Garabito showed significant coherence only with ET and precipitation before 2012 in the 1-yr period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' When creating the clusters, Atenas, Upala, and Quepos stayed isolated as the unique canton in their groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Quepos is a touristic canton located in the central Pacific where the rain is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence shows an area of joint significance only between dengue cases and ET and precipitation for a short period, with the dengue time series the leading one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Upala is located in the North in a subregion where the climate is classified as rainy with monsoon influence (in essence, a tropical monsoon climate tends to have more 13 precipitation and less pronounced dry seasons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence shows a region of high joint significance between 2012 and 2015 (the worst dengue outbreak in Upala was registered in 2013) with all variables except EVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The leading time series is dengue except in the coherence with ET, where the time series were in phase with a vanish phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Atenas is located in the Central of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The wavelet coherence shows two areas of joint significance in the 1-yr period, with a disruption between 2007-2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The time series are in-phase with environmental variables leading, except between 2009-2013, when the dengue time series started to be the leading time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The lag change between 0-3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Discussion Climate variables Our results show that the 3-year dengue outbreaks were highly coherent with El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and TNA indicators, mainly in cantons located in the North, South Pacific coastal, and Center region of Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' When analyzing phase differences between the time series in these regions, synchrony between dengue cases and climatic variables was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The time series of El Ni˜no and TNA used to be ahead of dengue cases by nine and less than three months, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In some cantons, TNA and dengue cases are almost in perfect sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The synchronization of the time series began to be observed mainly after 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The periods in which areas of high significance become evident and the year in which the time series starts to be synchronized change between cantons and over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Those timing periods and the series leading are the main differences between the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Our results also showed that annual dengue outbreaks were highly coherent with TNA in cantons located in the North Pacific and the Central region of the country, with the dengue and TNA time series synchronized for short periods with a vanishing phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In some cantons, associations between climate and dengue cases were also observed in the 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='5, and 2-year periods, with dengue time series leading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, dengue cases influencing climate variables are a biologically unlikely relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' It is more plausible that changes in El Ni˜no 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='4 and TNA increase subsequent dengue transmission [59], given that the anomalies in the ocean temperatures affect local temperature and precipitation, which are directly associated with the virus ecology and transmission [8, 7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Significant coherence between climate variables and dengue cases was observed mainly in the cantons in which there are periods of drought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' There is no evidence of a contemporary statistical association between the SST anomalies in the two oceans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' But, it has been found that seasonal precipitation is dependent upon the simultaneous interaction of both anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The draught, which typifies the Pacific slope during years of warm phase ENSO, is less severe when the Caribbean is warmer, but the reduction in rainfalls may spread over the entire country [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The North Pacific belongs to the Pacific precipitation regime, known for a well-defined dry and rainy period [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The geographical characteristics in the central and South Pacific generate a climate where the dry period is very favorable and short and the rainy intense [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The years of intense droughts in those regions, except for 2001, have coincided with El Ni˜no years [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' There has been observed that during El Ni˜no, there is a greater probability that the entire Pacific slope and the Central region of 14 Costa Rica will experience dry to extremely dry conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' At the same time, there is a greater probability of extreme rainy scenarios in the Caribbean [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Environmental variables The annual dengue cycle was highly coherent with the four local environmental variables in all cantons except those located in the central Pacific, South Pacific, and South Caribbean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' For those, dengue incidence showed a significant coherence with ET and precipitation over short periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' In those regions, the rain is abundant, and there is not a well-defined dry season [60] and the time series for the vegetation indices (EVI, NDWI) does not show an annual seasonality like that observed in the North Pacific and the Central valley of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Regarding synchronization, dengue and environmental time series were in phase with the environmental time series ahead of dengue cases by approximately 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Climate and environmental variables showed significant coherence with dengue epi- demics across the geographically diverse regions of Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, spatial hetero- geneity in their effects exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Even when the association was established, it was not possible to provide those variables’ causal effects on the dengue cases’ dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Epidemic data is typically noisy, complex, and non-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Changes in the periodicity over time may also be to external factors or inherent characteristics of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Alternative explanations for the multi-annual outbreaks of dengue epidemics have been proposed, such as partial cross- immunity among the four serotypes of the dengue virus [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Moreover, sociological factors such as population structure, unplanned urbanization, and international transportation of infected people and mosquitoes may also affect dengue transmission [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' The complex interaction of biological, socioeconomic, environmental, and climatic fac- tors creates a substantial spatiotemporal heterogeneity in dengue outbreak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Un- derstanding this heterogeneity in dengue transmission could improve epidemic prediction and disease control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' There are some previous works in which predictive models have been proposed for the country using climatic and vegetation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [65] pro- posed a simple structural model incorporating lagged SST and MODIS vegetation indices, explaining 83% of the variance in the total weekly cases of DF/FHD in Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' How- ever, the cantonal-level analysis conducted in this study highlights the spatial heterogeneity of the effect of climate and environmental factors on dengue incidence, which reveals that the effect of those variables on dengue transmission on a local scale might differ from global expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Thus, for the design of interventions and resource allocation, more localized predictions may be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Other studies show that when incorporating the same climatic variables in all cantons to predict relative risk areas of dengue outbreak, the models pro- jections may fail in some places [66, 67], which reinforces the importance of understanding the local correlations between dengue cases and external factor such as climate and socioe- conomic drivers to improve local dengue predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Many studies have been conducted in tropical and subtropical regions to elucidate the complex interactions between climate variables and dengue transmission [11, 12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, this study is unique in the localized analysis that takes into account variables such as TNA, EVI, NDWI, evapotranspiration, and precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Wavelet time serires analysis allows a retrospective study to characterize outbreaks over time, which provides important guidelines for future modeling approaches in which explicit mechanisms 15 can be incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' However, as climate factors are not the only predictors influencing the rise in dengue infection, future studies are needed to include other factors unique to this area, such as the predominant circulating dengue viruses, anthropogenic factors, and herd immunity, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Acknowledgments This research was supported by a Seed Grant for International Activities from Global Affairs and the School 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Assess- ing dengue fever risk in Costa Rica by using climate variables and machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='01483 [Preprint].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 2022 [cited 24 October 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Available from: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='org/abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='01483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [67] V´asquez Brenes PA, Lor´ıa Garc´ıa A, S´anchez Pe˜na FA, Barboza Chinchilla LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Climate-driven statistical models as effective predictors of local dengue incidence in Costa Rica: A generalized additive model and random forest approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' Revista de Matem´atica: Teor´ıa y Aplicaciones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='27(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='15517/rmta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='v27i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='39931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [68] Ministerio de Salud Costa Rica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' An´alisis de la Situaci´on de Salud 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' [cited 15 November 2022] Available from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='ministeriodesalud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='cr/ index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content='php/biblioteca-de-archivos-left/documentos-ministerio-de-salud/ vigilancia-de-la-salud/analisis-de-situacion-salud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE0T4oBgHgl3EQfXAB2/content/2301.02286v1.pdf'} diff --git a/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/2301.04768v1.pdf.txt b/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/2301.04768v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ef2273a10dfa8ec457e25945a2dbf49ea5ebe86 --- /dev/null +++ b/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/2301.04768v1.pdf.txt @@ -0,0 +1,1135 @@ +Closely estimating the entropy of sparse graph models +Edward D. Lee +Complexity Science Hub Vienna, Josefstædter Strasse 39, Vienna, Austria +We introduce an algorithm for estimating the entropy of pairwise, probabilistic graph models +by leveraging bridges between social communities and an accurate entropy estimator on sparse +samples. We propose using a measure of investment from the sociological literature, Burt’s structural +constraint, as a heuristic for identifying bridges that partition a graph into conditionally independent +components. We combine this heuristic with the Nemenman-Shafee-Bialek entropy estimator to +obtain a faster and more accurate estimator. We demonstrate it on the pairwise maximum entropy, +or Ising, models of judicial voting, to improve na¨ıve entropy estimates. We use our algorithm to +estimate the partition function closely, which we then apply to the problem of model selection, +where estimating the likelihood is difficult. This serves as an improvement over existing methods +that rely on point correlation functions to test fit can be extended to other graph models with a +straightforward modification of the open-source implementation. +Estimating the entropy of a probabilistic model is a +common and essential yet difficult task in information +theoretic analysis of collective behavior. +For example, +widely used maximum entropy models of neural firing [1– +5], political voting [6], social groups [7, 8], or collective +motion [9, 10] require an estimate of the entropy of the +model for assessing the fit with the multi-information. +Furthermore, the relative decrease in the entropy from +the independent model provides a general metric for col- +lectivity, or a distance from independent statistics, that +gives a sense of the constraints imposed by model as- +sumptions. The entropy is intimately connected to the +free energy, which describes the balance between energy +constraints and disorder and thus can be used to assess +the goodness of fit. Finally, the free energy is the loga- +rithm of the partition function, the derivatives of which +reveal physical properties such as the magnetization, sus- +ceptibility, and the distance to a critical point [11, 12]. +The general problem of entropy estimation even when +given a model is hard. Statistical physics approaches in- +cluding moment expansions like the Bethe free energy +approximation [13, 14] and cluster expansions [15] pro- +vide one approach along with thermodynamic integration +[16]. An alternative approach is to infer the entropy from +a sample of the distribution, which can be relatively quick +to generate using Monte Carlo Markov chain methods. +Most interesting systems, however, are of sufficient size +that the state space dwarfs the number of possible (or +reasonable) samples from simulation, a problem that is +especially intractable in the space of discrete outcomes. +Unfortunately, we know well that estimates of the en- +tropy in the undersampled limit are biased. Even worse, +starting with a prior in the space of probability distri- +butions in order to infer a good estimate of entropy can +backfire, introducing other biases that cannot be over- +come in the low data limit. A remarkably accurate esti- +mator, called NSB after the authors Nemenman-Shafee- +Bialek [17], instead attempts to be as unbiased in the +estimate by using a prior that is flat in the space of en- +tropies and as a result can perform well in the highly +undersampled limit. Like any other estimator, however, +it will still face problems in larger systems because of the +prior inevitably dominates and (in practical terms) the +computation is slower and numerically challenging. +Here, we are interested in the problem of estimating +the entropy when already given a model and leveraging +entropy estimators from small samples in the literature +like NSB such as when characterizing the statistics of +natural phenomena [18]. In short, we address a techni- +cal problem in the numerical estimation of the entropy +of a probabilistic model leveraging the fact that many +problems show sparse structure. In the context of physi- +cal models, this is the observation that components can +be split into communities that are sparsely connected to +one another. This the case for many political interac- +tion graphs such as for judicial courts or for legislatures, +where the vast majority of occupants over time have +never met or interacted with one another [19]. This is +also the case for some social graphs which tend to be +split into cliques that are connected only indirectly to +one another in small-world networks [20–22], but it is not +the case for small animal groups that display lattice-like, +topologically local interactions such as in bird flocks and +local interaction models for herding behavior [23]. +As +we describe below, we develop a heuristic that leverages +existing estimators and package it into a Python library +TreeEnt that can estimate faster and more accurately the +entropy of sparsely connected graph models in a discrete +state space. +ENTROPY & ITS ESTIMATION IN A NUTSHELL +In the mid 20th century, Claude Shannon sought a way +to characterize the statistical structure of signals sent +through AT&T’s telecommunication networks [24]. By +treating a message as a sequence of discrete characters +s each occurring with a probability p(s), he established +the “information entropy” measuring the surprise that +each new character in the received sequence would entail. +arXiv:2301.04768v1 [physics.comp-ph] 12 Jan 2023 + +2 +He furthermore showed that this was a unique measure +(barring a choice of units) adhering to three axioms [25]: +continuity with respect to p, maximization at maximal +uncertainty, and consistency under a hierarchical decom- +position of the symbols into sets. Formally, the informa- +tion entropy H of a probability distribution p(s) for a +configuration s from a discrete-state space S is [26] +H[p] ≡ − +� +s∈S +p(s) log p(s). +(1) +The entropy is maximized when the distribution over all +configurations s is uniform (and thus completely unpre- +dictable) and zero when only a single configuration oc- +curs with probability one (completely predictable). Thus, +the entropy presents a unique measure of the amount of +structure in the distribution and places limits on its sta- +tistical predictability. +Estimation of the entropy given a random sample from +p(s) is difficult because the state space is computationally +expensive, if not impossible, to sample comprehensively +even for systems of moderate size. If we, for example, try +estimating the entropy in the most na¨ıve way; that is, to +rely on the number of times ks that we see state s from +a random, finite sample of size K, we would posit that +ˆp(s) = ks/K. Then, the deviation from the true value +can be represented as an error term, p(s) = ˆp(s) + ϵ(s). +Expanding the entropy in terms of the errors, we find +ˆH[p] = H[p] − A +K − O +� 1 +K2 +� +, +(2) +where we have grouped into the last term all terms of +order (1/K)2 and higher and A is a positive constant. +Thus, it is the case that the na¨ıve estimator will return +a biased estimate that underpredicts the true entropy of +the distribution. +One way to ameliorate this problem may be a Bayesian +approach in which we define entropy estimation as an +inference task over a prior distribution of the probability +distributions from which the sample originates. To be +pedantic, a classic way to do this is to write a Dirichlet +family of models where we specify the probabilities qi +with which each unique state occurs out of the possible +set of S states (e.g. for a binary spin system of size N, +we would have S = 2N). Accounting for normalization +of the set of probabilities {qi}, we have +Pβ({qi}) = +1 +Z(β)δ +� +1 − +S +� +i=1 +qi +� S +� +i=1 +qβ−1 +i +. +(3) +This is known as the Dirichlet family of priors where β, +which determines how likely we consider larger proba- +bilities a priori; Laplace’s counting rule corresponds to +β = 1 and the maximum likelihood estimator β = 0. The +factor Z ensures that this is a normalized probability dis- +tribution. Given the prior in Eq 3, the distribution over +FIG. 1. Example of graph partition. (a) Graph before par- +tition. Edges denote interactions between nodes. We index +with increasing index i each generation of the tree away from +the root at i = 0. Generations also differentiated by shading. +When each branch is indexed by the alphabet, we specify the +set of nodes by the generation and branch index, e.g. sa +i . (b) +Bipartite graph from our partition algorithm. Downstream +nodes (e.g. b relative to a) are conditionally independent once +fixing the values of upstream nodes. This permits us to re- +duce the sample state space exponentially because the largest +subgraph determines the largest state space to sample. Graph +partitioned by setting maximum cluster size to K = 4. +the estimated entropy H will be given by Bayes’ theorem, +Pβ(H) = Pβ(H|{qi})Pβ({qi}) +Pβ({qi}|H) +. +(4) +Then, the averaged estimate of the entropy given β is +ξ(β) ≡ +� ∞ +0 +Pβ(H)H dH. +(5) +Crucially, the choice of β plays a crucial role in determin- +ing the final estimate of then entropy because the prior +is peaked and thus essentially determines the entropy in +the low data sample size limit [17]. +Instead, we might consider a meta-prior that consider +a mixture of all possible priors within this family, or the +range of possible β in a way that flattens our starting as- +sumptions about the entropy. This means that we should +move β over some weighted range P(β) in such a way that +corresponds to moving ξ(β) between 0 and log H. Then, +a candidate mixture prior proposed in reference 17 is +P({qi}; β) = +1 +Z(β)δ +� +1 − +S +� +i=1 +qi +� S +� +i=1 +qβ−1 +i +dξ(β) +dβ +P(β). +(6) +To perform this integral, one must also know how the +estimated entropy varies as a function of ξ in order to +flatten the prior in the estimated range, the relationship +between which is given in reference 17. As it turns out, +Eq 6 performs surprisingly well for entropy estimates on + +e +i=2 +a +a +d +d3 +tiny samples. Regardless, it cannot overcome the funda- +mental problem that entropy estimates are dominated by +the prior in any undersampled system. +One way out of this pickle is to leverage the structure +of the model’s probability distribution to simplify the +entropy estimation problem by effectively reducing the +state space over which one’s estimator needs to work. +Conveniently, Shannon’s last axiom tells us that if we +can group the states into independent subsets, then the +information entropy is the sum of both the uncertainty +of the labels from the clusters as well as the contribution +from within each subset given the labels. When a clever +decomposition is possible, it would allow us to consider +subsets of the system independently of one another. This, +for example, is possible if some components of s with in- +dex i, the set {si} of which we denote si for simplicity, +are conditionally independent of components indexed by +j once holding fixed components k. In other words, this +is the assertion that we can factorize the probability dis- +tribution +p(s) = p(si|sk)p(sj|sk)p(sk). +(7) +If this is the case, then the entropy of this set decomposes +in the summation of the information entropies +H[p] = H[p(sk)] − +� +sk +p(sk) (H[p(si|sk)]+ +H[p(sj|sk)]) . +(8) +We can think of each unique configuration of sk as a “la- +bel,” and once this is given we must compute the uncer- +tainty of the sets si and sj with their respective weights +given by the frequency of the sk on which they have been +conditioned. +When we generalize this basic example to a tree, we +denote a set again as si but now denote moving up gen- +eration in the tree as the index i − 1 and moving down a +generation as i+1. Labeling each branch with a different +letter of the alphabet as superscript, +p(s) = p(s0) +N +� +i=1 +� +a,a′ +p +� +sa′ +i +���sa +i−1 +� +, +(9) +where we again use the shorthand notation si to refer to +the set of components over index i. Then, the root is the +set of nodes s0, the first product is over each successive +layer in the tree indexed i down to the leaves in the Nth +generation, and the second product over the descendents +a′ of each branch a in the (i − 1)th layer, which is inde- +pendent once having conditioned on all parent branches. +As an example of such a factorization using this nota- +tion, we show a graph in Figure 1, where we indicate +the successive generation of the tree i generations away +from the root that are conditionally independent of one +another once conditioning on the parent branches i − 1. +When the probability distribution is factorizable in this +way, then it is possible to compute the entropy of the +entire system as a sum of entropies from the outside in +by computing the entropy of each leaf, which provides +an additive contribution to the final entropy. We express +this in the recursive form +H[p] = H[p(s0)] + +� +s0 +� +a +p(s0) +� +H[p(sa +1|s0)] + +� +sa +1 +� +a′ +p(sa +1|s0) +� +H[p(sa′ +2 |sa +1)] + · · · + +� +sa(n−1) +i +� +a(n) +p +� +sa(n) +i +|sa(n−1) +i−1 +� � +H +� +p +� +sa(n+1) +i+1 +|sa(n) +i +�� ++ · · · +��� +. +(10) +In short, we can implement the calculation in Eq 10 from +the leaves up to the root. We start with the conditional +entropy of a leaf, prune the leaf, upon which the branch +leading to the leaf in turn becomes a leaf. Continuing +in this recursive way, we eventually reach the root of the +graph. +Such a hierarchical decomposition presents a way to +ameliorate the problem of entropy estimation if we can +group components into subsets that are substantially +smaller than the full graph, which shrinks the state space +exponentially [27]. +If it is the case that the groups +are substantially smaller than the full graph, estimat- +ing the entropy in principles becomes more manageable +with an exponentially smaller Monte Carlo Markov chain +(MCMC) sample. +We present an algorithm that esti- +mates the entropy of probabilistic graph models that fac- +torizes the graph using a heuristic based on structural +holes and fast MCMC sampling of the conditioned prob- +ability distributions. +ALGORITHM +We provide an outline of algorithm in Table I and pro- +vide more details below. +The first step is to compute a factorization of the graph + +4 +that allows us to split the problem into more manage- +able entropy estimation problems. To do so, we rely on +the notion of structural holes from the sociological liter- +ature [20, 28]. Structural holes are inspired by a problem +of constrained action in a social network, where the as- +sumption is that the amount that any individual u is con- +strained by a neighbor v depends directly on two factors: +the relative investment that u dedicates in its the rela- +tionship with v and the simultaneous relative investment +of another neighbor w of u into v. The intuition under- +lying the latter step is that when investments overlap, +u is maximally constrained in terms of leverage because +u is redundantly influencing the local neighborhood of v +(they are competing for influence in the same sphere), +whereas with minimal overlap, v acts less as a constraint +and more as a bridge to other parts of the network to +which u does not have access. In other words, a neighbor +v with low structural constraint with respect to u is sur- +rounded by “structural holes” such as when it belongs to +a different clique. +For our purposes, the intuition is that the nodes with +low constraint are ones that are placed in between densely +connected communities; they should present a small set +that we can fix to render adjacent communities indepen- +dent of one another. More specifically, these nodes are +surrounded by holes and thus have small structural con- +straint cu, defined as the sum over local constraints l for +node u, +cu := +� +v∈N (u) +l(u, v), +(11) +which is the sum over all neighbors v of u, or N(u). The +local constraint given uniform edge weights is defined as +l(u, v) := +� +� +1 +|N(u)| + +� +w∈N (u)∩N (v) +1 +|N(u)| +1 +|N(w)| +� +� +2 +, +(12) +where the number of neighbors of node u is |N(u)|. The +first term in the parentheses accounts for u’s investment +in this neighbor v, which is uniform across all neighbors. +The second term accounts for the joint investment in +neighbor v from both u and a common neighbor w. Note +that Eq 12 is a simplified form for our scenario, where “in- +vestment,” or the weights, between pairs of nodes does +not generally need to be uniform [29]. +We start by removing nodes from the graph starting +with the node of minimal constraint and recomputing +the local constraints upon every removal step. Every re- +moved node is placed into the set X′. This will tend to +split apart connected components in the set of remaining +nodes X along the bridges that connected them. If we +continue indefinitely with this procedure, the subgraphs +living in the set of removed nodes X′ will eventually con- +TABLE I. Algorithm for entropy estimation. +In the case +the starting graph consists of multiple components, they are +treated separately. +1. Factorize graph to generate “contracted” tree. +a. Place all nodes into X. +b. +Calculate the structual constraint for every +node in X. +c. Remove from set X and place into X′ the node +with minimal constraint in X. +d. If the largest connected component in either X +or X′ is now larger than largest component +previous to step c or if the largest connected +component in X is smaller than or equal to +threshold M, then jump to step f. +e. Return to step b. +f. +Create a coarse-grained representation of the +graph, where a connected component in X +and X′ are connected to one another if there +is at least one interaction between the two +components. +2. Estimate conditional entropies. +a. Identify leaves in coarse-grained graph. +b. For each leaf, generate a Monte Carlo sample +of size K. +c. For each unique state in the sample of size K, +generate from the leaf a Monte Carlo sample +of size K′. +d. Estimate the entropy of the leaf and the condi- +tioned branch using NSB. +e. Prune the coarse-grained graph by removing all +leaves. +f. Return to first step. +3. Sum entropies and calculate errors. +a. Sum over the estimated entropies of each leaf +and its branch. +sist of most of the graph, and they may form large compo- +nents that have not simplified the problem at all. Thus, +we keep removing nodes with minimal constraint as long +as the largest components in the set of removed nodes as +well as the pruned graph are shrinking or until they have +all fallen below a size smaller than the specified threshold +M. +As a result of this procedure, we have two sets of nodes +X and X′. Within each set, we have disconnected com- +ponents labeled x and x′, respectively, that bridge nodes +in the other but are not directly connected, or a bipar- +tite structure as we show in Figure 1b. Thus, the graph +presents a structure in which fixing the components in x +renders components x′ independent of one another and +vice versa. +As the final step, we calculate the entropy of the re- +sulting factorized tree. We do so identifying a leaves, or +components in either X or X′ that are conditionally in- +dependent when holding a single component in the com- + +5 +2 +1 +0 +1 +2 +coupling J +10 +15 +20 +25 +30 +entropy estimate (bits) +TreeEnt K = Kcond = 103 +naive K = 104 +NSB K = 104 +exact +FIG. 2. Comparison of TreeEnt with na¨ıve and NSB entropy +estimators on five replicas of an “pointy” triangle (inset on top +left). Our algorithm does better faster and with fewer samples +by recognizing the interaction structure of the model. +plementary set fixed. While we cannot guarantee that a +leaf can be found at the end of our structural constraint +removal procedure, it is the case that removing nodes of +minimal constraint will tend to fragment the graph into a +chain of conditionally independent pieces. Furthermore, +there always exists a partition of the graph such that a +leaf node can be designated (in the trivial case a single +node can be put into X′). +Finally, we then sample from the set of conditionally +independent “leaves” and the unique set of nodes that +lead to them, or their “branch.” First, we generate an +MCMC sample of the distribution of the a’th branch sa +i−1 +from a sample of the entire system. +Then, we iterate +through the unique states and sample for the a′’th leaf sa′ +i +given each of the possible states of the branch. The size +of the sample set also allows us to estimate the standard +deviation of the sampled entropy. This gives us the terms +in Eq 10. Then, we prune the leaf from the graph and +iterate this process recursively until we only have the root +of the tree. At each step, we calculate the entropy using +the NSB estimator. The total entropy of the tree is the +summation of the calculated entropies. +ERROR ESTIMATION +For the algorithm, we must account for two sources of +error including from the finite sample size of the condi- +tioned set and the NSB estimator. +For the finite sample contribution, the contribution to +the entropy for a single term (the entropy of a leaf) in +Eq 10 is the combination of terms +� +H[p(sa′′ +i+1)] +� += +� +sa′ +i +p(sa′ +i )H[p(sa′′ +i+1|sa′ +i )] +(13) +where the sum is over all the unique states that occur +for the branch sa′ +i and for the particular leaf a′′. For this +calculation, we are not considering the sum over all the +parents of the branch sa′ +i because we can MCMC sample +directly from the subset. This means that we can also +calculate the variance in the entropy of this leaf, +σ2 +sa′′ +i+1 = +� +sa′ +i +p(sa′ +i ) +� +H[p(sa′′ +i+1|sa′ +i )]− +H[p(sa′′ +i+1|sa′ +i )] +�2 +. +(14) +As in the usual sense, we normalize the variance by the +number of samples K to estimate the standard error of +the mean. Since the NSB error σNSB(sa′ +i )2 on each term +in the sums of Eqs 13 and 14 are computed independently +of the finite-sample error, we add the norms of both the +errors together to determine the total error on the en- +tropy estimate. For a single leaf, we have the sum of the +variance, or the norm-squared error +Σ2 +sa′′ +i+1 = σ2 +sa′′ +i+1 +� +K + +� +sa′ +i +p(sa′ +i )σNSB(sa′ +i )2. +(15) +ASSESSING PERFORMANCE +We compare our algorithm with the na¨ıve and NSB es- +timator in an example where the entropy can be exactly +calculated by enumeration of all configurations. We find +that leveraging the factorized structure of the graph al- +lows us to recover a nearly perfect estimate to the exact +entropy. As an example, we show the results for an en- +semble consisting of multiple, disconnected graphs, each +one consisting of a triangle with a single additional node +coming off of each vertex (thus a “pointy” triangle) for +a total of six spins. +The spins are coupled with uni- +form strength J that we vary. When we sample from five +replicas of the graph at the same time, the state space of +K = 230 ∼ 109 is difficult to sample well on a desktop +machine. Our estimator, as we show in Figure 2, per- +forms well across different scales of the coupling J even +with a relatively small number of samples. It is also sub- +stantially faster than applying the NSB estimator to the +entire ensemble at once because the subsets on which we +calculate the estimator are smaller. +APPLICATION TO MAXENT VOTING MODEL +We use our algorithm to estimate the entropy of a +sparse voting model of judge voting on the US Circuit +courts. The probabilistic models that we consider derive +from the maximum entropy principle, which is an algo- +rithm for determining minimal statistical models of data +[30]. Importantly, the quantity of interest in this frame- +work is the entropy of the model, which is maximized + +6 +FIG. 3. Examples of graph partitions using the minimal constraint heuristic applied to sparse US Appeals Courts interaction +graphs from reference . (a) DC Circuit Court separates into four connected components when max cluster size is set to M = 14. +Largest component is of size N = 19 compared to N = 47 for the entire graph. Each color is a different subgraph. Inset shows +interaction structure on coarse-grained graph which is a tree, i.e. the green, red, and orange clusters are independent of one +another once fixing the blue cluster. (b) First Circuit is partitioned into sets that form a line. Largest component is of size +N = 13 vs. total graph size of N = 23. +while constraining a few, crucial properties of the system +such as the average vote of each judge and their pairwise +correlations [31]. When these two sets of constraints are +imposed, we obtain the pairwise maxent model, which +has been shown to model to high accuracy voting pat- +terns on the US Supreme Court [6, 19]. +According to the model, the probability distribution +takes the Boltzmann form +p(s) = 1 +Z e−E(s) +(16) +with normalization term Z known as the “partition func- +tion” in statistical physics. To each configuration s de- +fined as a vector of −1 and 1 for the votes of the judges, +an “energy” E(s) is assigned, where lower energy implies +that the configuration is more likely. In the case of the +pairwise maxent model, it takes the form +E(s) = − +N +� +i=1 +hisi − +N +� +i 0 will tend to take on value of 1. The couplings Jij +describe the tendencies of components to align with each +other. Similar to the bias, a positive coupling lowers the +energy when two components are aligned and a negative +coupling lowers the energy when the components are mis- +aligned. Thus, the pairwise maxent model captures both +an independent tendency to be biased in one direction or +FIG. 4. Probability of state s using our entropy estimate for +the partition function vs. from Monte Carlo Markov chain +sampling for the DC Circuit. We show points that appeared +at least twice in the sample. As in Eq 18, we must also es- +timate the average energy ⟨E⟩, which is straightforward to +obtain with high precision using an MCMC sample unless the +distribution of the energy displays a heavy tail such as near +a critical point—but this will also pose a problem for the en- +tropy estimate. MCMC sample size of K = 106. +another and a pairwise interaction tendency for how one +component interacts with every other. +Importantly, the couplings describe a statistical inter- +action network akin to the edges displayed in Figure 1. +This is visible from looking at the factorization of the +probability distribution described in Eqs 16 and 17. It +is clear from the additivity of the energy function that a + +10-2 +10-5 +TreeEnt p(s) +10-8 +10-11 +10-17 +10-5 +10-4 +10-3 +MC sample p(s)(a) +(b) +15 +17 +37 +28 +30 +(12 +26 +16 +34 +35 +21 +38 +5 +37 +16 +27 +1929 +42 +4436 +20 +6 +11 +18 +8 +45 +41 +2 +7 +9 +46 +25 +12 +45 +14 +22 +0 +31 +39 +137 +FIG. 5. Probability p(s) of state s using our entropy estimate +for the partition function vs. from MCMC sampling for the +First Circuit. See Figure 4 for more details. MCMC sample +size of K = 105. +separate term in the product for p(s) appears for every +pair of voters i and j related by Jij ̸= 0. Thus, we only +need use the connectivity of the matrix of couplings to +break the graph into conditionally independent compo- +nents. +In the US Court of Appeals, the corresponding pair- +wise maxent model is sparse because the number of +judges sitting on a court is capped at any given time. +This means that many judges in the past have had no +interaction with judges in the future and therefore there +is no coupling between them (more details see the sup- +plementary information in reference 19). As we show in +Figure 3a, our algorithm factorizes the interaction graph +for the District of Columbia (DC) circuit into four com- +ponents: a simple tree, where the green, red, and or- +ange sets are rendered conditionally independent given +the blue cluster. While the na¨ıve approach would have +required estimating in a state space corresponding to +N = 47 voters, or of size 247 ∼ 1015, the largest clus- +ter after our partition is of N = 19, or a state space of +about 106, which is feasible to sample well numerically +and for which we expect the NSB estimator to work well +with many fewer samples than the full state space. In +Figure 3b, we show the interaction graph of the First +Circuit, where N = 23, and we again find a partition +into three clusters, two of which in green and orange +are conditionally independent of one another once fix- +ing the center blue cluster. Thus, TreeEnt provides an +accelerated and more accurate method for estimating the +entropies of these voting models. +For the best fit models, our estimator finds an en- +tropy of SDC = 17.84 ± 0.05 bits for the DC Circuit and +S1 = 11.34 ± 0.03 bits for the First Circuit once hav- +ing set branch sample size K = 104 and leaf sample size +K′ = 103. Since the entropy estimates represent substan- +tial decreases in the entropy from the independent model +of SDC = 47 bits and S1 = 23 bits, they show that the +voting behavior of the judges is consistent with strong +correlations. In comparison, the pairwise maxent model +of the US Supreme Court from 1994–2005 is about 5 bits +out of the possible maximum of 9 bits. In terms of the en- +tropy per voter, the appeals courts are more constrained +than the US Supreme Court, which is sensible because +they are generally obligated to follow precedents set by +the latter. +The entropy is intimately related to the partition func- +tion, allowing us to estimate the partition function from +the entropy in a reversal of the usual procedure. Starting +with the logarithm of Eq 16, +log p(s) = −E(s) − log Z, +− log Z = ⟨E⟩ − H[p], +(18) +where in the last step we took the average of both sides +over the distribution p(s). This leads to the Helmholtz +free energy relation in Eq 18. Thus, we can use our esti- +mate for the entropy to calculate the partition function, +which is an important quantity that, besides determining +the normalization, is directly involved in calculations of +physical properties of the model. As we show in Figures 4 +and 5, we obtain excellent agreement with the probabili- +ties of states p(s) estimated from MCMC sampling using +this estimate of the partition function to normalize the +probabilities. +In the case of the solution landscape for the Court of +Appeals, the solution landscape is degenerate because of +the set of consistency conditions used to impute miss- +ing votes [19]. As a result, we require a way of choosing +amongst the multiple solutions that we recover. +Each +solution has the same interaction structure, which is de- +termined by the pairs of judges we have observed voting +together or not voting together, but the particular values +of the fields hi or couplings Jij will change. As a standard +information theoretic measure of the goodness-of-fit, we +rely on the KL divergence between the distribution of the +data and the model, +DKL = +� +s +pdata(s) log +�pdata(s) +p(s) +� +. +(19) +This reduces to a “pseudo likelihood” after ignoring the +entropy of the data, which is a constant that does not +matter for minimizing the divergence, or +˜DKL ∼ ⟨E⟩pdata + log Z. +(20) +We call Eq 20 a pseudo likelihood because it leads to +the same relative outcome as when maximizing the likeli- +hood. Note that we have inserted the form for the maxent +model for p(s). Then, the first term is the average energy +weighted by the data distribution (once marginalized over +any unobserved voters) and the second the free energy as +given in Eq 18. As an alternative measure for goodness of + +10-2. +10-4 +TreeEnt p(s) +10-6 +10-8 +10-10 +10-5 +10-4 +10-3 +10-2 +MC sample p(s)8 +18 +19 +20 +21 +pseudo l.h. +free energy +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +solution index +10.8 +11.0 +11.2 +11.4 +52 +50 +26.0 +25.8 +pseudo likelihood +free energy +FIG. 6. Goodness of fit measured by pseudo likelihood (blue +and defined in Eq 20) and free energy (orange) for ten differ- +ent, degenerate solutions to the DC and First Circuits. The +collective measure reveals some of the solutions to be superior +to others along both counts. Errors from entropy estimation +as described in the main text and standard errors of the mean +from energy estimates are summed by norm (the squared er- +rors are summed and then a square root taken). +fit, we can consider the free energy alone because it bal- +ances the energy (which more constrained distributions +minimize) and the entropy (which more random distri- +butions and thus of higher multiplicity maximize). As +we show in Figure 6, these measure distinguish certain +solutions, and the ones of best fit agree with qualitative +checks on the ranked order of judges according to the +imputed correlation matrix [32]. +Importantly, the likelihood is difficult to compute, and +as a result a typical way to assess the fit of maxent mod- +els is to use correlations that have not been explicitly +fed into the model. For example, the Ising model should +reproduce exactly the mean vote and the pairwise corre- +lations, but there is no guarantee that it can reproduce +higher-order correlations. These statistics are often used +for quality of fit in models of neural statistics and voting +[1, 6]. In voting data sets, relying on such checks poses +serious problem when votes are missing such as when +only subsets of individuals vote together (e.g. a standard +bench is only of three judges in the appeals courts) and +checking correlations is not possible. Here, we show that +it is feasible to do an accurate and better comparison +between models using our heuristic. +DISCUSSION +We propose a heuristic algorithm for calculating an +essential collective property of graph models of social be- +havior, the information entropy. +In contrast with the +usual estimates of collective properties and measures of +goodness-of-fit with prediction of lower-order statistics +[1, 6], we show that leveraging sparse graph structure +along with good estimators of the entropy can allow us +compute collective statistics that implicitly incorporate +correlations of all orders. As examples of our algorithm, +we present two judicial voting models for the US Court +of Appeals. We find that we can obtain precise estimates +of the entropy that we then use to identify an optimal +solution from an ensemble of degenerate solutions. From +our calculation, we also discover that the relative entropy +per judge is higher than that of the US Supreme Court, a +useful observation step for further work on comparison of +institutional properties and constraints from behavioral +data. +For the problems that we consider, the sociological +measure for detecting bridges, the structural contraint +work well, but our work could be extended by consider- +ing alternative algorithms for factorizing the graph. In +experimenting, we found that other common measures +such as between centrality or Louvain community clus- +tering were not as good at factorizing the graphs for our +recursive entropy calculation. One potential fruitful di- +rection would be to consider structural holes of higher +order. Furthermore, incorporating effective resistance as +edge weights for the structural constraint did not improve +matters, although surely the success of an algorithm will +depend on the properties of the graph. As a step towards +future work, our current work provides a useful step in +assessing both collective properties and model fit in the +context of graph models for social interaction. +ARCHITECTURE +An open-source package for our algorithm TreeEnt will +be available on https://github.com/eltrompetero/ +treeEnt, where TreeEnt is shorthand for Tree Entropy +(as well as a reference to Lord of the Rings, where mobile +trees are known as Ents). +The core of the package consists of two Python mod- +ules contained in “measures.py,” “test measures.py,” and +“NSB toolbox.py”. As the names indicate, the algorithm +described in the main text is implemented in the mea- +sures module as part of the TreeEntropy class. The test- +ing module is contained in the test module that provides +some automated tests that can be run with pytest as +well as routines for checking the validity of the algo- +rithm as is implemented in the accompanying Jupyter +notebook. +The final module contains an implementa- +tion of the NSB estimator, which is heavily borrowed +from an existing codebase written by Bryan Daniels at +https://github.com/bcdaniels/toolbox.git. +The TreeEnt class takes an model instance that de- +scribes the probabilistic graph model. This instance must +have routines for sampling from the probabilistic model + +9 +of interest. In the current implementation, we base this +class on the Ising model class implemented in the ConIII +coding project. Running the computation with ConIII +entails a much larger number of dependencies than those +explicitly identified as part of TreeEnt. +ACKNOWLEDGEMENTS +EDL acknowledges funding from the Austrian Science +Fund under grant number ESP 127-N. 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John Wiley & Sons, Hoboken, second +edition, 2006. +[27] In a similar sense, one can also show that the complexity +of the partition function for the Ising model on a square +lattice does not go as 2N but rather as 2L, where L is +the length of one side of the lattice, because one can +condition on a line of spins that cuts the problem in half +and do this recursively. +[28] Ronald S. Burt. Structural Holes and Good Ideas. Amer- +ican Journal of Sociology, 110(2):349–399, September +2004. +[29] Burt proposes another measure for structural constraint, +where instead the last term in Eq 12 goes with the weight +that w puts on v normalized by the maximum edge weight +for w. Since the weights are uniform here, this would +mean that the term becomes unity [20]. +[30] E. T. Jaynes. +Information Theory and Statistical Me- +chanics. Phys. Rev., 106(4):620–630, May 1957. +[31] Edward D. Lee and Bryan C. Daniels. Convenient Inter- +face to Inverse Ising (ConIII): A Python 3 Package for +Solving Ising-Type Maximum Entropy Models. JORS, + +10 +7(1):3, March 2019. +[32] We use the principal dimension of the imputed correla- +tion matrix to define a relative ranking of judges, and the +ranking is consistent with qualitative checks with a legal +scholar on a conservative-liberal dimension. + diff --git a/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/load_file.txt b/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08b21e464eb81ff78c458ab5941f0b89621bd8bb --- /dev/null +++ b/g9E3T4oBgHgl3EQf4AtG/content/tmp_files/load_file.txt @@ -0,0 +1,472 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf,len=471 +page_content='Closely estimating the entropy of sparse graph models Edward D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Lee Complexity Science Hub Vienna, Josefstædter Strasse 39, Vienna, Austria We introduce an algorithm for estimating the entropy of pairwise, probabilistic graph models by leveraging bridges between social communities and an accurate entropy estimator on sparse samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We propose using a measure of investment from the sociological literature, Burt’s structural constraint, as a heuristic for identifying bridges that partition a graph into conditionally independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We combine this heuristic with the Nemenman-Shafee-Bialek entropy estimator to obtain a faster and more accurate estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We demonstrate it on the pairwise maximum entropy, or Ising, models of judicial voting, to improve na¨ıve entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We use our algorithm to estimate the partition function closely, which we then apply to the problem of model selection, where estimating the likelihood is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This serves as an improvement over existing methods that rely on point correlation functions to test fit can be extended to other graph models with a straightforward modification of the open-source implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Estimating the entropy of a probabilistic model is a common and essential yet difficult task in information theoretic analysis of collective behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For example, widely used maximum entropy models of neural firing [1– 5], political voting [6], social groups [7, 8], or collective motion [9, 10] require an estimate of the entropy of the model for assessing the fit with the multi-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Furthermore, the relative decrease in the entropy from the independent model provides a general metric for col- lectivity, or a distance from independent statistics, that gives a sense of the constraints imposed by model as- sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The entropy is intimately connected to the free energy, which describes the balance between energy constraints and disorder and thus can be used to assess the goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Finally, the free energy is the loga- rithm of the partition function, the derivatives of which reveal physical properties such as the magnetization, sus- ceptibility, and the distance to a critical point [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The general problem of entropy estimation even when given a model is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Statistical physics approaches in- cluding moment expansions like the Bethe free energy approximation [13, 14] and cluster expansions [15] pro- vide one approach along with thermodynamic integration [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' An alternative approach is to infer the entropy from a sample of the distribution, which can be relatively quick to generate using Monte Carlo Markov chain methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Most interesting systems, however, are of sufficient size that the state space dwarfs the number of possible (or reasonable) samples from simulation, a problem that is especially intractable in the space of discrete outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Unfortunately, we know well that estimates of the en- tropy in the undersampled limit are biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Even worse, starting with a prior in the space of probability distri- butions in order to infer a good estimate of entropy can backfire, introducing other biases that cannot be over- come in the low data limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' A remarkably accurate esti- mator, called NSB after the authors Nemenman-Shafee- Bialek [17], instead attempts to be as unbiased in the estimate by using a prior that is flat in the space of en- tropies and as a result can perform well in the highly undersampled limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Like any other estimator, however, it will still face problems in larger systems because of the prior inevitably dominates and (in practical terms) the computation is slower and numerically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Here, we are interested in the problem of estimating the entropy when already given a model and leveraging entropy estimators from small samples in the literature like NSB such as when characterizing the statistics of natural phenomena [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In short, we address a techni- cal problem in the numerical estimation of the entropy of a probabilistic model leveraging the fact that many problems show sparse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In the context of physi- cal models, this is the observation that components can be split into communities that are sparsely connected to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This the case for many political interac- tion graphs such as for judicial courts or for legislatures, where the vast majority of occupants over time have never met or interacted with one another [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This is also the case for some social graphs which tend to be split into cliques that are connected only indirectly to one another in small-world networks [20–22], but it is not the case for small animal groups that display lattice-like, topologically local interactions such as in bird flocks and local interaction models for herding behavior [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As we describe below, we develop a heuristic that leverages existing estimators and package it into a Python library TreeEnt that can estimate faster and more accurately the entropy of sparsely connected graph models in a discrete state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' ENTROPY & ITS ESTIMATION IN A NUTSHELL In the mid 20th century, Claude Shannon sought a way to characterize the statistical structure of signals sent through AT&T’s telecommunication networks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' By treating a message as a sequence of discrete characters s each occurring with a probability p(s), he established the “information entropy” measuring the surprise that each new character in the received sequence would entail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='04768v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='comp-ph] 12 Jan 2023 2 He furthermore showed that this was a unique measure (barring a choice of units) adhering to three axioms [25]: continuity with respect to p, maximization at maximal uncertainty, and consistency under a hierarchical decom- position of the symbols into sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Formally, the informa- tion entropy H of a probability distribution p(s) for a configuration s from a discrete-state space S is [26] H[p] ≡ − � s∈S p(s) log p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (1) The entropy is maximized when the distribution over all configurations s is uniform (and thus completely unpre- dictable) and zero when only a single configuration oc- curs with probability one (completely predictable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Thus, the entropy presents a unique measure of the amount of structure in the distribution and places limits on its sta- tistical predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Estimation of the entropy given a random sample from p(s) is difficult because the state space is computationally expensive, if not impossible, to sample comprehensively even for systems of moderate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' If we, for example, try estimating the entropy in the most na¨ıve way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' that is, to rely on the number of times ks that we see state s from a random, finite sample of size K, we would posit that ˆp(s) = ks/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Then, the deviation from the true value can be represented as an error term, p(s) = ˆp(s) + ϵ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Expanding the entropy in terms of the errors, we find ˆH[p] = H[p] − A K − O � 1 K2 � , (2) where we have grouped into the last term all terms of order (1/K)2 and higher and A is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Thus, it is the case that the na¨ıve estimator will return a biased estimate that underpredicts the true entropy of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' One way to ameliorate this problem may be a Bayesian approach in which we define entropy estimation as an inference task over a prior distribution of the probability distributions from which the sample originates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' To be pedantic, a classic way to do this is to write a Dirichlet family of models where we specify the probabilities qi with which each unique state occurs out of the possible set of S states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' for a binary spin system of size N, we would have S = 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Accounting for normalization of the set of probabilities {qi}, we have Pβ({qi}) = 1 Z(β)δ � 1 − S � i=1 qi � S � i=1 qβ−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (3) This is known as the Dirichlet family of priors where β, which determines how likely we consider larger proba- bilities a priori;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Laplace’s counting rule corresponds to β = 1 and the maximum likelihood estimator β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The factor Z ensures that this is a normalized probability dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Given the prior in Eq 3, the distribution over FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Example of graph partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (a) Graph before par- tition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Edges denote interactions between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We index with increasing index i each generation of the tree away from the root at i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Generations also differentiated by shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When each branch is indexed by the alphabet, we specify the set of nodes by the generation and branch index, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' sa i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (b) Bipartite graph from our partition algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Downstream nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' b relative to a) are conditionally independent once fixing the values of upstream nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This permits us to re- duce the sample state space exponentially because the largest subgraph determines the largest state space to sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Graph partitioned by setting maximum cluster size to K = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' the estimated entropy H will be given by Bayes’ theorem, Pβ(H) = Pβ(H|{qi})Pβ({qi}) Pβ({qi}|H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (4) Then, the averaged estimate of the entropy given β is ξ(β) ≡ � ∞ 0 Pβ(H)H dH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (5) Crucially, the choice of β plays a crucial role in determin- ing the final estimate of then entropy because the prior is peaked and thus essentially determines the entropy in the low data sample size limit [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Instead, we might consider a meta-prior that consider a mixture of all possible priors within this family, or the range of possible β in a way that flattens our starting as- sumptions about the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This means that we should move β over some weighted range P(β) in such a way that corresponds to moving ξ(β) between 0 and log H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Then, a candidate mixture prior proposed in reference 17 is P({qi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' β) = 1 Z(β)δ � 1 − S � i=1 qi � S � i=1 qβ−1 i dξ(β) dβ P(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (6) To perform this integral, one must also know how the estimated entropy varies as a function of ξ in order to flatten the prior in the estimated range, the relationship between which is given in reference 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As it turns out, Eq 6 performs surprisingly well for entropy estimates on e i=2 a a d d3 tiny samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Regardless, it cannot overcome the funda- mental problem that entropy estimates are dominated by the prior in any undersampled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' One way out of this pickle is to leverage the structure of the model’s probability distribution to simplify the entropy estimation problem by effectively reducing the state space over which one’s estimator needs to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Conveniently, Shannon’s last axiom tells us that if we can group the states into independent subsets, then the information entropy is the sum of both the uncertainty of the labels from the clusters as well as the contribution from within each subset given the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When a clever decomposition is possible, it would allow us to consider subsets of the system independently of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This, for example, is possible if some components of s with in- dex i, the set {si} of which we denote si for simplicity, are conditionally independent of components indexed by j once holding fixed components k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In other words, this is the assertion that we can factorize the probability dis- tribution p(s) = p(si|sk)p(sj|sk)p(sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (7) If this is the case, then the entropy of this set decomposes in the summation of the information entropies H[p] = H[p(sk)] − � sk p(sk) (H[p(si|sk)]+ H[p(sj|sk)]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (8) We can think of each unique configuration of sk as a “la- bel,” and once this is given we must compute the uncer- tainty of the sets si and sj with their respective weights given by the frequency of the sk on which they have been conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When we generalize this basic example to a tree, we denote a set again as si but now denote moving up gen- eration in the tree as the index i − 1 and moving down a generation as i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Labeling each branch with a different letter of the alphabet as superscript, p(s) = p(s0) N � i=1 � a,a′ p � sa′ i ���sa i−1 � , (9) where we again use the shorthand notation si to refer to the set of components over index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Then, the root is the set of nodes s0, the first product is over each successive layer in the tree indexed i down to the leaves in the Nth generation, and the second product over the descendents a′ of each branch a in the (i − 1)th layer, which is inde- pendent once having conditioned on all parent branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As an example of such a factorization using this nota- tion, we show a graph in Figure 1, where we indicate the successive generation of the tree i generations away from the root that are conditionally independent of one another once conditioning on the parent branches i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When the probability distribution is factorizable in this way, then it is possible to compute the entropy of the entire system as a sum of entropies from the outside in by computing the entropy of each leaf, which provides an additive contribution to the final entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We express this in the recursive form H[p] = H[p(s0)] + � s0 � a p(s0) � H[p(sa 1|s0)] + � sa 1 � a′ p(sa 1|s0) � H[p(sa′ 2 |sa 1)] + · · · + � sa(n−1) i � a(n) p � sa(n) i |sa(n−1) i−1 � � H � p � sa(n+1) i+1 |sa(n) i �� + · · · ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (10) In short, we can implement the calculation in Eq 10 from the leaves up to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We start with the conditional entropy of a leaf, prune the leaf, upon which the branch leading to the leaf in turn becomes a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Continuing in this recursive way, we eventually reach the root of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Such a hierarchical decomposition presents a way to ameliorate the problem of entropy estimation if we can group components into subsets that are substantially smaller than the full graph, which shrinks the state space exponentially [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' If it is the case that the groups are substantially smaller than the full graph, estimat- ing the entropy in principles becomes more manageable with an exponentially smaller Monte Carlo Markov chain (MCMC) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We present an algorithm that esti- mates the entropy of probabilistic graph models that fac- torizes the graph using a heuristic based on structural holes and fast MCMC sampling of the conditioned prob- ability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' ALGORITHM We provide an outline of algorithm in Table I and pro- vide more details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The first step is to compute a factorization of the graph 4 that allows us to split the problem into more manage- able entropy estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' To do so, we rely on the notion of structural holes from the sociological liter- ature [20, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Structural holes are inspired by a problem of constrained action in a social network, where the as- sumption is that the amount that any individual u is con- strained by a neighbor v depends directly on two factors: the relative investment that u dedicates in its the rela- tionship with v and the simultaneous relative investment of another neighbor w of u into v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The intuition under- lying the latter step is that when investments overlap, u is maximally constrained in terms of leverage because u is redundantly influencing the local neighborhood of v (they are competing for influence in the same sphere), whereas with minimal overlap, v acts less as a constraint and more as a bridge to other parts of the network to which u does not have access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In other words, a neighbor v with low structural constraint with respect to u is sur- rounded by “structural holes” such as when it belongs to a different clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For our purposes, the intuition is that the nodes with low constraint are ones that are placed in between densely connected communities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' they should present a small set that we can fix to render adjacent communities indepen- dent of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' More specifically, these nodes are surrounded by holes and thus have small structural con- straint cu, defined as the sum over local constraints l for node u, cu := � v∈N (u) l(u, v), (11) which is the sum over all neighbors v of u, or N(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The local constraint given uniform edge weights is defined as l(u, v) := � � 1 |N(u)| + � w∈N (u)∩N (v) 1 |N(u)| 1 |N(w)| � � 2 , (12) where the number of neighbors of node u is |N(u)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The first term in the parentheses accounts for u’s investment in this neighbor v, which is uniform across all neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The second term accounts for the joint investment in neighbor v from both u and a common neighbor w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Note that Eq 12 is a simplified form for our scenario, where “in- vestment,” or the weights, between pairs of nodes does not generally need to be uniform [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We start by removing nodes from the graph starting with the node of minimal constraint and recomputing the local constraints upon every removal step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Every re- moved node is placed into the set X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This will tend to split apart connected components in the set of remaining nodes X along the bridges that connected them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' If we continue indefinitely with this procedure, the subgraphs living in the set of removed nodes X′ will eventually con- TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Algorithm for entropy estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In the case the starting graph consists of multiple components, they are treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Factorize graph to generate “contracted” tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Place all nodes into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Calculate the structual constraint for every node in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Remove from set X and place into X′ the node with minimal constraint in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' If the largest connected component in either X or X′ is now larger than largest component previous to step c or if the largest connected component in X is smaller than or equal to threshold M, then jump to step f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Return to step b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Create a coarse-grained representation of the graph, where a connected component in X and X′ are connected to one another if there is at least one interaction between the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Estimate conditional entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Identify leaves in coarse-grained graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For each leaf, generate a Monte Carlo sample of size K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For each unique state in the sample of size K, generate from the leaf a Monte Carlo sample of size K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Estimate the entropy of the leaf and the condi- tioned branch using NSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Prune the coarse-grained graph by removing all leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Return to first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Sum entropies and calculate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Sum over the estimated entropies of each leaf and its branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' sist of most of the graph, and they may form large compo- nents that have not simplified the problem at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Thus, we keep removing nodes with minimal constraint as long as the largest components in the set of removed nodes as well as the pruned graph are shrinking or until they have all fallen below a size smaller than the specified threshold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As a result of this procedure, we have two sets of nodes X and X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Within each set, we have disconnected com- ponents labeled x and x′, respectively, that bridge nodes in the other but are not directly connected, or a bipar- tite structure as we show in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Thus, the graph presents a structure in which fixing the components in x renders components x′ independent of one another and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As the final step, we calculate the entropy of the re- sulting factorized tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We do so identifying a leaves, or components in either X or X′ that are conditionally in- dependent when holding a single component in the com- 5 2 1 0 1 2 coupling J 10 15 20 25 30 entropy estimate (bits) TreeEnt K = Kcond = 103 naive K = 104 NSB K = 104 exact FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Comparison of TreeEnt with na¨ıve and NSB entropy estimators on five replicas of an “pointy” triangle (inset on top left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Our algorithm does better faster and with fewer samples by recognizing the interaction structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' plementary set fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' While we cannot guarantee that a leaf can be found at the end of our structural constraint removal procedure, it is the case that removing nodes of minimal constraint will tend to fragment the graph into a chain of conditionally independent pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Furthermore, there always exists a partition of the graph such that a leaf node can be designated (in the trivial case a single node can be put into X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Finally, we then sample from the set of conditionally independent “leaves” and the unique set of nodes that lead to them, or their “branch.” First, we generate an MCMC sample of the distribution of the a’th branch sa i−1 from a sample of the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Then, we iterate through the unique states and sample for the a′’th leaf sa′ i given each of the possible states of the branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The size of the sample set also allows us to estimate the standard deviation of the sampled entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This gives us the terms in Eq 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Then, we prune the leaf from the graph and iterate this process recursively until we only have the root of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' At each step, we calculate the entropy using the NSB estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The total entropy of the tree is the summation of the calculated entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' ERROR ESTIMATION For the algorithm, we must account for two sources of error including from the finite sample size of the condi- tioned set and the NSB estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For the finite sample contribution, the contribution to the entropy for a single term (the entropy of a leaf) in Eq 10 is the combination of terms � H[p(sa′′ i+1)] � = � sa′ i p(sa′ i )H[p(sa′′ i+1|sa′ i )] (13) where the sum is over all the unique states that occur for the branch sa′ i and for the particular leaf a′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For this calculation, we are not considering the sum over all the parents of the branch sa′ i because we can MCMC sample directly from the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' This means that we can also calculate the variance in the entropy of this leaf, σ2 sa′′ i+1 = � sa′ i p(sa′ i ) � H[p(sa′′ i+1|sa′ i )]− H[p(sa′′ i+1|sa′ i )] �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (14) As in the usual sense, we normalize the variance by the number of samples K to estimate the standard error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Since the NSB error σNSB(sa′ i )2 on each term in the sums of Eqs 13 and 14 are computed independently of the finite-sample error, we add the norms of both the errors together to determine the total error on the en- tropy estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' For a single leaf, we have the sum of the variance, or the norm-squared error Σ2 sa′′ i+1 = σ2 sa′′ i+1 � K + � sa′ i p(sa′ i )σNSB(sa′ i )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (15) ASSESSING PERFORMANCE We compare our algorithm with the na¨ıve and NSB es- timator in an example where the entropy can be exactly calculated by enumeration of all configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' We find that leveraging the factorized structure of the graph al- lows us to recover a nearly perfect estimate to the exact entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' As an example, we show the results for an en- semble consisting of multiple, disconnected graphs, each one consisting of a triangle with a single additional node coming off of each vertex (thus a “pointy” triangle) for a total of six spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The spins are coupled with uni- form strength J that we vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When we sample from five replicas of the graph at the same time, the state space of K = 230 ∼ 109 is difficult to sample well on a desktop machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Our estimator, as we show in Figure 2, per- forms well across different scales of the coupling J even with a relatively small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' It is also sub- stantially faster than applying the NSB estimator to the entire ensemble at once because the subsets on which we calculate the estimator are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' APPLICATION TO MAXENT VOTING MODEL We use our algorithm to estimate the entropy of a sparse voting model of judge voting on the US Circuit courts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' The probabilistic models that we consider derive from the maximum entropy principle, which is an algo- rithm for determining minimal statistical models of data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Importantly, the quantity of interest in this frame- work is the entropy of the model, which is maximized 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Examples of graph partitions using the minimal constraint heuristic applied to sparse US Appeals Courts interaction graphs from reference .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (a) DC Circuit Court separates into four connected components when max cluster size is set to M = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Largest component is of size N = 19 compared to N = 47 for the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Each color is a different subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Inset shows interaction structure on coarse-grained graph which is a tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' the green, red, and orange clusters are independent of one another once fixing the blue cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' (b) First Circuit is partitioned into sets that form a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' Largest component is of size N = 13 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' total graph size of N = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' while constraining a few, crucial properties of the system such as the average vote of each judge and their pairwise correlations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' When these two sets of constraints are imposed, we obtain the pairwise maxent model, which has been shown to model to high accuracy voting pat- terns on the US Supreme Court [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' According to the model, the probability distribution takes the Boltzmann form p(s) = 1 Z e−E(s) (16) with normalization term Z known as the “partition func- tion” in statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' To each configuration s de- fined as a vector of −1 and 1 for the votes of the judges, an “energy” E(s) is assigned, where lower energy implies that the configuration is more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E3T4oBgHgl3EQf4AtG/content/2301.04768v1.pdf'} +page_content=' In the case of the pairwise maxent model, it takes the form E(s) = − N � i=1 hisi − N � i [Accessed +9 May 2022]. +Auerbach, M. & Uygun, Y. 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(2020): The Current State of Research & Application of Industry 4.0 in Germany. +In: Industry 4.0: Principles, Effects and Challenges. Y. Uygun (Ed.). Nova Science: +Hauppauge, NY, pp. 3-15. +Uygun, Y. & Ahsan, J. (2021): Analyzing the Railway Network of the Belt and Road +Initiative. In: Cogent Business & Management, Volume 8 Issue 1, +https://doi.org/10.1080/23311975.2021.1932066 +Uygun, Y. & Jafri, A. (2020): Controlling Risks in Sea Transportation of Cocoa Beans. In: +Cogent Business & Management. Open Access. Vol. 7, No. 1. +https://doi.org/10.1080/23311975.2020.1778894 + + +Razan Elzain | Large-Scale 3D Printing Market Analysis +42 + + +Uygun, Y. & Kuhn, A. (2010). Life-cycle Oriented Postponement in International Supply +Chains. In: K.S. Pawar & A.T. Potter (Eds.): Proceedings of the 15th International +Symposium on Logistics: Configuring Next Generation Supply Chains, pp. 13 - 21. +Uygun, Y. & Luft, N. (2010). Vorgehensmodell zur Maßnahmenselektion - Das +Maßnahmenfilter-Modell. In: G. Bandow & H. H. Holzmüller (Eds.): "Das ist gar kein +Modell" - Unterschiedliche Modelle und Modellierungen in Betriebswirtschaftslehre und +Ingenieurwissenschaften. Gabler: Wiesbaden, pp. 213-232. +Uygun, Y. & Reynolds, E. B. (2016). Advanced Manufacturing Ecosystems. In: C. Brecher & +S. Jeschke (Eds.): Industrial Internet of Things - Cybermanufacturing Systems. Springer: +Berlin, pp. 691-715. +Uygun, Y. & Rustemaj, A. (2021): Ant colony optimisation for milk-runs in manufacturing +systems. In: International Journal of Advanced Operations Management. pp 167-181, +https://doi.org/10.1504/IJAOM.2022.123267 +Uygun, Y. & Schmidt, A. (2011). „Performance Measurement for Interorganisational +Collaborations of SMEs“. In: H.-J. Kreowski, B. Scholz-Reiter, K.-D. Thoben (Eds.): +Dynamics in Logistics. Springer: Berlin et al., pp. 169-190, 2011. +Uygun, Y. & Straub, N. (2012). Human-centred Model for Application of Lean Production in +Networks. In: H. ElMaraghy (Ed.): Enabling Manufacturing Competitiveness and Economic +Sustainability. Springer: New York, pp. 660-665. +Uygun, Y. & Wagner, S. U. (2011). Guidelines for Human-based Implementation of Lean +Production. In: N. Duffie (Ed.): Proceedings of 44th CIRP International Conference on +Manufacturing Systems - New Worlds of Manufacturing. Omnipress: Madison, Wisconsin. +Uygun, Y. & Wötzel, A. (2009). Antizipative Veränderungsplanung intralogistischer Systeme +- Eigenschaften und Handlungsfelder. In: ZWF Zeitschrift für wirtschaftlichen Fabrikbetrieb, +Volume 104 Issue 12, pp. 1131-1134, https://doi.org/10.3139/104.110223 + + +Razan Elzain | Large-Scale 3D Printing Market Analysis +43 + + +Uygun, Y., Ringeln, M. & Straub, N. (2015): Pull-Prinzip. In: U. Dombrowski & T.Mielke +(Eds.): Ganzheitliche Produktionssysteme – Aktueller Stand und zukünftige Entwicklungen. +Springer: Berlin, pp. 110-128. +Uygun, Y., Schupp, F., Gotsadze, N., Gzirishvili, L., Tindjou, S. (2022): A Holistic Model for +Understanding the Dynamics of Outsourcing. In: International Journal of Production +Research. Taylor & Francis. Open Access. https://doi.org/10.1080/00207543.2022.2031330 +Uygun, Y.; Luft, N.; Woetzel, A. (2012). A Model to Select Specific Measures for +Adaptability of Logistics and Production Systems. In: K.S. Pawar & A.T. Potter (Eds.): +Proceedings of the 17th International Symposium on Logistics: New Horizons in Logistics +and Supply Chain Management. Centre for Concurrent Enterprise: Nottingham, pp. 195-203. +www.fortunebusinessinsights.com. (n.d.). 3D Printing Market Size, Share, Analysis | +Forecast Report, 2026. [online] Available at: +https://www.fortunebusinessinsights.com/industry-reports/3d-printing-market-101902. +www.futuremarketinsights.com. (n.d.). 3D Printing Market. [online] Available at: +https://www.futuremarketinsights.com/reports/3d-printing-market [Accessed 13 May 2022]. +www.linkedin.com. (n.d.). The Implications of 3D printing for Manufacturing. [online] +Available at: https://www.linkedin.com/pulse/implications-3d-printing-manufacturing- +dominic-parsonson/ [Accessed 13 May 2022]. +XponentialWorks. 2022. What is Large Format 3D Printing and Who Needs It? - +XponentialWorks. [online] Available at: [Accessed 26 April 2022]. + diff --git a/kNAyT4oBgHgl3EQfyPlA/content/tmp_files/load_file.txt b/kNAyT4oBgHgl3EQfyPlA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9857249eb6439dfcff9808ef4a52dd462e97468b --- /dev/null +++ b/kNAyT4oBgHgl3EQfyPlA/content/tmp_files/load_file.txt @@ -0,0 +1,1308 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf,len=1307 +page_content='Large Scale 3D Printing Market Analysis Razan Abdelazim Idris Alzain Campus Ring 7, 28759 Bremen, Germany 1 Table of Contents Table of Contents 1 List of figures 2 List of tables 3 List of abbreviations 4 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Problem of the paper 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Aims of the paper 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Course of research 2 2 3D Printing 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Type of Technology 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Benefits 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Disadvantages 7 3 Large-Scale Printers 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Background 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Today’s Large scale printers 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Manufacturers 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 Future Developments 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 Benefits 17 4 3D Printing Market Analysis 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Market Players 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Growth Impact made by Market Players due to their Strategies 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Largest Global Share in the Market 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 3D Printing in Germany and UK 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 3D Printing Market Size Forecast with COVID-19 Impact 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='6 3D Printing Application Analysis 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='7 Rise in Investment from Governmental bodies in 3D Printing Projects 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='8 Things hindering the 3D Printing Market Growth 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='9 Market Opportunities 30 5 Final Consideration 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Results and critical reflections 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Implications for further research 32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Implications for practice 33 Table of references 35 ' metadata={'source': 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Sintering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='Electron Beam Melting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='Big Area Additive Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3D Systems Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Proto Labs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' FARO Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Materialise NV The ExOne Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Application Programming Interface Additive Manufacturing Big Area Additive Machine Razan Elzain | Large-Scale 3D Printing Market Analysis 1 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Problem of the paper One of the main problems brought by large-scale 3D printers is the lack of standardization of machines and the potential of low-quality products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' If a company invests in an inexpensive 3D printer then the risk of a bad quality product increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' On the other hand, a high-end Large-scale 3D printer would cost millions of dollars to produce a trustworthy result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Regardless, the traditional manufacturing route will always be preferred by production companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Many different 3D printers produce very different products, making there a lack of universal standards in 3D printing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Therefore, manufacturers compare their products with other manufacturers' methods worrying that they would vary in terms of quality, strength, and reliability." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This causes a continuous wariness in the 3D printing technology, making companies always judge the risks compared to the benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Another problem that comes with large-scale 3D printers is the short product lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Yes, having the ability to print a large number of spare parts on-demand can help to prolong product warranties while also being significantly more ecologically friendly, but it is not a smart move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Unfortunately, many businesses rely on a business plan that revolves around low-quality items and product rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing is a concern because it reduces product obsolescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Firms will need to develop new business models that ensure quality and product lifespans and do not rely on continually creating new items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, 3D printing allows customers to create their own spare parts, which is bad for businesses1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As mentioned above, 3D printing companies may be able to begin printing their own product parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is exacerbated by the fact that smaller enterprises may begin printing product components and entire goods, stealing the intellectual property of larger corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 1 Cf (The big challenges of 3D printing 2019) Razan Elzain | Large-Scale 3D Printing Market Analysis 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Aims of the paper The aim of this research is to get a better understanding of the future of large-scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' By developing the market analysis, it will be clear whether large-scale 3D printing is becoming more of a preferred way of printing custom-made parts for production companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Companies can then choose whether to change their ways, for a more profitable less costly method, or stay on the route they are on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' By getting deep into this topic, a new world of technology is then being discovered and familiarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' With a mix of theoretical and practical relevance, a complete coverage could be made on large-scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This paper could then cover all aspects of this topic, and the reader could then make their own judgment if large-scale 3D printing would be the best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Course of research The way that is planned on approaching this research is by introducing the topic and familiarizing the reader with Large-scale 3D printers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Then, to move into the literature review, where some reports and academic publications will be covered on large-scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Then proceed with mentioning the benefits and risks that come with 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A comparison will also be done between small-scale and large-scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A market analysis is then made for large-scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Deep research will be done to see what the market wants and expects from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Also a few mentions about the companies that create large-scale 3D printers and the companies that rent/buy/use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Also make a cost analysis, of the revenue and expenses that come with large-scale printers, to better understand their market value of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Afterward, the discussion follows and a limitation section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The limitation section will include features that prevent the large-scale 3D printer from achieving a higher market value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Within this section, a paragraph will be dedicated to the research gap that comes with the internet search limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Then, the research will move to the conclusion and then references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As previously mentioned, this research will include a literature review, which will explain a big part of this research’s aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Then followed by the market analysis, which will be made, by providing an overall industry market than focusing on the targeted market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Also by Razan Elzain | Large-Scale 3D Printing Market Analysis 3 distinguishing the customer characteristics, we can understand the exact appeal of large-scale 3D printers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' An important step in the market analysis is to compare large-scale 3D printers with their competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' To see which dominates the market now, and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Later on, all the data must be gathered and then analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Razan Elzain | Large-Scale 3D Printing Market Analysis 4 2 3D Printing What is 3D printing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' To answer that question in the most basic way, it is said to be a manufacturing technique where the material is layered onto each other layer by layer to build a three-dimensional part/item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' With the help of a computer-aided design (CAD) or computer- aided manufacturing (CAM), programmers can create and convert the digital files containing the 3D data into physical objects2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' For decades, the car industry has been exploring the possibility of 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing is particularly beneficial for quick prototyping and has demonstrated the ability to dramatically reduce design and lead times on new automobile models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The manufacturing process has also been improved by 3D printing in the sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Specialized jigs, fixtures, and another tooling that would be required for a single automobile part, particularly for high- performance machines, used to necessitate a slew of custom tools, adding expense and complicating the process as a whole3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Custom jigs and other low-volume items may be made directly for the production line using 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Manufacturers can reduce lead times by up to 90% and reduce risk by integrating 3D printing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The manufacturing process as a whole becomes more efficient and lucrative by simplifying with in-house production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing is causing a design revolution in the jewelry industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It used to be difficult to create 3D printed items that looked and felt like conventional handmade and cast jewelry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, with the most recent wave of advancements in specialized high-end 3D modeling tools and more printable materials available, an increasing number of jewelry designers are now preferring to 3D model and print their creations over conventional handcrafted methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Weight reduction is one of the key ways that 3D printing has helped the aircraft industry to save money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" The lower volume of components required in 3D printed construction of a part results in lighter overall parts—this seemingly minor change in production positively 2 Cf (3D Printing: What You Need to Know, 2022) 3 Cf (25 (Unexpected) 3D Printing Use Cases, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 5 affects an aircraft's payload, emissions, and fuel consumption, as well as speed and safety, while significantly reducing production waste." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The approach, like in many other sectors, enables the manufacturing of components that are just too complicated for traditional ways to manage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Type of Technology Different technologies are used by 3D printers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The most well-known is fused deposition modeling (FDM), often known as fused filament manufacturing (FFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is manufactured from acrylonitrile butadiene styrene (ABS), polylactic acid (PLA), or another thermoplastic filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They are then melted together and discharged in layers through a heated extrusion tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The first 3D printers to hit the market, created by Stratasys in the mid- 1990s, utilized FDM, as do most 3D printers aimed at consumers, enthusiasts, and schools today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' FDM is utilized in 3D printed structures by extruding clay or concrete, 3D printed desserts by extruding chocolate, 3D printed organs by extruding living cells in a bio gel, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' If anything can be extruded, it can almost certainly be 3D printed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Stereolithography is another 3D printing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In it, a UV laser is shone into a vat of ultraviolet-sensitive photopolymer, tracing the surface of the item to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The polymer hardens wherever the beam comes into contact with it, and the beam "prints" the item layer by layer according to the instructions in the CAD or CAM file from which it is working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A variant of this is a digital light projector (DLP) 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A liquid polymer is exposed to light from a digital light processing projector in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This hardens the polymer layer by layer until the item is completed, at which point the residual liquid polymer is drained away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" SLA has the distinction of being the world's first 3D printing technology." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Chuck Hull devised stereolithography in 1986, filing a patent on the technology and establishing 3D Systems to market it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 An SLA printer employs mirrors known as galvanometers or galvos, one on the X-axis and one on the Y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These galvos fire a laser beam over a vat of resin, 4 Cf (All3DP, 2018) Razan Elzain | Large-Scale 3D Printing Market Analysis 6 selectively curing and hardening a cross-section of the item inside the construction area, layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Multi-jet modeling is a 3D printing technology that sprays a colored, glue-like binder over consecutive layers of powder where the item is to be produced, similar to an inkjet printer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is one of the quickest ways, as well as one of the few that offers multicolor printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It is feasible to alter a regular inkjet printer so that it can print using materials other than ink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Enterprising do-it-yourselfers have developed or modified print heads, primarily piezoelectric heads, to operate with a variety of materials—in some cases printing the print heads themselves on other 3D printers!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' MicroFab Technologies, for example, sells 3D- capable print heads (as well as complete printing systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' There is also selective laser sintering (SLS), which is a high-powered laser used to fuse particles of plastic, metal, ceramic, or glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' On the other hand, Electron beam melting (EBM) melts metal powder layer by layer using an electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Titanium is frequently combined with EBM to create medical implants and aeronautical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Benefits Designers may use 3D printing to swiftly convert thoughts into 3D models or prototypes (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' "rapid prototyping") and execute rapid design revisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It enables producers to make things on demand rather than in huge batches, which improves inventory management and reduces warehousing space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' People in faraway areas may now create items that would otherwise be unavailable to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In practice, 3D printing may save money and material over subtractive processes since very little raw material is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It also promises to transform the nature of production by Razan Elzain | Large-Scale 3D Printing Market Analysis 7 someday allowing people to download data for printing even complicated 3D products, such as electrical gadgets, in their own homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Another advantage of 3D printing is that it enables the creation and production of more complicated designs than traditional manufacturing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Traditional techniques have design constraints that are no longer applicable with the usage of 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing can also produce parts in a matter of hours, which expedites the prototype process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This allows each step to be completed more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' When compared to machining prototypes, 3D printing is less expensive and faster at generating components since the part may be produced in hours, allowing each design alteration to be performed at a much faster rate5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Another advantage of print on demand is that, unlike traditional printing procedures, it does not need a large amount of storage space for inventories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This saves space and money by eliminating the need to print in bulk until absolutely essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The 3D design files are all maintained in a virtual library and may be searched and printed as needed since they are produced using a 3D model as a CAD or STL file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Design modifications may be made at a low cost by modifying individual files rather than discarding out-of-date items and investing in new equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing is being utilized in the medical field to save lives by printing human organs such as livers, kidneys, and hearts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Further advancements and applications are being explored in the healthcare sector, which will provide some of the most significant benefits of adopting the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Disadvantages While 3D printing can make products from a variety of polymers and metals, the accessible raw materials are not exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is because not all metals or polymers can be 5 Cf (What are the Advantages and Disadvantages of 3D Printing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 8 thermally regulated sufficiently to allow 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, many of these printing materials are not recyclable, and just a handful are food safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Currently, 3D printers feature limited print chambers that limit the size of items that can be manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Anything larger will have to be printed in separate sections and then assembled afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Because the printer needs to produce more components before manual labor is employed to connect the parts together, this can raise costs and time for bigger parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Although big pieces, as previously stated, require post-processing, most 3D printed items require some type of cleaning up to remove support material from the build and smooth the surface to obtain the desired quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Waterjetting, sanding, a chemical soak and rinse, air or heat drying, assembling, and other procedures are used for post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The quantity of post-processing required is determined by a variety of factors, including the size of the item being produced, the intended application, and the type of 3D printing technique utilized in manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, while 3D printing allows for the rapid creation of parts, post- processing might impede the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As 3D printing becomes more popular and accessible, there is a higher potential that individuals may make false and counterfeit items that will be nearly difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This has obvious implications for both copyright and quality control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Another potential issue with 3D printing is directly tied to the type of machine or method utilized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' certain printers have lesser tolerances, which means that finished items may deviate from the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This can be corrected in post-production, but keep in mind that it will increase the time and expense of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Razan Elzain | Large-Scale 3D Printing Market Analysis 9 3 Large-Scale Printers To be said simply, large-scale 3D printing is the industrial-scale 3D printing of previously molded or machined products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Consider printing a life-size mannequin, a larger- than-life Coke bottle for advertising, a car's whole bumper, or even an airplane wing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Large- scale 3D printing is not just a less expensive alternative to machining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' it can also manufacture complicated shapes that would otherwise need several pieces and assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This reduces both production time and end-product costs6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Large-scale 3D printing is a fantastic alternative for manufacturers when it comes to developing molds and tooling since it reduces lead times and costs compared to traditional production setup while also reducing the constraints associated with traditional design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Background Oak Ridge National Labs, a US Department of Energy research site, pioneered large- scale 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In terms of sheer scale, they continue to lead the way: the latest Big Area Additive Machine (BAAM) can deposit over 36 kg of material per hour, producing pieces up to 13 feet long, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 feet wide, and 8 feet tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" BAAM's initial edition was released in 2014 as part of a collaboration with the City of Cincinnati." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The most recent iteration, which was used to print a full 3D-printed automobile in 2017, features two hoppers, dryers, and lines to the extruder to enable printing with diverse materials — especially beneficial for creating objects with different qualities on the surface than within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' BAAM and related machines are most typically employed to make big molds — for things like airplane wings or the cladding that many high-rise buildings have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Traditionally, these things would have been made using massive wood molds over which material would be 6 Cf (What is Large Format 3D Printing and Who Needs It?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' - XponentialWorks, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 10 molded or poured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, making these molds might take months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' BAAM permits the creation of a mold in a matter of days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Today’s Large scale printers The most recent generation of commercial large-format 3D printing technologies, which have already been widely used in a variety of sectors, are not as huge as BAAM – but they are significantly faster and more sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The Nexa3D NXE400 printer, for example, can print up to 16 liters of component volume per minute at rates of up to 1Z centimeter per minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' By contrast, every other similar 3D printer on the market has six times the speed and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 times the volume of this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The Nexa3D printer can also print with strong materials, making it perfect for high- speed printing of functional prototypes, manufacturing tools, full-scale end-use parts, and casting patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The printers have intelligent software and integrated sensors, which optimize production performance while also providing thorough diagnostics and continuous monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Following that, there's no doubting the insaneness of a 600 x 600 x 660 mm construction space for less than $4,0007." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The Modix Big60 V3 is a true workhorse printer with the tools to make the most of its size, including an E3D Volcano hot end for fast printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The thinking is handled by a Duet 2 Wi-Fi-enabled mainboard, which is paired with a Duet touchscreen controller, which gives the operator access to macros and other live-print controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" It lacks an enclosure (which costs extra) and requires self-assembly, but it's difficult not to be lured to that build volume." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Finally, we'll take a look at Vivedino (previously Formbot) and its Vivedino Troodon CoreXY 3D printer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" The build space of 400 x 400 x 500 mm and Wi-Fi-enabled mainboard are two of the main draws, coupled with the fact that it is based on the Voron project's highly programmable CoreXY 3D printer architecture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" At $1,600, it isn't the cheapest, but it has a 7 Cf (All3DP, 2018) Razan Elzain | Large-Scale 3D Printing Market Analysis 11 substantial build volume and a great feature set." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" It's also nearly ready to use right out of the box." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Table 1: List of best Large scale printers in 2022 3D Printer Build Volume (mm) Price (USD, approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Check Price (Commissions Earned) Creality Ender 5 Plus 350 x 350 x 400 mm $582 Creality3D Official Store Modix Big-40 400 x 400 x 800 mm $4,900 Modix Modix Big60 V3 600 x 600 x 660 mm $3,900 Modix Tronxy X5SA-500 Pro 500 x 500 x 600 mm $820 AliExpress Vivedino Troodon 400 x 400 x 500 mm $1,675 AliExpress gCreate gMax 2 457 x 457 x 609 mm $3,995 gCreate (no commission) Source: (All3DP, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Manufacturers 3D Systems Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (DDD) 3D Systems pioneered 3D printing in 1989 with the development and patenting of their stereolithography technique, which employs ultraviolet lasers to aid in the creation of highly exact parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' DDD expanded on this by innovating additional technologies such as selective laser sintering, multi-jet printing, film-transfer imaging, color jet printing, direct metal printing, and plastic jet printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D Systems operates in three divisions: products, Razan Elzain | Large-Scale 3D Printing Market Analysis 12 materials, and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The items category covers tiny desktop and commercial printers that print in plastics and other materials, as well as 3D printers and software8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Proto Labs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (PRLB) Proto Labs was established in 1999 with the goal of developing automated solutions for the development of plastic and metal parts utilized in the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The firm grew to create an industrial-grade 3D printing service, allowing developers and engineers to transfer prototypes into the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Injection molding, sheet metal fabrication, and 3D printing are the core commercial services provided by the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' FARO Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (FARO) FARO specializes in 3D measuring as well as other services for the design, engineering, and construction industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" FARO' has a 40-year history that predates the emergence of 3D printing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Coordinate measuring equipment, laser trackers and projectors, mappers, scanners, and software are among the company's products." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' FARO also provides services to the aerospace, automotive, and power generating industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Materialise NV (MTLS) Materialise, a Belgian firm, has been supplying 3D printing technologies and accompanying software for 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It provides platforms for the development of 3D printing applications in fields such as healthcare, automotive, aerospace, and art & design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Anatomical 8 Cf (5 Biggest 3D Printing Companies, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 13 models in dentistry and hearing aid goods were among the company's earliest 3D printing efforts." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Materialise also makes eyeglasses and automobiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The ExOne Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (XONE) ExOne specializes in producing 3D printing equipment for customers in a variety of sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It also manufactures 3D printed things to order for industrial clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Binder jetting technology is used by ExOne 3D printers to fuse powder particles of materials such as metal or sand into molds, cores, and other products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Table 2: List of 3D printing Companies in descending order 3D Printing Companies Revenue (TTM) Net Income (TTM) Market Cap: 1-Year Trailing Total Return Exchange 3D Systems Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (DDD) $566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='6 million -$78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 million $632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 million -24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='60% New York Stock Exchange Proto Labs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (PRLB) $451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='0 million $58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='6 million $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='9 billion 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='20% New York Stock Exchange FARO Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (FARO) $334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='7 million -$79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='7 million $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='0 billion 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='60% NASDAQ Materialise NV (MTLS) $205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 million -$2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='7 million $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='9 billion 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='80% NASDAQ The ExOne Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (XONE) $52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='9 million -$14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 million $238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 million 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='30% NASDAQ Razan Elzain | Large-Scale 3D Printing Market Analysis 14 Source: (5 Biggest 3D Printing Companies, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 Future Developments Which 3D printing materials are predicted to flourish in the coming decade?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Will we see freshly found 3D printing materials as a result of AI-enabled computational alloy discovery, with end-users able to define desired properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' What about the widespread use of metamaterials, which are materials having features that do not occur naturally?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' How will sustainability be considered, such as an increase in attempts to make bio-based polymers or recapture waste plastics and metals, therefore returning resources to a circular economy9?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Some of the efforts may be done by the market through end consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Purchasing power is used to find successful technological platforms among a sea of imitators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Conversely, as demand for additive manufacturing rises, a swollen market may lift all ships, giving both newcomers and incumbents access to a larger pie to share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Below is a quote was given by the president and CEO of 3D Systems, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Jeffery Graves, giving an insight into what he thinks the future will be for 3D printing: “I expect mass customization will not only be an important trend for 2022 but the coming decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' While many organizations aspire to take advantage of additive manufacturing’s ability to produce large quantities of distinct parts, I don’t think every organization has fully understood how to integrate AM into its workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As we see a broader acceptance of additive alongside traditional technologies, I anticipate we’ll also see manufacturers of all sizes embracing AM for mass customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' To facilitate the integration of AM into existing workflows, I believe machine learning will play a critical role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It is not enough to introduce design flexibility, speed to market, or supply chain efficiency offered by additive manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' For companies to maintain their competitive position, they need to have a smart manufacturing strategy to introduce AM 9 Cf (Sertoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 15 effectively and efficiently into their overall manufacturing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As more companies adopt smart manufacturing solutions, I expect they will see how machine learning can enable autonomous manufacturing – thus helping improve productivity, and enhance capacity to introduce scalability and flexibility into processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Focusing our perspective on the next decade even further, I expect that we’ll continue to see additive manufacturing drive remarkable advancements in the transformation of healthcare delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' AM has already demonstrated its power in this industry to enable patient-specific healthcare with unique solutions to create surgical plans and medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' I’m very excited about the next frontier in healthcare where bioprinting plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Over the past year, there has been a dramatic increase in the number of entrants to this field, whether it be research organizations or private and public companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' We’ve seen the scope of the research efforts broaden to include new printing technologies and new materials designed to assist in drug discovery, the creation of tissues, and hopefully one day, producing transplantable human organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' I believe we’re on the precipice of amazing advancements in this arena and look forward to what we’re able to influence and achieve as an industry.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='10 In the field of architectural design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' there is a significant trade-off between the desire to save money,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' which entails producing simple and straight walls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' and the desire to create novel and nonstandard buildings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' which entails producing customized molds and formwork for one- time use,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' resulting in large amounts of waste materials and unforeseen construction delays11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Large-scale 3D printing has the ability to produce complicated geometries that cannot be defined by fundamental geometrical concepts and have previously only been achieved for a few projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Now that it has been mathematically shown that any 3D building may be formed layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Structures may be built in practically any shape, regardless of complexity, cost, or potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing technology is rapidly evolving, becoming larger, quicker, and more affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The spectrum of materials in AM will expand as the need for specialized materials to meet the requisite qualities of end-parts grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The capacity of the current generation of 10 (Sertoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022) 11 Cf (Mourad, Aljassmi and Al Najjar, 2018) Razan Elzain | Large-Scale 3D Printing Market Analysis 16 printers that can handle a wider range of sophisticated materials is critical since it allows businesses to profit from AM where they previously could not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Metal filaments for FDM printing are an excellent example of novel materials12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Although machine prices remain high, part costs are decreasing as speed increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These developments will increase as more firms use 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" With technologies like dual extrusion, 3D printing's adaptability is increasing and is being seen used in a larger range of sectors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Another noteworthy breakthrough noticed is printing without the usage of support structures, which broadens the spectrum of applications for AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Could be said, the potential for cost and time savings is quite favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" It's not simply about having new and improved printers for manufacturers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Manufacturers would require a wide choice of printers, materials, and, most crucially, contacts with other industry specialists to reap the most benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, interoperability across all of the various systems is becoming a crucial problem in order to fully utilize 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Automation in manufacturing and post-processing, as well as integrated usability, will be major trends in 2022 and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A sustainable manufacturing and supply chain is increasingly important, driven not just by end-customer demand, but also by official requirements and personal convictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This tendency is also influencing 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Because 3D printing is an additive manufacturing technique, it may minimize waste during production in the correct application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' By precisely designing a part for AM, the weight of the finished item may be dramatically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, employing on-demand and decentralized 3D printing might minimize the number of components in inventory and related waste, as well as CO2 emissions during transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" As a result, an anticipation increase in the use of 3D printing as part of firms' sustainability strategies in 2022." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" To boost AM's sustainability, even more, energy usage 12 Cf (Where is 3D printing heading towards?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 6 trends to watch in 2022 - Replique, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 17 during production must be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, an increase in the use of sustainable AM materials such as recycled, reused, and biodegradable plastics are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 Benefits When the size is no longer an issue, massive pieces may be made at a low cost to replicate enormous goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Post-processing will make the finished model closer to the original product, and some of the printed pieces will be usable as end-use products (The Ocke Stool, Bathtub after Post Processing, and Children Lamps)13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' When compared to non-3D printing manual procedures, large-scale 3D printing reduces the time it takes from a concept to a full- size prototype (working with wood, foams, and fiberglass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It also significantly reduces expenditures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, the freedom to develop unique shapes and tailor-made goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' There is also the printing of a single enormous full-scale item or multiple discrete pieces that are joined to form a prototype of a major product or gadget, such as a driver seat or an MRI machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The advantage is that a prototype of the final part/product may be produced to do fit-testing with other components, design verification, and basic functional testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' When compared to other methods, large-scale 3D printing speeds up design and manufacturing processes while saving money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is another example of an industry that has traditionally produced models and things by hand utilizing various processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The use of a large-scale 3D printer allows for the manufacture of one-of-a-kind marketing and promotional materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Traditionally, in this sector, all components and models are handcrafted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' When compared to traditional techniques, large-scale 3D printing significantly lowers manual labor, saves time, and cuts prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It also allows for the freedom of design and production of bespoke goods and elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The following is a common argument in favor of 3D-printed parts over traditional metal-based ones: You may manufacture components with complicated geometry, porous 13 Cf (BigRep GmbH, 2017) Razan Elzain | Large-Scale 3D Printing Market Analysis 18 interiors, lattice structures, and membrane structures using 3D printing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' as a consequence, you can make lighter parts with less material, resulting in waste reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, this is a naive picture that ignores how 3D printing materials are sourced and what the long-term impacts will be when they become a component of hundreds of thousands of daily things ranging from electronics, wearables, home decor, and furniture to automotive and aircraft parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Parts are made from nylon, ABS, thermoplastic polyurethane, and other thermoplastics using the two most prevalent 3D printing processes, FFM and SLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Technically, pieces made from such materials may be melted and recycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Razan Elzain | Large-Scale 3D Printing Market Analysis 19 4 3D Printing Market Analysis Market participants are constantly improving 3D printing technology in response to the increased demand for 3D printing applications in the automotive, healthcare, aerospace, and military sectors for manufacturing reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The leading companies are recognizing Razan Elzain | Large-Scale 3D Printing Market Analysis 20 business transformation opportunities by using additive manufacturing in new product development processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Table 3: 3D Printing Market Report Scope Source: (3D Printing Market Size & Share Report, 2022-2030, 2022) The research anticipates revenue growth at the global, regional, and national levels, as well as an examination of the most recent market trends and prospects in each sub-segment from 2017 to 2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The research also includes shipping figures and predictions, as well as ASP qualitative analysis from 2017 to 2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Grand View Research has segmented the 3DPrintingMarketReportScope Report Attribute Details Market size value in 2022 USD 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='75billion Revenue forecast in 2030 USD 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='17 billion Growth Rate CAGRof20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='8%from2022to2030 Base year for estimation 2021 Historical data 2017 -2020 Forecast period 2022 -2030 Revenue in USD million/billion and CAGR from 2022 to Quantitative units 2030 Revenue forecast, company ranking, competitive Report coverage landscape, growth factors, and trends Component, printer type, technology, software, Segments Covered application, vertical, material, region North America;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Europe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Asia Pacific;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' South America;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Regional scope MEA U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Canada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Mexico;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' South Korea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Singapore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Country scope Brazil Stratasys, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Materialise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' EnvisionTec, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D Systems, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' GE Additive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Autodesk Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Made In Key companiesprofiled Space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Canon Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Voxeljet AG Free report customization (equivalent up to 8 analysts working days) with purchase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Addition or alteration to Customizationscope country,regional & segment scope Avail customized purchase options to meet your exact Pricing and purchase options research needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Explore purchase options Razan Elzain | Large-Scale 3D Printing Market Analysis 21 worldwide 3D printing market research based on component, printer type, technology, software, application, vertical, material, and region14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Spare parts providers are now unable to meet customer demand in the manufacturing and industrial products industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Industrial products are complicated and comprise a number of distinct parts, the majority of which will need to be changed during the equipment's lifespan." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' To suit the demands of the consumer, a replacement parts provider is expected to handle a complex network of suppliers, production, sales, and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' To keep the spare parts running, several strategic decisions must be made, as well as higher expenditures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='15 As a result, firms tend to retire an increasing number of parts each year, giving consumers inconvenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Given the constraints and expense pressures that spare parts buyers face, a growing number of businesses that purchase spare parts are turning to 3D printing to make their own components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, a growing number of customers are turning to 3D printing design and production services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing suppliers are forming alliances and collaborations with regional players and software vendors to strengthen and collaboratively develop their product portfolios, with the goal of meeting the expanding global demand for sophisticated 3D printing and providing clients with distinctive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In June 2019, for example, the firm announced a collaboration with Siemens, a worldwide technology powerhouse in automation and digitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" The goal of this collaboration is to connect ExOne's S-Max Pro with Siemens' Digital Enterprise Portfolio of software and automation technology in order to reap the benefits of Industry 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It also intends to broaden its collaboration with Siemens to include its industry-leading industrial 3D printers for tooling and production metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Aside from that, in October 2019, 3D Systems announced a cooperation deal with Antleron to offer innovative solutions and speed 14 Cf (3D Printing Market Size & Share Report, 2022-2030, 2022) 15 Cf (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='futuremarketinsights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=") Razan Elzain | Large-Scale 3D Printing Market Analysis 22 the development of bioprinting solutions in the biomedical market using 3D Systems' printing technology." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It has been noted that 3D printing solutions and service providers are focused on forming partnerships and collaborations with other regional businesses and raw material providers in the industry to enhance and jointly build their product portfolio throughout the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These partnerships and collaborations are emerging all the time in order to meet the rising demand for superior 3D printing technologies throughout the world and to offer clients quick access to 3D printing solutions and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content="1 Market Players The 3D printing market is concentrated since the bulk of the market share is held by the industry's top players." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Small and medium-sized businesses are updating their cloud services, but the top companies have captured a considerable number of users and are spending heavily on new advancements and innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Stratasys Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=',3D Systems Corporation, EOS GmbH, Electro-Optical Systems, Concept – Laser GmbH, Sisma SpA, Razan Elzain | Large-Scale 3D Printing Market Analysis 23 ExOne Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', Arcam AB (GE Aviation), SLM Solutions Group AG, Hewlett Packard Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', Materialise NV (ADR), and Proto Labs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' are only a few of the main companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" Figure 1: Major Players Source: (3D Printing Market Size, Growth, Trends (2022 - 27) | Industry Forecast, 2022) The market's leading competitors are focused on providing improved and new solutions to meet the rising demands of industries." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These prominent firms are investing in research and development to create novel services and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They are forming strategic alliances and collaborating to develop next-generation solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These businesses provide consumer-centric solutions to assist enhance corporate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Similarly, the leading companies are eager to provide a diverse selection of 3D materials in order to expand across every industrial application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In February 2022, for example, Imaginarium teamed with Ultimaker to launch a desktop and industrial 3D printer range in the Indian market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This collaboration will assist Major Players MarketConcentration Consolidated Market dominated by 1-5 major Stratasys Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' players 3DSystems Corporation 3D Printing Market ExOneCo General Electric Company (GE 4 Additive) Proto Labs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Fragmented Highly competitive market without dominant players Razan Elzain | Large-Scale 3D Printing Market Analysis 24 Ultimaker in expanding its company in India, where additive manufacturing is expected to reach a tipping point in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D Systems is then expected to further additive manufacturing innovation through collaborations and product launches in November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The firm introduced new 3D printing tools and partnered with the UK-based startup Additive Manufacturing Technologies to provide unique 3D printing workflows and develop AM software, which is appropriate for automobile components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Growth Impact made by Market Players due to their Strategies To increase their market presence, market participants are diversifying their product offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, developing enterprises are heavily spending on new product development and launches in order to maintain their market position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, numerous suppliers are concentrating on increasing the desirability of their solutions among diverse consumers through innovation and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Over the projected period, the 3D printing industry is expected to be extremely fragmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The income generated by large giants and new start-ups, as well as other established small- and medium-sized 3D printing suppliers active in the industry, is attributable to the expected high fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Companies with a market share of more than 10% are considered market leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This category comprises industry titans like 3D Systems and Stratasys, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', which are the largest and most experienced in the business and have extensive regional presence worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Companies having a market share of more than 5% but less than 10% are promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These firms are anticipated to see rapid expansion and capitalize on the possibilities provided Razan Elzain | Large-Scale 3D Printing Market Analysis 25 by the global market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These firms control around 25-30% of the worldwide market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' ExOne, SLM Solutions Group AG, Renishaw plc, and Nanoscribe are among these firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Companies with a relatively low share value of less than 5% are attempting to recruit new clients in overseas markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These firms control between 40 percent to 42 percent of the worldwide 3D printing market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Largest Global Share in the Market The North American area is projected to dominate the 3D printing sector, just as it has dominated technological adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' According to research, the United States is the most advanced country in 3D printing since November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A slew of new product releases, as well as product improvements and advancements, are likely to boost market growth even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Several 3D printing solution suppliers are extending their presence in the North American market in order to strengthen their market position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' For example, in May 2019, Italian 3D printer maker Roboze debuted its high- temperature ROBOZE Xtreme line in North America for the first time at RAPID+TCT 201916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" The area is also seeing a surge in investments in North America's healthcare, aerospace and defense, industrial, and consumer products industries, which are likely to increase considerably in the coming years." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Various government agencies, such as NASA, have noticed that significant expenditures on 3D printing technologies may significantly contribute to space applications and the development of zero-G technologies, boosting market growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Fitness trackers and smart clothing are also predicted to drive 3D printing technology in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It is expected that around 19% of broadband homes owned a wearable fitness gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, changing customer tastes and an increasing desire for 16 Cf (3D Printing Market Size, Growth, Trends (2022 - 27) | Industry Forecast, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 26 customization have created a need for a flexible band and electronics systems that might be produced utilizing 3D printing technology, propelling its expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Figure 2: 3D Printing Market - Growth Rate by Region (2021-2026) Source: (3D Printing Market Size, Growth, Trends (2022 - 27) | Industry Forecast, 2022) Over the predicted period, Asia Pacific is expected to expand the most past North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This is due to government-funded research into 3D printed solutions, as well as international investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This area is predicted to expand significantly due to unmet requirements of a large population base and improved manufacturing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='4 3D Printing in Germany and UK In the United Kingdom and Germany, the 3D printing business is especially vulnerable to economic swings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In the region, it is largely employed as a prototyping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 17 Cf (Research, 2022) Regional Growth Rates High Mid Low Source: Mordor Intelligence Razan Elzain | Large-Scale 3D Printing Market Analysis 27 Over the previous two decades, 3D printing industry participants have lowered R&D spending, resulting in a decline in demand for prototype services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' By bringing 3D printing processes in-house, the production time for 3D printing manufacturing processes may be cut to days, if not hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a consequence, the whole product design and production cycle are reduced, and repetitive redesigning, as well as lengthier wait periods, are avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, the aerospace and defense industries are expected to continue to see increased use of 3D printing software and solutions in Europe throughout the projection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Bringing 3D printing techniques in-house, on the other hand, maybe difficult and costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, not all businesses can afford to manufacture their own 3D printed items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='5 3D Printing Market Size Forecast with COVID-19 Impact The introduction of COVID-19 has had a significant impact on the 3D printing sector, owing to a scarcity of competent individuals to operate the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, the pandemic had an influence on the economy and the operation of manufacturing sectors, which considerably stimulated market growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In contrast, the market had a revenue impact as a result of worldwide manufacturing unit shutdowns18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" The COVID-19 pandemic has disrupted the ecosystem's supply chains." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In terms of the market, the pandemic has had a wide-ranging impact on industries such as healthcare, automotive, aerospace, consumer electronics, retail, energy and power, oil & gas, construction, jewelry, food & culinary, and education, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' During the pandemic, the healthcare industry experienced an unprecedented surge in demand for personal protection equipment including face masks, shields, and ear bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, the demand for venturi valves and regulators that help patients breathe has increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Aside from that, e-commerce has thrived due to regional lockdowns and logistical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing firms may directly print and distribute items and have total control over the materials used in 18 Cf (3D Printing Market Size, Share | Trends & Forecast - 2030, 2022) Razan Elzain | Large-Scale 3D Printing Market Analysis 28 manufacturing, storage conditions, and distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They can also offer digital files that allow customers to print their own devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The COVID-19 pandemic has also hampered multiple enterprises in a variety of industries, including automotive, aerospace and military, consumer electronics, food and drinks, and retail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Import and export restrictions from China and other APAC nations have prevented industrial facilities in North America and Europe from operating19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This has resulted in considerable fluctuations in overhead costs and overall product quality for these producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The International Monetary Fund (IMF) forecast a more than 8% drop in GDP owing to the coronavirus pandemic in April 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' One of the leading manufacturers of Binder Jetting systems reported a huge drop of 27% in the second quarter of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='6 3D Printing Application Analysis Figure 3: Global 3D Printing Market Share - by Application 2021 Source: (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='fortunebusinessinsights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Prototype held a substantial market share in 2021 due to the broad acceptance of the prototyping process across many vertical sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Prototyping enables firms to attain higher 19 Cf (Market, 2022) Prototyping Production Proof of Concept Others (R&D, Tooling) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3% Razan Elzain | Large-Scale 3D Printing Market Analysis 29 precision and consistently generate finished goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This technology aids in the production of 3D CAD models and prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' During the projection period, the output is expected to expand rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' While businesses move their traditional production units to advanced manufacturing processes, the use of this technology to produce complicated and low-volume products is likely to increase throughout the projection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='7 Rise in Investment from Governmental bodies in 3D Printing Projects Because 3D printers are considered a growing global market, government organizations all around the world are backing 3D printer producers with specific policies to help them overcome financial difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In addition, the government is offering specific educational workshops to help people overcome technical and economic difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" These authorities' policies contribute to a healthy environment for the design, development, and deployment of 3D printers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing, also known as additive manufacturing, is a method of creating prototypes or working models of products by layering materials such as plastic, resin, thermoplastic, metal, fiber, or ceramic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The model to be printed is created by the computer using software, which then sends instructions to the 3D printer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The main problem that producers may encounter is the availability of raw materials, as it is regarded as a highly specialized industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='8 Things hindering the 3D Printing Market Growth One of the primary issues now confronting 3D printer producers, particularly those in developing nations, is a scarcity of competent individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The cost of raw materials is another aspect that is predicted to stymie the industry, as the bulk of materials is created by patent-holding corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, nations like Brazil, India, Australia, and others may 20 Cf (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='fortunebusinessinsights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Razan Elzain | Large-Scale 3D Printing Market Analysis 30 encounter cost issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' These are some of the issues that the Global 3D Printer Market is now facing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='9 Market Opportunities The ongoing transition to industrial automation will provide future prospects for macro 3D printers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' For example, the future of the construction sector will feature 3D printers, drones, and robotic bulldozers, which will aid in the creation of robotic construction sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Furthermore, Local Motors, a US-based business, has already produced Strati, a 3D-printed automobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Spark, an open collaboration platform for 3D printing, assists firms in developing various types of 3D printing technologies by offering extensible Application Programming Interface (API) for all phases of the 3D printing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Companies may use this program to create 3D models for any 3D printer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This sort of platform is assisting in raising awareness and promoting macro 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='21 21 Cf (Persistence Market Research, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Razan Elzain | Large-Scale 3D Printing Market Analysis 31 5 Final Consideration 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='1 Results and critical reflections With the aid of 3D printing, the world is constantly evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The usage of 3D printing for therapeutic purposes today is astounding, but what the future holds is uncertain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' yet, additive layer manufacturing will play a significant role in fixing our issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing truly is boundless, and we have just scratched the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' there is much more to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As seen across the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Although 3D printing bones is still in its early stages and is constantly being improved and adjusted, it has already improved the lives of many people throughout the world, particularly in Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It is obvious that the more money and research put into 3D printing, the further it will lead us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D is inherently unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='22 The year 2022 appears to be “The year of 3D Printing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' A slew of new discoveries will propel AM industrialization ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Printing times are very long, which is one of the key reasons why AM has not yet achieved large-scale manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Injection molding enables the production of standardized components in a consistent and timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, 3D printing is still a relatively young technology that is rapidly evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The cost of such printers is falling as manufacturing speed, quality, and build plate size improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing will also enable a more resilient and sustainable supply chain strategy, making the technology more appealing to businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Security, quality assurance, and interoperability advancements will make the technology more accessible to a broader spectrum of companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Even though we are still a long way from mass manufacturing, we anticipate potential production quantities for 3D printing increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It is widely assumed that 3D printing will be a transformative force in manufacturing, whether positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Despite worries about counterfeiting, numerous firms are 22 Cf (3d printing is limitless, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Razan Elzain | Large-Scale 3D Printing Market Analysis 32 already employing the technology to make sophisticated components in a repeatable manner, such as in automotive and aerospace production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='23 As 3D printers grow more inexpensive, they will eventually be employed for local, small-scale production, obviating the need for many forms of supply networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Consumer units for home usage will even be possible, allowing end customers to simply download and print a design for the product they desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The traditional manufacturing industry will have significant hurdles in adapting to these developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, the prospects for technology and engineering are certainly enormous, as are the creative possibilities in product design and printing material formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Recent scientific breakthroughs and uses of 3D printing indicate that the technology has the potential to transform many aspects of daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" AM's influence on supply chains, for example, takes various forms, including simpler production processes, decreased material waste for leaner manufacturing, improved flexibility, lower prices, faster response to demand, and the capacity to decentralize production." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='2 Implications for further research For further research, 3D printing technology must first overcome a major challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' There are still perspectives on 3D printing from its beginnings more than 5 years ago to supply more knowledge among people about all of its capabilities and advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Today, the reality is different, and the argument that 3D printing increases raw material use could be deceptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The exact opposite is said to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D printing could have a significant influence on trash logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' However, favorably, a well-designed 3D printing production process leads to considerable reductions in resource consumption and waste generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' because manufacturing will be closer to the consumer, packing and transportation materials will be used less often than in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The basic nature of the additive process results in reduced waste in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 23 Cf (AZoM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com, 2012) 24 Cf (Kubáč and Kodym, 2017) Razan Elzain | Large-Scale 3D Printing Market Analysis 33 Another incorrect perception of 3D printing is that its primary application is for plastic things, generally as components to finished products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' That is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The key advantage of the technology is the ability to employ a variety of materials other than plastics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Metal 3D Printing, for example, has the capability and capacity to build delicate, streamlined components with physical qualities that can sometimes exceed those of parts made traditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' As a result, the technology has the potential to totally transform the way we manufacture crucial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It could be used to make lightweight items with distinctive shapes that reduce material waste and energy usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In other words, there is a "knowledge gap" between the existing technology and the people that must be bridged before 3D printing can become a mainstream technique of product creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='3 Implications for practice Not all 3D printers are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They vary depending on the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They differ depending on the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' They differ in terms of output and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=" As a result, while embarking on a 3D Printing journey, it is critical to have a clear grasp of what your supply chain looks like, what the customer's expectations are, and where it receives its information from in order to pick the appropriate equipment and procedures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Discussions about design, iteration, production, warehousing, shipping, warehousing, and distribution to customers in a classic manufacturing setting and supply chain have been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' The 3D printed supply chain has the potential to be considerably shorter, more efficient, and tailored to the needs of the consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' It reduces the design cycle, optimizes the iteration cycle, improves the manufacturing cycle by generating unique items as needed, eliminates the warehouse, and transforms the distribution routes into a 3D Printed Supply Chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' This often necessitates a full rethinking of your business as well as a possibly more collaborative engagement with your consumer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' In fact, a company model may alter so drastically that it begins licensing the intellectual property or data, allowing the client to manufacture the goods on their premises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Razan Elzain | Large-Scale 3D Printing Market Analysis 34 The ramifications are enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' If a mere examination has trainers, sneakers, and shoes, have an influence on the materials produced by major chemical plants, logistics, the way money is exchanged, the retail environment, and the size and type of the factories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Changes the labor component, possibly altering the dynamics and ethical sourcing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Trade and tariffs have macroeconomic implications, as does the collecting of duty money for national governments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='25 For further reading of the Logistics Engineering and Technologies Group please refer to Auerbach & Uygun, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Keßler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Keßler & Uygun, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Kortmann & Uygun, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Droste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Wötzel, 2009, Jungmann & Uygun, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Keßler & Uygun, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Kuhn, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Luft 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Schmidt, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Wagner, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Liesebach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2012c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Straub, 2012, Besenfelder et al, 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Besenfelder et al 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Güller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Scholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Straub, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Güller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Mevenkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Karakaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Reynolds, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Güller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Reynolds, & Uygun, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Ilie, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Lyutov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Nosheen & Uygun, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Sommerfeld & Uygun, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Jafri, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Özgür et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Lyutov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Ahsan, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun & Rustemaj, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Uygun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022, Merten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022a, Merten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022b, Lyutov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 25 Cf (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='linkedin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=') Razan Elzain | Large-Scale 3D Printing Market Analysis 35 Table of references 3d printing is limitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3d printing is limitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' [online] Available at: https://3dprintingislimitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='weebly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com/ [Accessed 13 May 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' All3DP (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2019 Best Large-Format 3D Printers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' [online] All3DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Available at: https://all3dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com/1/best-large-3d-printer-large-format-scale-3d-printers/ [Accessed 28 Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' All3DP (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' All 10 Types of 3D Printing Technology in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' [online] All3DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Available at: https://all3dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content='com/1/types-of-3d-printers-3d-printing-technology/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' Allied Market Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' 3D Printing Market Size, Share | Trends & Forecast - 2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAyT4oBgHgl3EQfyPlA/content/2301.00680v1.pdf'} +page_content=' [online] Available at: 0 +where P(c1, · · · , cs) is a positive polynomial in the Chern classes c1, · · · , cs +of any quotient bundle Q of E|V , V ⊂ X is any complex analytic subvariety. +Block-Gieseke [BG71] proved all Chern classes of an ample vector bundle +satisfy (1.1), Fulton-Lazarsfeld [FL83, Theorem I] extended Block-Gieseke’s +result and proved all Schur polynomials are numerically positive for am- +ple vector bundles. For nef vector bundles over compact Kähler manifolds, +Demailly-Peternell-Schneider [DPS94, Theorem 2.5] proved the numerical +semi-positivity of all Schur polynomials. +Griffiths [Gri70, Page 247] also conjectured (1.1) holds on the level of the +differential forms, which can be reformulated as follows, see [Fin22, Page +1541, Question of Griffiths]. +Question 1.1 (Griffiths). Let P ∈ R [c1, . . . , cr] be a non-zero non-negative +linear combination of Schur polynomials of weighted degree k. Are the forms +P +� +c1(E, hE), . . . , cr(E, hE) +� +weakly positive for any Griffiths positive vector +bundle (E, hE) over a complex manifold X of dimension n, n ⩾ k? + +POSITIVITY OF SCHUR FORMS +3 +Recall that a real (k, k)-form u is called weakly positive (resp. +non- +negative) if u ∧ (√−1)(n−k)2β ∧ β > 0 (resp. +≥ 0) for any non-zero de- +composable (k, 0)-form β = β1 ∧ · · · ∧ βn−k, where βi, 1 ≤ i ≤ n − k, are +(1, 0)-forms, see Definition 3.1 for the definitions of (weakly) positive (resp. +non-negative) vector bundles. +Griffiths [Gri70, Page 249] proved that the second Chern form of a Griffiths +vector bundle is positive by using Schwarz inequality. Guler [Gul12, Theo- +rem 1.1] verified Question 1.1 for all signed Segre forms, and Diverio-Fagioli +[DF22] showed the positivity of several other polynomials in the Chern forms +of a Griffiths (semi)positive vector bundle by considering the pushforward +of a flag bundle, including the later developments [Fag22a, Fag22b]. +See +Xiao [Xia22] and Ross-Toma [RT19] for other related results of ample vector +bundles. +For Bott-Chern non-negative vector bundles, Bott-Chern [BC65, Lemma +5.3, (5.5)] proved that all Chern forms are non-negative, Li [Li21, Proposition +3.1] extended Bott-Chern’s result and obtained all Schur forms are non- +negative. +Later, Finski [Fin22, Theorem 2.15] proved the equivalence of +Bott-Chern non-negativity and dual Nakano non-negativity. Moreover, using +a purely algebraic method, Finski [Fin22, Section 3.4] proved that all Schur +forms of a Nakano non-negative vector bundle are non-negative. For (dual) +Nakano positive vector bundles, Finski [Fin22, Theorem 1.1] proved that all +Schur forms are positive by the refinement of the determinantal formula of +Kempf-Laksov on the level of differential forms. However, as pointed out +by Finski [Fin22, Remark 3.18], the above algebraic method can be used +to deal with the case of non-negativity, while for the positivity statement, +it is not clear if one can refine the algebraic method because there is no +similar criterion for (dual) Nakano positivity (see [Fin22, Remark 2.16]) and +there is little known about the specific structure of the forms defined in +[Fin22, (3.83)]. This motivates the author to study the question of Griffiths +(Question 1.1) by developing the purely algebraic method. +In [Fin22, Section 2.3], Finski introduced the definition of decomposably +positive vector bundles, see Definition 2.2, which is a generalization of both +Nakano positivity and dual Nakano positivity, and coincides with Griffiths +positivity for n · r ≤ 6. So it is natural to wonder if Question 1.1 holds for +such positive vector bundles. In this paper, we introduce two new notions of +positivity of vector bundles, called strongly decomposable positivity of type +I and type II, see Definition 2.4 and Definition 2.7. They fall in between +(dual) Nakano positivity and decomposable positivity. Roughly speaking, +(E, hE) is strongly decomposably positive of type I if, for any x ∈ X, there +is a decomposition T 1,0 +x X = Ux ⊕ Vx such that it is Nakano positive in the +subspaces Ex⊗Ux and dual Nakano positive in the subspace Ex⊗Vx, and the +cross curvature terms vanish, see Definition 2.4. Using the purely algebraic +method, we answer Question 1.1 affirmatively for strongly decomposably +positive vector bundles of type I. + +4 +XUEYUAN WAN +Theorem 1.2. Let (E, hE) be a strongly decomposably positive vector bundle +over a complex manifold X, rankE = r, and dim X = n. Then the Schur +form Pλ(c(E, hE)) is weakly positive for any partition λ ∈ Λ(k, r), k ≤ n +and k ∈ N. +From [HK74, Theorem 1.2], a real (k, k)-form u is non-negative if and only +if u can be written as +(1.2) +u = +N +� +s=1 +( +√ +−1)k2αs ∧ αs +for some (k, 0)-forms αs, 1 ≤ s ≤ N. By (4.13) and (4.14), the Schur form +has the following form +(1.3) +Pλ(c(E, hE)) = +� 1 +2π +�k � 1 +k! +�2 +( +√ +−1)k2 � +ρ,t,c,ϵ +(−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ. +where ψρtcϵ is a (|ϵ|, k − |ϵ|)-form and is defined by +(1.4) +ψρtcϵ := +� +σ∈Sk +qσt +k� +j=1 +(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj. +It is not clear to express (1.3) as in (1.2) in the general case. Hence, it seems +hard to prove that the Schur form Pλ(c(E, hE)) is a positive (k, k)-form by +using the algebraic method. However, if (E, hE) is Nakano positive or dual +Nakano positive, it is equivalent to A = 0 or B = 0. For example, for A = 0, +(1.4) gives +ψρtcϵ1 = +� +σ∈Sk +qσt +k� +j=1 +Bρσ(j)cj, +ϵ1 = (1, · · · , 1) +and ψρtcϵ = 0 for any ϵ ̸= ϵ1. Then the Schur form is given by +Pλ(c(E, hE)) = +� 1 +2π +�k � 1 +k! +�2 +( +√ +−1)k2 � +ρ,t,c +ψρtcϵ1 ∧ ψρtcϵ1, +which satisfies (1.2) because ψρtcϵ1 is a (k, 0)-form. As a result, we can give +an algebraic proof of the following positivity of Schur forms for (dual) Nakano +positive vector bundles. +Theorem 1.3 (Finski [Fin22, Theorem 1.1]). Let (E, hE) be a (dual) Nakano +positive vector bundle of rank r over a complex manifold X of dimension n. +Then for any k ∈ N, k ⩽ n, and λ ∈ Λ(k, r), the (k, k)-form Pλ +� +c +� +E, hE�� +is positive. +Inspired by the definition of the strongly decomposable positivity of type +I, it is natural to define the strongly decomposable positivity of type II by +decomposing the vector bundle, which is the direct sum of Naknao positive +and dual Nakano positive vector bundles point-wisely, see Definition 2.7. By + +POSITIVITY OF SCHUR FORMS +5 +Littlewood-Richardson rule (see [Ful97, Chapter 5]), the Schur class of direct +sum E ⊕ F can be given by +Pλ(c(E ⊕ F)) = +� +µ,ν +cλ +µνPµ(c(E))Pν(c(F)), +where cλ +µν(≥ 0) is a Littlewood-Richardson coefficient. In this paper, by re- +fining the above identity on the level of differential forms and using Theorem +1.3, we obtain +Theorem 1.4. Let (E, hE) be a strongly decomposably positive vector bundle +of type II over a complex manifold X, rankE = r, and dim X = n. Then +the Schur form Pλ(c(E, hE)) is positive for any partition λ ∈ Λ(k, r), k ≤ n +and k ∈ N. +Remark 1.5. Comparing Theorem 1.2 with Theorem 1.4, it is natural to ask +if the Schur forms are positive for a strongly decomposably vector bundle of +type I. +The article is organized as follows: In Section 2, we will define two types of +strongly decomposably positive vector bundles, which are the generalizations +of both Nakano positivity and dual Nakano positivity, and are stronger than +decomposable positivity. In Section 3, we will recall the positivity notions +for differential forms and show the positivity of the product of two positive +forms. +In Section 4, we will give a criterion of a strongly decomposably +positive vector bundle of type I, recall the definitions of Schur forms and +Griffiths cone, then prove the weak positivity of Schur forms, Theorem 1.2 +and Theorem 1.3 are established in this section. In Section 5, we will give +a criterion of a strongly decomposably positive vector bundle of type II and +prove the positivity of Schur forms, Theorem 1.4 is established in this section. +Acknowledgements. The author would like to thank Siarhei Finski for many +helpful discussions. +2. Strongly decomposably positive vector bundles +This section will define two types of strongly decomposably positive vector +bundles. +2.1. Connections and curvatures. In this subsection, we will recall the +definitions of the Chern connection and its curvature for a Hermitian holo- +morphic vector bundle. One can refer to [Kob87, Chapter 1] for more details. +We will use the Einstein summation convention in this paper. +Let π : (E, hE) → X be a Hermitian holomorphic vector bundle over a +complex manifold X, rankE = r and dim X = n. Let ∇E be the Chern +connection of (E, hE), which preserves the metric hE and is of (1, 0)-type. +With respect to a local holomorphic frame {ei}1≤i≤r of E, one has +(2.1) +∇Eei = θj +i ej, + +6 +XUEYUAN WAN +where θ = (θj +i ) (j row, i column) is the connection form of ∇E. +More +precisely, +θj +i = ∂hi¯kh +¯kj, +where hi¯k := h(ei, ek). In terms of matrix form, it is +θ⊤ = ∂h · h−1. +Considering e = (e1, · · · , er) as a row vector, then (2.1) can be written as +∇Ee = e · θ. +Let RE = (∇E)2 ∈ A1,1(X, End(E)) denote the Chern curvature of (E, hE), +and write +RE = Rj +iej ⊗ ei ∈ A1,1(X, End(E)), +where R = (Rj +i) (j row, i column) is the curvature matrix whose entries are +(1, 1)-forms, {ei}1≤i≤r denotes the dual frame of {ei}1≤i≤r. The curvature +matrix R = (Rj +i) is given by +Rj +i = dθj +i + θj +k ∧ θk +i = ¯∂θj +i . +If {˜ei}1≤i≤r is another local holomorphic frame of E with ˜ei = aj +iej, then +˜e = e · a, +where ˜e := (˜e1, · · · , ˜er) and a = (ai +j). Denote by �R the curvature matrix +with respect to the local frame {˜ei}1≤i≤r. Then +(2.2) +�R = a−1 · R · a. +Let {zα}1≤α≤n be local holomorphic coordinates of X. Write +Rj +i = Rj +iα¯βdzα ∧ d¯zβ +and denote +Ri¯j := Rk +i hk¯j = Ri¯jα¯βdzα ∧ d¯zβ, +so that +Ri¯jα¯β = Rk +iα¯βhk¯j = −∂α∂¯βhi¯j + h +¯lk∂αhi¯l∂¯βhk¯j, +where ∂α := ∂/∂zα and ∂¯β := ∂/∂¯zβ. +2.2. Strongly decomposable positivity. This subsection will define two +types of strongly decomposably positive vector bundles. Firstly, we recall +the following definitions of Nakano positive and dual Nakano positive vector +bundles. +Definition 2.1 ((Dual) Nakano positive). A Hermitian holomorphic vector +bundle (E, hE) is called Nakano positive (resp. non-negative) if +Ri¯jα¯βuiαujβ > 0 (resp. ≥ 0) +for any non-zero element u = uiαei ⊗∂α ∈ E ⊗T 1,0X. (E, hE) is called dual +Naknao positive (non-negative) if +Ri¯jα¯βv¯jαv¯iβ > 0 (resp. ≥ 0) + +POSITIVITY OF SCHUR FORMS +7 +for any non-zero element v = v¯jα¯ej ⊗ ∂α ∈ E ⊗ T 1,0X. +In [Fin22, Definition 2.18], S. Finski introduced the following new notion +of positivity for vector bundles: decomposable positivity. +Definition 2.2 (Decomposably positive). A Hermitian vector bundle +� +E, hE� +is called decomposably non-negative if for any x ∈ X, there is a number +N ∈ N and linear (respectively sesquilinear) forms l′ +p : T 1,0 +x X ⊗ Ex → C +(respectively lp : T 1,0 +x X ⊗ Ex → C), p = 1, . . . , N, such that for any v ∈ +T 1,0 +x X, ξ ∈ Ex, we have +1 +2π +� +RE +x (v, ¯v)ξ, ξ +� +hE = +N +� +p=1 +|lp(v, ξ)|2 + +N +� +p=1 +��l′ +p(v, ξ) +��2 . +We say that it is decomposably positive if, moreover, +� +RE +x (v, ¯v)ξ, ξ +� +hE ̸= 0 +for v, ξ ̸= 0. +Remark 2.3. From the above definition, a (dual) Nakano positive vector +bundle must be decomposably positive. From [Fin22, Proposition 2.21], for +n · r ≤ 6 decomposable positivity is equivalent to Griffiths positivity, i.e. +(2.3) +Ri¯jα¯βvi¯vjξα¯ξβ > 0 +for any non-zero ξ = ξα∂α ∈ T 1,0X and v = viei ∈ E. +Decomposable +positivity is strictly stronger than Griffiths positivity for all other n, r ̸= 1. +Next, we will introduce the notion of strongly decomposable positivity, +which falls in between (dual) Nakano positivity and decomposable positivity. +Definition 2.4 (Strongly decomposably positive of type I). A Hermitian +vector bundle (E, hE) is called a strongly decomposably positive (resp. non- +negative) vector bundle of type I if, for any x ∈ X, there exists a decompo- +sition T 1,0 +x X = Ux ⊕ Vx such that +Ri¯jα¯βuiαujβ > 0(resp. ≥ 0), Ri¯jα¯βuiαv′jβ = 0, Ri¯jα¯βv¯jαv¯iβ > 0(resp. ≥ 0) +for any non-zero elements u = uiαei⊗∂α ∈ Ex⊗Ux, v′ = v′jβej⊗∂β ∈ Ex⊗Vx +and v = v¯jα¯ej ⊗ ∂α ∈ Ex ⊗ Vx. +Remark 2.5. From the above definition, a strongly decomposably positive +vector bundle of type I means that it is Nakano positive in the subspace Ex⊗ +Ux, dual Nakano positive in the subspace Ex ⊗ Vx, and the cross curvature +terms vanish. +For any point x ∈ X, if Vx = {0} (resp. +Ux = {0}), then strongly +decomposable positivity of type I is exactly Nakano positive (resp. +dual +Nakano positive). +Example 2.6. Let π1 : (E1, hE1) → X1 be a Nakano positive vector bundle +and π2 : (E2, hE2) → X2 be a dual Nakano positive vector bundle. Denote by +pi : X1 × X2 → Xi, i = 1, 2, the natural projection, then +π : (p∗ +1E1 ⊕ p∗ +2E2, p∗ +1hE1 ⊕ p∗ +2hE2) → X1 × X2 + +8 +XUEYUAN WAN +is a strongly decomposably non-negative vector bundle of type I. +From Definition 2.4, a strongly decomposably positive vector bundle (E, hE) +of type I means that there is a decomposition of holomorphic tangent bundle +T 1,0 +x X = Ux ⊕ Vx, such that (E, hE) is Nakano positive in Ex ⊗ Ux and dual +Naknao positive in Ex ⊗ Vx. Naturally, one may define another strongly de- +composable positivity of vector bundles by decomposing the vector bundle. +More precisely, +Definition 2.7 (Strongly decomposably positive of type II). We call (E, hE) +a strongly decomposably positive (resp. non-negative) vector bundle of type +II if, for any x ∈ X, there is an orthogonal decomposition of (Ex, hE|Ex), +Ex = E1,x ⊕ E2,x, such that the Chern curvature RE +x has the following form +RE +x = +�RE +x |E1,x +0 +0 +RE +x |E2,x +� +, +and Ri¯jα¯βuiαujβ > 0 (resp. ≥ 0) for any non-zero u = uiαei ⊗ ∂α ∈ E1,x ⊗ +T 1,0 +x X, Ri¯jα¯βvi¯βvj ¯α > 0 (resp. ≥ 0) for any non-zero v = vi¯βei ⊗ ∂¯β ∈ +E2,x ⊗ T 0,1 +x X. +A simple example of a strongly decomposably vector bundle of type II is +as follows. +Example 2.8. Let (E, hE) be a Nakano positive vector bundle and (F, hF ) +be a dual Nakano positive vector bundle over a complex manifold X. Then +(E ⊕F, hE ⊕hF ) is a strongly decomposably positive vector bundle of type II. +By Definition 2.4 and Definition 2.7, a strongly decomposably positive +vector bundle is defined as follows. +Definition 2.9 (Strongly decomposably positive). A Hermitian vector bun- +dle (E, hE) is called strongly decomposably positive if it is a strongly decom- +posably positive vector bundle of type I or type II. +Similarly, one can define strongly decomposably negative (non-positive) +vector bundles. Note that the dual of the Nakano positive (negative) vector +bundle is dual Nakano negative (positive), so +Proposition 2.10. A Hermitian vector bundle (E, hE) is a strongly decom- +posably positive (non-negative) vector bundle of type I (type II) if and only +if (E∗, hE∗) is a strongly decomposably negative (non-positive) vector bundle +of type I (type II). +Remark 2.11. Let (E, hE) be a strongly decomposably positive vector bun- +dle, and Q be a quotient bundle of E. The curvature of the bundle Q is +given by +RQ = RE�� +Q + C ∧ C +⊤ +for some matrix C whose entries are (1, 0)-forms. +From the criteria of +strongly decomposably positive vector bundles, Theorem 4.2 and Theorem + +POSITIVITY OF SCHUR FORMS +9 +5.2, and the above curvature formula of quotient bundles, the quotient bun- +dle (Q, hQ) cease to be a strongly decomposably positive vector bundle in +general. +2.3. Relation to decomposable positivity. From the equivalent descrip- +tions of Nakano non-negative and dual non-negative due to S. Finski [Fin22, +Theorem 2.15 and 2.17], one has +Proposition 2.12. A Hermitian vector bundle (E, hE) is decomposably non- +negative if and only if the Chern curvature matrix has the following form +(2.4) +R = −B ∧ B +⊤ + A ∧ A +⊤ +with respect to a unitary frame, where A (resp. B) is a r × N matrix with +(1, 0)-forms (resp. (0, 1)-forms) as entries. +Proof. From Definition 2.2, (E, hE) is decomposably positive if for any x ∈ +X, there is a number N ∈ N and linear (respectively sesquilinear) forms +l′ +p : T 1,0 +x X ⊗ Ex → C (respectively lp : T 1,0 +x X ⊗ Ex → C), p = 1, . . . , N, such +that for any v ∈ T 1,0 +x X, ξ ∈ Ex, we have +(2.5) +� +RE +x (v, ¯v)ξ, ξ +� +hE = +N +� +p=1 +|lp(v, ξ)|2 + +N +� +p=1 +��l′ +p(v, ξ) +��2 . +We denote +lp(v, ξ) = lip¯βvi¯ξβ, +l′ +p(v, ξ) = l′ +ipαviξα, +and set A = (Ajp) and B = (Bjp) by +Ajp := Ajpαdzα = ljp¯αdzα, +Bjp := Bjp¯βd¯zβ = l′ +jpβd¯zβ. +Then (2.5) is equivalent to +Ri¯jα¯β = +N +� +p=1 +lip¯βljp¯α + +N +� +p=1 +l′ +ipαl′ +jpβ += +N +� +p=1 +AjpαAipβ + +N +� +i=1 +Bjp¯βBip¯α. +(2.6) +With respect to a unitary frame, Rj +iα¯β = Ri¯jα¯β and (2.6) is equivalent to +R = −B ∧ B +⊤ + A ∧ A +⊤, +which completes the proof. +□ +From Proposition 2.12 and Definition 2.2, (E, hE) is decomposably pos- +itive if and only if (2.4) holds and +� +RE +x (v, ¯v)ξ, ξ +� +hE ̸= 0 for v, ξ ̸= 0. By +Theorem 4.2 and Theorem 5.2, we have +Corollary 2.13. If (E, hE) is a strongly decomposably positive vector bundle +of type I or type II, then (E, hE) is decomposably positive. + +10 +XUEYUAN WAN +On the other hand, from Theorem 4.2 and Theorem 5.2, the two types of +strongly decomposably positive vector bundles can not contain each other. +Both are the generalizations of (dual) Nakano positive vector bundles and +are stronger than decomposable positivity. One can refer to the following +Figure 1. +Decomposably positive +Griffiths positive +Nakano positive +dual Nakano positive +Strongly decomposably positive (type I) +Strongly decomposably positive (type II) +Figure 1. Relations of several notions of positivity +Remark 2.14. Note that Griffiths positivity, decomposable positivity, and +(dual) Nakano positivity are stable under addition. In particular, decompos- +able positivity is spanned by Naknao positivity and dual Nakano positivity. +Hence, it is also spanned by the strongly decomposable positivity of type I +(type II). So the strongly decomposable positivity of type I and type II are +not stable under addition. +3. Positivity notions for differential forms +In this section, we will recall positivity notions for differential forms. For +more details, one can refer to [Fag22a, Section 1.1] and [HK74, Fin22]. +Let V be a complex vector space of dimension n and let (e1, · · · , en) be +a basis of V . +Denote by (e1, · · · , en) the dual basis of V ∗. +Let Λp,qV ∗ +denote the space of (p, q)-forms, and Λp,p +R V ∗ ⊂ Λp,pV ∗ be the subspace of +real (p, p)-forms. +Definition 3.1. A form ν ∈ Λn,nV ∗ is called a non-negative (resp. positive) +volume form if ν = τ√−1e1 ∧ e1 ∧ · · · ∧ √−1en ∧ en for some τ ∈ R, τ ≥ 0 +(resp. τ > 0). +Now we set q = n − p, a (q, 0)-form β is called decomposable if β = +β1 ∧ · · · ∧ βq for some β1, . . . , βq ∈ V ∗. +Definition 3.2. A real (p, p)-form u ∈ Λp,p +R V ∗ is called +• weakly non-negative (resp. +weakly positive), if for every non-zero +β ∈ Λq,0V ∗ decomposable, u ∧ (√−1)q2β ∧ ¯β is a non-negative (resp. +positive) volume form; +• non-negative (resp. positive), if for every non-zero β ∈ Λq,0V ∗, u ∧ +(√−1)q2β ∧ ¯β is a non-negative (resp. positive) volume form; + +POSITIVITY OF SCHUR FORMS +11 +• strongly non-negative (resp. strongly positive) if there are decompos- +able forms α1, . . . , αN ∈ Λp,0V ∗ such that u = �N +s=1(√−1)p2αs ∧αs. +Remark 3.3. Let WPpV ∗, PpV ∗ and SPpV ∗ denote respectively the closed +positive convex cones contained in Λp,p +R V ∨ spanned by weakly non-negative, +non-negative and strongly non-negative forms. Then +(3.1) +SPpV ∗ ⊆ PpV ∗ ⊆ WPpV ∗. +Note that the above two inclusions become equalities for p = 0, 1, n − 1, n, +and the inclusions are strict for 2 ≤ p ≤ n − 2, see e.g. [Fag22a, Remark 1.7, +1.8] and [HK74]. +Proposition 3.4. If u is a positive (k, k)-form and v is a positive (l, l)-form, +k + l ≤ n, then u ∧ v is a positive (k + l, k + l)-form. +Proof. By [HK74, Corollary 1.3 (a)], u∧v is a non-negative (k+l, k+l)-form, +i.e. for any non-zero β ∈ Λn−k−l,0V ∗, +(3.2) +u ∧ v ∧ ( +√ +−1)(n−k−l)2β ∧ ¯β ≥ 0. +By [HK74, Theorem 1.2], v has the following form +v = +N +� +s=1 +( +√ +−1)l2αs ∧ αs +for some (l, 0)-forms αs, 1 ≤ s ≤ N. So +u ∧ v ∧ ( +√ +−1)(n−k−l)2β ∧ ¯β = +N +� +s=1 +u ∧ ( +√ +−1)l2αs ∧ αs ∧ ( +√ +−1)(n−k−l)2β ∧ ¯β += +N +� +s=1 +u ∧ ( +√ +−1)l2αs ∧ β ∧ αs ∧ β. +Thus the equality in (3.2) holds if and only if +u ∧ ( +√ +−1)l2αs ∧ β ∧ αs ∧ β = 0, +1 ≤ s ≤ N, +which is equivalent to +αs ∧ β = 0, +1 ≤ s ≤ N. +Thus +v ∧ ( +√ +−1)(n−k−l)2β ∧ ¯β = +N +� +s=1 +( +√ +−1)l2αs ∧ β ∧ αs ∧ β = 0, +which contradicts the positivity of v. Hence +u ∧ v ∧ ( +√ +−1)(n−k−l)2β ∧ ¯β > 0 +for any non-zero β ∈ Λn−k−l,0V ∗, i.e. u ∧ v is positive. +□ + +12 +XUEYUAN WAN +Let X be a complex manifold of dimension n and denote by Ap,q(X) the +space of all smooth (p, q)-forms. +Definition 3.5. A real (p, p)-form α ∈ Ap,p(X) is called weakly non-negative +(weakly positive), non-negative (positive), or strongly non-negative (strongly +positive) if for any x ∈ X, αx ∈ Λp,p +R (T 1,0 +x X)∗ is weakly non-negative (weakly +positive), non-negative (positive), or strongly non-negative (strongly positive) +respectively. +4. Strongly decomposable positivity of type I +In this section, we will give a criterion of strongly decomposably positive +vector bundles of type I and prove the weak positivity of Schur forms. +4.1. A criterion of type I positivity. In this subsection, following S. +Finski’s approach [Fin22, Theorem 2.15, 2.17], we will give a criterion for +the strongly decomposable positivity (non-negativity) of type I by using M.- +D. Choi’s results. +Let (E, hE) be a strongly decomposably non-negative vector bundle of +type I. For any x ∈ X, there exists a decomposition T 1,0 +x X = Ux ⊕ Vx. One +can take local holomorphic coordinates {z1, · · · , zn} around x such that +Ux = spanC{∂1, · · · , ∂n0}, Vx = spanC{∂n0+1, · · · , ∂n}, +where n0 := dim Ux and recall that ∂α := ∂/∂zα. Let {ei}1≤i≤r be a local +holomorphic frame of E such that +hi¯j(x) = hE(ei(x), ej(x)) = δij. +With respect to {zα}1≤α≤n and {ei}1≤i≤r, the Chern curvature matrix R = +(Rj +i) at x ∈ X has the following expression +Rj +i = Rj +iα¯βdzα ∧ d¯zβ += +n0 +� +α,β=1 +Rj +iα¯βdzα ∧ d¯zβ + +n +� +α,β=n0+1 +Rj +iα¯βdzα ∧ d¯zβ ++ +n0 +� +α=1 +n +� +β=n0+1 +Rj +iα¯βdzα ∧ d¯zβ + +n +� +α=n0+1 +n0 +� +β=1 +Rj +iα¯βdzα ∧ d¯zβ. +(4.1) +By assumption, (E, hE) is strongly decomposably non-negative of type I, so +n0 +� +α=1 +n +� +β=n0+1 +Ri¯jα¯βuiαv′jβ = 0 +for any �n0 +α=1 uiαei ⊗ ∂α and �n +β=n0+1 v′jβej ⊗ ∂β, which follows that +(4.2) +Ri¯jα¯β = 0, +1 ≤ α ≤ n0, n0 + 1 ≤ β ≤ n. +By conjugation, one gets +(4.3) +Ri¯jα¯β = Rj¯iβ ¯α = 0, +n0 + 1 ≤ α ≤ n, 1 ≤ β ≤ n0. + +POSITIVITY OF SCHUR FORMS +13 +Substituting (4.2), (4.3) into (4.1), one has +Rj +i = +n0 +� +α,β=1 +Rj +iα¯βdzα ∧ d¯zβ + +n +� +α,β=n0+1 +Rj +iα¯βdzα ∧ d¯zβ. +For the local frame {∂α}1≤α≤n, we define a local metric g around x by +gα¯β = g(∂α, ∂β) := δαβ. +Now we define a linear map +HV +x : End(Vx) → End(Ex) +by +(4.4) +HV +x (∂α ⊗ dzγ) = Rj +iα¯βg +¯βγej ⊗ ei = Rj +iα¯γej ⊗ ei +for any n0 + 1 ≤ α, γ ≤ n. With respect to the basis {∂α}n0+1≤α≤n, the +matrix of ∂α ⊗ dzγ ∈ End(Ux) is Eαγ, which is the (n − n0) × (n − n0) +matrix with 1 at the (α, γ)-component and zeros elsewhere. The matrix of +Rj +iα¯γej ⊗ ei ∈ End(Ex) is given by (Rj +iα¯γ)1≤j,i≤r (j row, i column). In terms +of matrices, (4.4) becomes +(4.5) +HV +x (Eαγ) = (Rj +iα¯γ)1≤j,i≤r = (Ri¯jα¯γ)1≤j,i≤r. +Then (HV +x (Eαγ))n0+1≤α,γ≤n is a (n − n0) × (n − n0) block matrix with r × r +matrices as entries, and +(HV +x (Eαγ))n0+1≤α,γ≤n = +� +(Ri¯jα¯γ)1≤j,i≤r +� +n0+1≤α,γ≤n +(j α row, i γ column). +Since (E, hE) is strongly decomposably non-negative of type I, so +Ri¯jα¯βv¯jαv¯iβ ≥ 0 +for any non-zero v = v¯jαei ⊗ ∂α ∈ Ex ⊗ Vx, which follows that the matrix +(HV +x (Eαγ))n0+1≤α,γ≤n +is positive semi-definite. By using [Cho75, Theorem 2 and Theorem 1], there +exist (n − n0) × r matrices Vp, 1 ≤ p ≤ N1 (one can choose N1 = (n − n0) · r) +such that +HV +x (Eαγ) = +N1 +� +p=1 +Vp +⊤ · Eαγ · Vp +for any n0 + 1 ≤ α, γ ≤ n. Combining with (4.5) and considering the (j, i) +entry, one has +Rj +iα¯β = +N1 +� +p=1 +(Vp +⊤ · Eαβ · Vp)j,i = +N1 +� +p=1 +(Vp)αj(Vp)βi. + +14 +XUEYUAN WAN +Hence +n +� +α,β=n0+1 +Rj +iα¯βdzα ∧ d¯zβ = +n +� +α,β=n0+1 +N1 +� +p=1 +(Vp)αj(Vp)βidzα ∧ d¯zβ += +N1 +� +p=1 +Ajp ∧ Aip, +where Ajp := �n +α=n0+1 (Vp)αjdzα, and one has +� +� +n +� +α,β=n0+1 +Rj +iα¯βdzα ∧ d¯zβ +� +� +1≤j,i≤r += A ∧ A +⊤, +where A = (Ajp) is a r × N1 matrix with (1, 0)-forms in V ∗ +x as entries. +Similarly, by considering the linear map +HU +x : End(Ux) → End(E∗ +x), +HV +x (∂α ⊗ dzγ) = Rj +iα¯γei ⊗ ej. +One can obtain that +(HV +x (Eαγ))1≤α,γ≤n0 = +� +(Ri¯jα¯γ)1≤j,i≤r +� +1≤α,γ≤n0 +(i α row, j γ column), +which follows that +� +� +n0 +� +α,β=1 +Rj +iα¯βdzα ∧ d¯zβ +� +� +1≤j,i≤r += −B ∧ B +⊤, +where B = (Bjp) is a r × N2 (one can choose N2 = n0 · r) matrix with +(0, 1)-forms in U ∗x as entries. +Thus, if (E, hE) is strongly decomposably non-negative of type I, then for +any x ∈ X, the Chern curvature matrix at this point has the following form +(4.6) +R = −B ∧ B +⊤ + A ∧ A +⊤ +with respect to a unitary frame, where B is a r×N2 matrix with (0, 1)-forms +as entries, A is a r × N1 matrix with (1, 0)-forms as entries, and +(4.7) +spanC{B} ∩ spanC{A} = {0}, +where +{B} := {Bip, 1 ≤ i ≤ r, 1 ≤ p ≤ N2} +and +{A} := {Aip, 1 ≤ i ≤ r, 1 ≤ p ≤ N1}. +Remark 4.1. It is noted that the above argument is independent of the choice +of unitary frames. If ˜e = e · a is also a unitary frame, then a is a unitary +matrix. By (2.2), one has +�R = a−1 · (−B ∧ B +⊤ + A ∧ A +⊤) · a += −a−1B ∧ a−1B +⊤ + a−1A ∧ a−1A +⊤, + +POSITIVITY OF SCHUR FORMS +15 +which has the form (4.6). Moreover, spanC{B} = spanC{a−1B} and spanC{A} = +spanC{a−1A}, (4.7) is equivalent to +spanC{a−1B} ∩ spanC{a−1A} = {0}. +Conversely, we assume (4.6) and (4.7) hold. For any x ∈ X, taking local +holomorphic coordinates {zα}1≤α≤n around x ∈ X such that +spanC{dz1|x, · · · , dzn0|x} = spanC{B} +and +spanC{dzn0+1|x, · · · , dzn1|x} = spanC{A}. +Now we set +Ux := spanC{∂1|x, · · · , ∂n0|x}, +Vx = spanC{∂n0+1|x, · · · , ∂n|x}. +Then Ux ⊕ Vx = T 1,0 +x X. Using (4.6) and (4.7), one can check that (E, hE) +is strongly decomposably non-negative. +Hence (E, hE) is strongly decomposably non-negative of type I if and only +if the Chern curvature matrix of (E, hE) satisfies (4.6) and (4.7). +Next, we assume that (E, hE) is strongly decomposably positive of type +I, i.e., it is strongly decomposably non-negative of type I and +(4.8) +Ri¯jα¯βuiαujβ = 0 =⇒ uiα = 0, for all 1 ≤ i ≤ r, 1 ≤ α ≤ n0 +and +(4.9) +Ri¯jα¯βv¯jαv¯iβ = 0 =⇒ v¯jα = 0, for all 1 ≤ j ≤ r, n0 + 1 ≤ β ≤ n. +By the equivalent description of strongly decomposably non-negative of type +I, i.e., (4.6) and (4.7), one has +Ri¯jα¯βuiαujβ = +N2 +� +p=1 +|Bip¯αuiα|2. +Hence (4.8) is equivalent to the equation Bx = 0 has only zero solution, +where +B := (Bip¯α)p,iα = +� +� +� +� +� +B111 +B112 +· · · +Br1n0 +B121 +B112 +· · · +Br2n0 +... +... +... +... +B1N21 +B1N22 +· · · +BrN2n0 +� +� +� +� +� +N2×rn0 +is a N2 × rn0 matrix. This is also equivalent to the matrix rank(B) = rn0. +Similarly, (4.9) is equivalent to the following matrix +A := (Ajpα)p,jα = +� +� +� +� +� +A11(n0+1) +A11(n0+2) +· · · +Ar1n +A12(n0+1) +A11(n0+2) +· · · +Ar2n +... +... +... +... +A1N1(n0+1) +A1N1(n0+2) +· · · +ArN1n +� +� +� +� +� +N1×r(n−n0) + +16 +XUEYUAN WAN +has rank r(n − n0). +In a word, we obtain +Theorem 4.2. +• A Hermitian vector bundle (E, hE) is a strongly de- +composably non-negative of type I if and only if (4.6) and (4.7) hold. +• A Hermitian vector bundle (E, hE) is strongly decomposably positive +of type I if and only if (4.6) and (4.7) hold, and rank(A) = r dim Vx, +rank(B) = r dim Ux. +As a result, we obtain the following criteria of (dual) Nakano positive +vector bundles. +Corollary 4.3. +• A Hermitian vector bundle (E, hE) is Nakano posi- +tive if and only if the Chern curvature matrix has the form +R = −B ∧ B +⊤ +with respect to some unitary frame, where B is a r × N matrix with +(0, 1)-forms as entries, and rank(B) = rn. +• A Hermitian vector bundle (E, hE) is dual Nakano positive if and +only if the Chern curvature matrix has the form +R = A ∧ A +⊤ +with respect to some unitary frame, where A is a r × N matrix with +(1, 0)-forms as entries, and rank(A) = rn. +4.2. Weak positivity of Schur forms. In this subsection, we will prove +the weak positivity of Schur forms for strongly decomposably positive vector +bundles of type I. +4.2.1. Schur forms. Let Λ(k, r) be the set of all the partitions of k by non- +negative integers less than or equal to r, i.e., any element λ ∈ Λ(k, r) is a +sequence +r ⩾ λ1 ⩾ λ2 ⩾ · · · ⩾ λk ⩾ 0 +satisfying |λ| = �k +i=1 λi = k. Each partition λ ∈ Λ(k, r) gives rise to a Schur +polynomial Pλ ∈ Q [c1, . . . , cr] of degree k, defined as k × k determinant +Pλ (c1, . . . , cr) = det (cλi−i+j)1⩽i,j⩽k += +��������� +cλ1 +cλ1+1 +. . . +cλ1+k−1 +cλ2−1 +cλ2 +. . . +cλ2+k−2 +... +... +... +... +cλk−k+1 +cλk−k+2 +. . . +cλk +��������� +, +where by convention c0 = 1 and ci = 0 if i /∈ [0, r]. +Denote by Mr(C) and GLr(C) the vector spaces of r × r matrices and the +general linear group of degree r. A map P : Mr(C) → C is called GLr(C)- +invariant if it is invariant under the conjugate action of GLr(C) on Mr(C). + +POSITIVITY OF SCHUR FORMS +17 +Now we define the following GLr(C)-invariant function ci : Mr(C) → C, i = +1, . . . , r by +det (Ir + tX) = +r +� +i=0 +ti · ci(X), +where Ir is the identity matrix in Mr(C). Then the graded ring of GLr(C)- +invariant homogeneous polynomials on Mr(C), which we denote here by +I(r) = �+∞ +k=0 I(r)k, is multiplicatively generated by c1, . . . , cr. +Let (E, hE) be a Hermitian vector bundle, the i-th Chern form ci(E, hE) +is defined by +ci +� +E, hE� += ci +�√−1 +2π RE +� +. +For each λ ∈ Λ(k, r), the associated Schur form is then defined as +Pλ(c(E, hE)) := Pλ(c1(E, hE), . . . , cr(E, hE)), +which represents the Schur class +Pλ(c(E)) := Pλ(c1(E), . . . , cr(E)) ∈ H2k(X, Z). +4.2.2. Griffiths cone. By [Gri70, Page 242, (5.6)], each P ∈ I(r)k can be +written as +(4.10) +P(B) = +� +σ,τ∈Sk +� +ρ∈[1,r]k +pρστBρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k), +where Bλµ, λ, µ = 1, . . . , r are the components of the matrix B, Sk is the +permutation group on k indices and [1, r] := {1, . . . , r}. An element P ∈ +I(r)k is called Griffiths non-negative if it can be expressed in the form (4.10) +with +pρστ = +� +t∈T +λρt · qρσt¯qρτt, +for some finite set T, some real numbers λρt ⩾ 0, and complex numbers qρσt. +The Griffiths cone Π(r) ⊂ I(r) is defined as the cone of Griffiths non- +negative polynomials. +Proposition 4.4 (Fulton-Lazarsfeld [FL83, Proposition A.3]). Let +P = +� +λ∈Λ(k,r) +aλ(P)Pλ +(aλ(P) ∈ Q) +be a non-zero weighted homogeneous polynomial in Q [c1, . . . , cr]. Then P +lies in the Griffiths cone Π(r) if and only if each of the Schur coefficients +aλ(P) is non-negative. +In particular, for each λ ∈ Λ(k, r), one has +Pλ(B) = +� +σ,τ∈Sk +� +ρ∈[1,r]k +pρστBρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k), + +18 +XUEYUAN WAN +where pρστ = � +1≤i,j≤m +� 1 +k! +�2 aij(τ)aij(σ) with (aij(τ)) ∈ U(m), see [FL83, +(A.6)]. Denote T = [1, m]2 and qσt := at(σ) for any t ∈ T, then +(4.11) Pλ(B) = +� 1 +k! +�2 � +σ,τ∈Sk +� +ρ∈[1,r]k +�� +t∈T +qσtqτt +� +Bρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k). +4.2.3. Weak positivity of Schur forms. We assume that (E, hE) is a strongly +decomposably positive vector bundle of type I over a complex manifold X. +By Theorem 4.2, for any x ∈ X, there exists a decomposition T 1,0 +x X = +Ux ⊕ Vx such that the Chern curvature matrix R of (E, hE) has the form +(4.12) +R = −B ∧ B +⊤ + A ∧ A +⊤ +with respect to a unitary frame, where B is a r ×N matrix with (0, 1)-forms +in U ∗x as entries, A is a r × N matrix with (1, 0)-forms in V ∗ +x as entries. +Moreover, rank(A) = r · dim Vx, rank(B) = r · dim Ux. +For each λ ∈ Λ(k, r), by (4.11), the Schur form Pλ(c(E, hE)) is given by +Pλ(c(E, hE)) = +�√−1 +2π +�k +1 +(k!)2 +� +σ,τ∈Sk +� +ρ∈[1,r]k +�� +t∈T +qσtqτt +� +· +k� +j=1 +Rρσ(j)ρτ(j). +By (4.12), the Chern curvature matrix satisfies +Rρσ(j)ρτ(j) = (Bρτ(j)cj ¯ +βjBρσ(j)cjαj + Aρτ(j)cjαjAρσ(j)cjβj)dzαj ∧ d¯zβj += +N +� +cj=1 +(Bρσ(j)cj ∧ Bρτ(j)cj + Aρτ(j)cj ∧ Aρσ(j)cj) += +N +� +cj=1 +� +ϵj∈[0,1] +(Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj, + +POSITIVITY OF SCHUR FORMS +19 +which follows that +k� +j=1 +Rρσ(j)ρτ(j) = +k� +j=1 +N +� +cj=1 +� +ϵj∈[0,1] +(Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj += +� +c∈[1,N]k +� +ϵ∈[0,1]k +k� +j=1 +(Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj += +� +c∈[1,N]k +� +ϵ∈[0,1]k +k� +j=1 +(−1)ϵj+1(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj ∧ (Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj += +� +c∈[1,N]k +� +ϵ∈[0,1]k +(−1)|ϵ|+k(−1) +k(k−1) +2 +· +( +k� +j=1 +(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj) ∧ ( +k� +j=1 +(Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj), +where |ϵ| := �k +j=1 ϵj. +Recall that ρ ∈ [1, r]k, t ∈ T, c ∈ [1, N]k and ϵ ∈ [0, 1]k, we obtain that +Pλ(c(E, hE)) = +�√−1 +2π +�k +1 +(k!)2 (−1) +k(k−1) +2 +� +ρ,t,c,ϵ +(−1)|ϵ|+k· +( +� +σ∈Sk +qσt +k� +j=1 +(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj) ∧ ( +� +τ∈Sk +qτt +k� +j=1 +(Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj). +Now we set +(4.13) +ψρtcϵ := +� +σ∈Sk +qσt +k� +j=1 +(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj, +which is a (|ϵ|, k − |ϵ|)-form. Hence +Pλ(c(E, hE)) = +� 1 +2π +�k � 1 +k! +�2 +( +√ +−1)k2 � +ρ,t,c,ϵ +(−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ. +(4.14) +For any non-zero decomposable (n − k, 0)-form η = η1 ∧ · · · ∧ ηn−k, where +ηi, 1 ≤ i ≤ n − k, are (1, 0)-forms, we assume that η1, · · · , ηi0 ∈ U ∗ +x and +ηi0+1, · · · , ηn−k ∈ V ∗ +x . +Now we can take local holomorphic coordinates +{zα}1≤α≤n around x ∈ X such that +U ∗ +x = spanC{dz1|x, · · · , dzn0|x}, with dzj|x = ηj, +1 ≤ j ≤ i0 +and +V ∗ +x = spanC{dzn0+1|x, · · · , dzn|x}, with dzn0−i0+j|x = ηj, +i0+1 ≤ j ≤ n−k, + +20 +XUEYUAN WAN +and so ψρtcϵ can be written as the following form +ψρtcϵ = +� +1≤α1<···<α|ϵ|≤n0 +n0+1≤β1<···<βk−|ϵ|≤n +ψα1···α|ϵ| ¯β1···¯βk−|ϵ|dzα1 ∧ · · · ∧ dzα|ϵ| +∧ d¯zβ1 ∧ · · · ∧ d¯zβk−|ϵ|. +Then +( +√ +−1)k2 � +ρ,t,c,ϵ +(−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ ∧ ( +√ +−1)(n−k)2η ∧ η += ( +√ +−1)k2 +� +ρ,t,c,|ϵ|=n0−i0 +(−1)n0−i0+k( +√ +−1)(n−k)2dz1 ∧ · · · ∧ dzi0∧ +dzn0+1 ∧ · · · ∧ dzn−i0+n−k ∧ d¯z1 ∧ · · · ∧ d¯zi0 ∧ d¯zn0+1 ∧ · · · ∧ d¯zn−i0+n−k∧ +� +ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯ndzi0+1 ∧ · · · ∧ dzn0 ∧ d¯zn0−i0+n−k+1 ∧ · · · ∧ d¯zn� +∧ +� +ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯nd¯zi0+1 ∧ · · · ∧ d¯zn0 ∧ dzn0−i0+n−k+1 ∧ · · · ∧ dzn� += +� +ρ,t,c,|ϵ|=n0−i0 +|ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯n|2· +( +√ +−1)n2dz1 ∧ · · · ∧ dzn ∧ d¯z1 ∧ · · · ∧ d¯zn, +(4.15) +which is a non-negative volume form. By (4.14), the Schur form Pλ(c(E, hE)) +is weakly non-negative. +By (4.15), one knows that +Pλ(c(E, hE)) ∧ ( +√ +−1)(n−k)2η ∧ ¯η = 0 +if and only if +(4.16) +ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯n = 0 +for any ρ, t, c, |ϵ| = n0 − i0. +Now we take a special vector ϵ = (1, · · · , 1 +� �� � +n0−i0 +, 0 · · · , 0) and denote j0 = +n0 − i0, by (4.13), then +ψρtcϵ = +� +σ∈Sk +qσtBρσ(1)c1 ∧ · · · ∧ Bρσ(j0)cj0 ∧ Aρσ(j0+1)cj0+1 ∧ · · · ∧ Aρσ(k)ck. +Combining with (4.16), one has +0 = ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯n += +� +τ1∈Sj0 +� +τ2∈Sk−j0 +� +σ∈Sk +sgn(τ1)sgn(τ2)qσt· +Bρσ(1)c1τ1(i0+1) · · · Bρσ(j0)cj0τ1(n0) · Aρσ(j0+1)cj0+1τ2(j0+n−k+1) · · · Aρσ(k)ckτ2(n). +(4.17) + +POSITIVITY OF SCHUR FORMS +21 +Since rank(A) = r · dim Vx, rank(B) = r · dim Ux, without loss of generality, +we assume that the submatrices +B′ = (Bc,iα)1≤c≤r dim Ux,1≤i≤r,1≤α≤dim Ux +and +A′ = (Ac,jα)1≤c≤r dim Vx,1≤j≤r,1≤α≤dim Vx +of B and A are inverse. By (4.17) and note that B′ +c,iα = Bic¯α and A′ +c,jα = +Ajcα, one has +0 = +� +τ1∈Sj0 +� +τ2∈Sk−j0 +rn0 +� +c1,··· ,cj0=1 +r(n−n0) +� +cj0+1,··· ,ck=1 +� +σ∈Sk +sgn(τ1)sgn(τ2)qσt· +B′ +c1,ρσ(1)τ1(i0+1) · · · B′ +cj0,ρσ(j0)τ1(n0) · A′ +cj0+1,ρσ(j0+1)τ2(j0+n−k+1) · · · A′ +ck,ρσ(k)τ2(n)· +A′−1 +lkβk,ck · · · A′−1 +lj0+1βj0+1,cj0+1B′−1 +lj0βj0,cj0 · · · B′−1 +l1β1,c1 += +� +τ1∈Sj0 +� +τ2∈Sk−j0 +� +σ∈Sk +sgn(τ1)sgn(τ2)qσt · δρσ(k)lkδτ2(n)βk · · · δρσ(j0+1)lj0+1· +δτ2(j0+n−k+1)βj0+1δρσ(j0)lj0δτ1(n0)βj0 · · · δρσ(1)l1δτ1(i0+1)β1 +(4.18) +for any (β1, · · · , βj0) ∈ [1, n0]j0, (βj0+1, · · · , βk) ∈ [n0+1, n]n−j0 and (l1, · · · , lk) ∈ +[1, r]k. +By taking +βs = +� +i0 + s +1 ≤ s ≤ j0, +n − k + s +j0 + 1 ≤ s ≤ k, +then (4.18) becomes +(4.19) +� +σ∈Sk +qσtδρσ(1)l1 · · · δρσ(k)lk = 0 +for any ρ, l ∈ [1, r]k and t ∈ T. +Remark 4.5. Note that (4.19) holds if and only if ψρtcϵ = 0 for any ρ, t, c, ϵ. +In fact, if ψρtcϵ = 0, then (4.16) holds, and follows (4.19). Conversely, if +(4.19) holds, then +ψρtcϵ = +� +l∈[1,r]k +( +� +σ∈Sk +qσtδρσ(1)l1 · · · δρσ(k)lk) +k� +j=1 +Bljcj +ϵj ∧ Aljcj +1−ϵj = 0. +For k ≤ r, one can take ρi = li = i for 1 ≤ i ≤ k. Thus +0 = +� +σ∈Sk +qσtδρσ(1)l1 · · · δρσ(k)lk = qId,t +for any t ∈ T, which is a contradiction since (qId,t)t∈T ∈ U(m) is a unitary +matrix. Hence all (k, k)-Schur forms Pλ(c(E, hE)) are weakly positive for + +22 +XUEYUAN WAN +any k ≤ r. In particular, all Chern forms ci(E, hE), 1 ≤ i ≤ r, are weakly +positive. +For general k and r, we take l1, · · · , lk in (4.19) to be +li = ρi, for 1 ≤ i ≤ k, +and (4.19) implies that +(4.20) +� +ρ∈[1,r]k +� +σ∈Sk +χλ(σ)δρ1ρσ(1) · · · δρkρσ(k) = 0, +where +χλ(σ) = Tr(qσt) = +m +� +i=1 +aii(σ) +is the character of the representation φλ(σ) = (aij(σ)) ∈ U(m) corresponding +to the partition λ. From [FL83, (A.5)], (4.20) is equivalent to +(4.21) +Pλ(Ir) = 0. +Here Pλ(•) denotes the invariant polynomial corresponding to the Schur +function Pλ under the isomorphism I(r) ∼= Q(c1, · · · , cr). +Denote by x1, · · · , xr the Chern roots, which are defined by +r +� +j=0 +cjtj = (1 + tx1)(1 + tx2) · · · (1 + txr). +Recall that the Schur polynomial is defined by +Pλ (c1, . . . , cr) = det +� +cλj−j+l +� +1⩽j,l⩽k , +where λ = (λ1, · · · , λk) ∈ Λ(k, r) is a partition satisfying +k +� +i=1 +λi = k and r ≥ λ1 ≥ · · · ≥ λk ≥ 0. +Denote by λ′ the conjugate partition to the partition λ, see e.g. +[FH91, +Section 4.1, Page 45], then +(4.22) +λ′ = (λ′ +1, · · · , λ′ +r), +with +r +� +i=1 +λ′ +i = k and λ′ +1 ≥ · · · ≥ λ′ +r ≥ 0. +The second Jacobi-Trudi identity (or Giambell’s formula) gives +Pλ (c1, . . . , cr) = det +� +cλj−j+l +� +1⩽j,l⩽k = sλ′(x1, · · · , xr), +(4.23) +see e.g. [FH91, Page 455, (A.6)], where +(4.24) +sλ′(x1, · · · , xr) := +���������� +xλ′ +1+r−1 +1 +xλ′ +1+r−1 +2 +. . . +xλ′ +1+r−1 +r +xλ′ +2+r−2 +1 +xλ′ +2+r−2 +2 +. . . +xλ′ +2+r−2 +r +... +... +... +... +xλ′ +r +1 +xλ′ +r +2 +. . . +xλ′ +r +r +���������� +� +1≤i 0 +for any non-zero decomposable (n − k, 0)-form η, and so Pλ(c(E, hE)) is a +weakly positive (k, k)-form. +In a word, we obtain +Theorem 4.6. Let (E, hE) be a strongly decomposably positive vector bundle +of type I over a complex manifold X, rankE = r, and dim X = n. Then +the Schur form Pλ(c(E, hE)) is weakly positive for any partition λ ∈ Λ(k, r), +k ≤ n and k ∈ N. +In particular, if (E, hE) is Nakano positive, then the Chern curvature +matrix has the form +R = −B ∧ B +⊤. +By considering A = 0 in (4.13), then +ψρtcϵ1 = +� +σ∈Sk +qσt +k� +j=1 +Bρσ(j)cj, +ϵ1 = (1, · · · , 1) +and +ψρtcϵ = 0, for any ϵ ̸= ϵ1, +By (4.14), one has +Pλ(c(E, hE)) = +� 1 +2π +�k � 1 +k! +�2 +( +√ +−1)k2 � +ρ,t,c +ψρtcϵ1 ∧ ψρtcϵ1, +(4.26) +where ψρtcϵ1 is a (k, 0)-form. For any non-zero (n − k, 0)-form η, one has +Pλ(c(E, hE)) ∧ ( +√ +−1)(n−k)2η ∧ η += +� 1 +2π +�k � 1 +k! +�2 +( +√ +−1)n2 � +ρ,t,c +ψρtcϵ1 ∧ η ∧ ψρtcϵ1 ∧ η, +which is a non-negative volume form, and so Pλ(c(E, hE)) is non-negative. +Moreover, +(4.27) +Pλ(c(E, hE)) ∧ ( +√ +−1)(n−k)2η ∧ η = 0 +if and only if +(4.28) +ψρtcϵ1 ∧ η = 0 + +24 +XUEYUAN WAN +for any ρ ∈ [1, r]k, t ∈ T, c ∈ [1, N]k. By the expression of ψρtcϵ1, (4.28) +becomes +0 = ψρtcϵ1 ∧ η += +� +σ∈Sk +qσtBρσ(1)c1 ∧ · · · ∧ Bρσ(k)ck ∧ η += +� +σ∈Sk +qσtBρσ(1)c1 ¯α1 · · · Bρσ(k)ck ¯αkdzα1 ∧ · · · ∧ dzαk ∧ η. +(4.29) +Since (E, hE) is Nakano positive, by Corollary 4.3, we can take B such that +B is inverse. Multiplying (4.29) by (B−1)c1,l1β1 · · · (B−1)ck,lkβk and summing +on c1, · · · , ck, one has +� +� � +σ∈Sk +qσtδρσ(1)l1 · · · δρσ(k)lk +� +� dzβ1 ∧ · · · ∧ dzβk ∧ η = 0 +for any l = (l1, · · · , lk) ∈ [1, r]k and β = (β1, · · · , βk) ∈ [1, n]k. By choosing +β1, · · · , βk such that dzβ1 ∧ · · · ∧ dzβk ∧ η ̸= 0, so +(4.30) +� +σ∈Sk +qσtδρσ(1)l1 · · · δρσ(k)lk = 0, +which is exactly (4.19). By Remark 4.5, (4.30) is equivalent to ψρtcϵ1 = 0. +Hence (4.27) is equivalent to (4.30), which follows that Pλ(Ir) = 0, see +(4.21). By (4.25), Pλ(Ir) ̸= 0, so we get a contradiction. Thus, Pλ(c(E, hE)) +is a positive (k, k)-form. Similarly, if (E, hE) is dual Nakano positive, then +Pλ(c(E, hE)) is also a positive (k, k)-form. +Hence, we can give an algebraic proof of the following positivity of Schur +forms for (dual) Nakano positive vector bundles. +Theorem 4.7 (Finski [Fin22, Theorem 1.1]). Let (E, hE) be a (dual) Nakano +positive (respectively non-negative) vector bundle of rank r over a complex +manifold X of dimension n. Then for any k ∈ N, k ⩽ n, and λ ∈ Λ(k, r), +the (k, k)-form Pλ +� +c +� +E, hE�� +is positive (respectively non-negative). +Remark 4.8. Note that S. Finski proved the above positivity of Schur forms +by using the following two steps: the first one is a refinement of the deter- +minantal formula of Kempf-Laksov on the level of differential forms, which +expresses Schur forms as a certain pushforward of the top Chern form of +a Hermitian vector bundle obtained as a quotient of the tensor power of +(E, hE), and the second one is to show the positivity of the top Chern form +of a (dual) Nakano positive vector bundle. Our method here is an algebraic +proof by analyzing the vanishing of Schur forms, which is very different from +S. Finski’s approach. + +POSITIVITY OF SCHUR FORMS +25 +5. Strongly decomposable positivity of type II +In this section, we will consider the strongly decomposable positivity of +type II, which is the direct sum of Nakano positive and dual Nakano positive +vector bundles point-wisely. +5.1. A criterion of type II positivity. Let (E, hE) be a strongly decom- +posably positive vector bundle of type II, see Definition 2.7. By Corollary +4.3, with respect to a unitary frame {e1, · · · , er1} of (E1,x, hE|E1,x), and a +unitary frame {er1+1, · · · , er} of (E2,x, hE|E2,x) at x ∈ X, one has +RE +x |E1,x = −B1 ∧ B1 +⊤, +RE +x |E2,x = A2 ∧ A2 +⊤, +with rank(B1) = dim(E1,x) · n and rank(A2) = dim(E2,x) · n, where B1 = +((B1)ip)1≤i≤r1,1≤j≤N1 is a matrix with (0, 1)-forms as entries and A2 = +((A2)ip)r1+1≤i≤r,1≤j≤N2 is a matrix with (1, 0)-forms as entries. +Now we +define the matrices Ar×N and Br×N (N = max{N1, N2}) by +(5.1) +B = +� +B1 +0 +0 +0 +� +, +A = +� +0 +0 +A2 +0 +� +. +Then +rank(B) = rank(B1) = r1 · n, +rank(A) = rank(A2) = (n − r1) · n +and +RE +x = +� +−B1 ∧ B1 +⊤ +0 +0 +A2 ∧ A2 +⊤ +� += −B ∧ B +⊤ + A ∧ A +⊤. +For the matrix B = (Bip¯α)iα,p, we can associate it with another matrix B by +B = (Bαp) = +� r +� +i=1 +Bip¯αei +� +αp +, +which is a n × N matrix with elements of E as entries. Similarly, we can +define a n × N matrix by +A = (Aαp) = +� r +� +i=1 +Aipαei +� +αp +. +We define +{B} := +� r +� +i=1 +Bip¯αei, 1 ≤ p ≤ N, 1 ≤ α ≤ n +� +and +{A} := +� r +� +i=1 +Aipαei, 1 ≤ p ≤ N, 1 ≤ α ≤ n +� +. +By the definitions of the matrices A and B, one has +spanC{A} ⊥ spanC{B}. + +26 +XUEYUAN WAN +Hence, if (E, hE) is a strongly decomposably positive vector bundle of type +II, then there are two r × N-matrices A, B of (1, 0)-forms and (0, 1)-forms +respectively, such that with respect to a unitary frame {ei}1≤i≤r of Ex, +(5.2) +RE +x = −B ∧ B +⊤ + A ∧ A +⊤, +and +(5.3) +spanC{A} ⊥ spanC{B}. +Moreover, the ranks of A and B satisfy +(5.4) +rank(B) = r1 · n, +rank(A) = (r − r1) · n. +Remark 5.1. If we consider a new unitary frame �e = e·a for a unitary matrix +a ∈ U(r), by Remark 4.1, one has +(5.5) +� +REx = − �B ∧ �B +⊤ ++ �A ∧ �A +⊤ +, +with �B = a−1 · B and �A = a−1 · A. Moreover, one has +�Bαp = +r +� +i=1 +�Bip¯α�ei = +r +� +i,j=1 +(a−1)ijBjp¯α�ei = +r +� +j=1 +Bjp¯αej = Bαp. +Similarly, � +Aαp = Aαp. Hence A and B are independent of the unitary frame. +One can also check that +rank(�B) = rank(B) = r1 · n, +rank( �A) = rank(A) = (r − r1) · n. +In a word, we show that (5.2)–(5.4) hold for any unitary frame. +Conversely, we assume that (5.2)–(5.4) hold for some unitary frame of Ex, +x ∈ X. Set +E1,x := spanC{B}, +E2,x := spanC{A}. +Let {e1, · · · , er′ +1} be a unitary frame of E1,x and {er′ +1+1, · · · , er′} be a unitary +frame of E2,x. Since E1,x ⊥ E2,x, so {ei}1≤i≤r′ is a unitary frame of E1,x ⊕ +E2,x. +Now we can extend the frame {ei}1≤i≤r′ and get a unitary frame +{ei}1≤i≤r of Ex. By Remark 5.1, (5.2)–(5.4) also hold for this unitary frame +{ei}1≤i≤r. Hence +Ri¯jα¯βej ⊗ ei = +N +� +p=1 +(−Bβp ⊗ Bαp + Aαp ⊗ Aβp). +So +RE +x |E1,x = −B ∧ B +⊤, RE +x |E2,x = A ∧ A +⊤ +and +Ri¯jα¯β = 0 for any (i, j) or (j, i) ∈ [1, r′ +1] × [r′ +1 + 1, r]. +By (5.4), one has +r′ +1 ≥ r1, +r − r′ +1 ≥ r′ − r′ +1 ≥ r − r1, +which follows that +r1 = r′ +1, +r′ = r. + +POSITIVITY OF SCHUR FORMS +27 +Hence Ex = E1,x ⊕ E2,x. By Corollary 4.3, we obtain that Ri¯jα¯βuiαujβ > 0 +for any non-zero u = uiαei⊗∂α ∈ E1,x⊗T 1,0 +x X, Ri¯jα¯βvi¯βvj ¯α > 0 for any non- +zero v = vi¯βei⊗∂¯β ∈ E2,x⊗T 0,1 +x X. Thus, (E, hE) is a strongly decomposably +positive vector bundle of type II. +In a word, we obtain a criterion of a strongly decomposably positive vector +bundle of type II. +Theorem 5.2. (E, hE) is a strongly decomposably positive vector bundle of +type II if and only if it satisfies (5.2)–(5.4). +5.2. Positivity of Schur forms. In this subsection, we will consider the +positivity of Schur forms for strongly decomposably positive vector bundles +of type II. +Let E and F be two holomorphic vector bundles over a complex manifold +X, rank(E) = r and rank(F) = q. Let x1, · · · , xr denote the Chern roots of +E. For any partition λ′ satisfying (4.22), we denote +sλ′(c(E)) := sλ′(x1, · · · , xr) ∈ H2k(X, R), +where sλ′(x1, · · · , xr) is defined in (4.24), which is also called a Schur poly- +nomial. +Similarly, one can define the cohomology classes sλ′(c(F)) and +sλ′(c(E ⊕F)). For these cohomology classes, by Littlewood-Richardson rule, +see [BRT21, Proposition 3.3 (3.14)], one has +sλ′(c(E ⊕ F)) = +� +µ′,ν′ +cλ′ +µ′ν′sµ′(c(E))sν′(c(F)), +where cλ′ +µ′ν′ is a Littlewood-Richardson coefficient. One can refer to [Ful97, +Chapter 5] for more details on the Littlewood-Richardson coefficients. By +(4.23), the Schur class Pλ(c(E ⊕ F)) of the direct sum E ⊕ F satisfies +(5.6) +Pλ(c(E ⊕ F)) = +� +µ′,ν′ +cλ′ +µ′ν′Pµ(c(E))Pν(c(F)) = +� +µ,ν +cλ +µνPµ(c(E))Pν(c(F)), +where λ, µ, ν are the conjugate partitions to λ′, µ′, ν′ respectively, the last +equality follows from the conjugation symmetry cλ′ +µ′ν′ = cλ +µν, see e.g. [Ste01, +Page 115]. +Let hE and hF be Hermitian metrics on E and F, respectively. The direct +sum E ⊕ F is equipped with the natural metric hE ⊕ hF . Now we can prove +(5.6) on the level of differential forms. +Proposition 5.3. For any λ ∈ Λ(k, r), one has +Pλ(c(E ⊕ F, hE ⊕ hF )) = +� +µ,ν +cλ +µνPµ(c(E, hE)) ∧ Pν(c(F, hF )). +Proof. We will follow the method in the proofs of [Gul12, Proposition 3.1] +and [DF22, Theorem 3.5]. From the definition of total Chern form, one has +c(E ⊕ F, hE ⊕ hF ) = c(E, hE) ∧ c(F, hF ). + +28 +XUEYUAN WAN +Recall that Pλ (c1, . . . , cr) = det (cλi−i+j)1⩽i,j⩽k, so that +Pλ(c(E ⊕ F, hE ⊕ hF )) − +� +µ,ν +cλ +µνPµ(c(E, hE)) ∧ Pν(c(F, hF )) += +� +i1+2i2+···+rir ++j1+2j2+···+rjr=k +fi1···irj1···jqc1(E, hE)i1 ∧ · · · ∧ cr(E, hE)ir +∧ c1(F, hF )j1 ∧ · · · ∧ cr(F, hF )jq. +where the universal coefficients fi1···irj1...jq do not depend on E, F and X, +just depend r, q, Pλ. By (5.6), then the cohomology class satisfies +� +�� +� +i1+2i2+···+rir ++j1+2j2+···+rjr=k +fi1···irj1···jqc1(E, hE)i1 ∧ · · · ∧ cr(E, hE)ir +∧c1(F, hF )j1 ∧ · · · ∧ cr(F, hF )jq� += 0 +Now we can take X to be any n-dimensional projective manifold and fix an +ample line bundle A on X. Let ωA be a metric on A with positive curvature. +For m1, · · · , mr, mr+1, · · · , mr+q positive integers, we define +E = A⊗m1 ⊕ · · · ⊕ A⊗mr, +F = A⊗mr+1 ⊕ · · · ⊕ A⊗mr+q. +By the same proof as in [DF22, Page 14], one can show all universal coeffi- +cients fi1···irj1···jq vanish, which follows that +Pλ(c(E ⊕ F, hE ⊕ hF )) − +� +µ,ν +cλ +µνPµ(c(E, hE)) ∧ Pν(c(F, hF )) = 0, +which completes the proof. +□ +Now we assume (E, hE) is a strongly decomposably positive vector bundle +of type II, for any x ∈ X, there exists an orthogonal decomposition of Ex, +Ex = E1,x ⊕ E2,x, +and the Chern curvature RE +x has the form +RE +x = +�RE +x |E1,x +0 +0 +RE +x |E2,x +� +. +Let {e1, · · · , er1} be a unitary frame of (E1,x, hE|E1,x) and {er1+1, · · · , er} be +a unitary frame of (E2,x, hE|E2,x). Let (U, {zα}1≤α≤n) be a local coordinate +neighborhood arround x, and denote by E1 = U × E1,x the locally trivial +bundle, and {ei}1≤i≤r1 also gives a frame of E1. Now we define the following +Hermitian metric on E1 by +hE1(ei, ej) := δij − Ri¯jα¯βzα¯zβ, +1 ≤ i, j ≤ r1, +which is a Hermitian metric by taking U small enough. Then (E1, hE1) is a +Hermitian vector bundle around x and satisfies +RE1 +x += RE +x |E1,x. + +POSITIVITY OF SCHUR FORMS +29 +Similarly, one can define a Hermitian vector bundle (E2, hE2) such that +RE2 +x += RE +x |E2,x. Hence +c(E1 ⊕ E2, hE1 ⊕ hE2)|x = det +� +Idr + +√−1 +2π +� +RE1 +x +0 +0 +RE2 +x +�� += det +� +Idr + +√−1 +2π +�RE|E1,x +0 +0 +RE|E2,x +�� += det +� +Idr + +√−1 +2π RE +x +� += c(E, hE)|x, +which follows that +Pλ(c(E, hE))|x = Pλ(c(E1 ⊕ E2, hE1 ⊕ hE2))|x. +By Proposition 5.3, one has +Pλ(c(E, hE))|x = +� +µ,ν +cλ +µνPµ(c(E1, hE1))|x ∧ Pν(c(E2, hE2))|x. +(5.7) +Since (E1, hE1) is Naknao positive and (E2, hE2) is dual Nakano positive at +x, so Pµ(c(E1, hE1))|x and Pν(c(E2, hE2))|x are positive forms. Since the +Littlewood-Richardson coefficients cλ +µν are non-negative integers, see [Ful97, +Corollary 1 in Chapter 5], so the each summand +cλ +µνPµ(c(E1, hE1))|x ∧ Pν(c(E2, hE2))|x +in RHS of (5.7) is non-negative. On the other hand, for any λ, µ, ν satisfying +λi = µi + νi for all i, then cλ +µ,ν = 1, see [Ful97, Page 66], and +Pµ(c(E1, hE1))|x ∧ Pν(c(E2, hE2))|x +is a positive (|λ|, |λ|)-form by Proposition 3.4. By (5.7), we show that the +Schur form Pλ(c(E, hE))|x is a positive (|λ|, |λ|)-form. +Theorem 5.4. Let (E, hE) be a strongly decomposably positive vector bundle +of type II over a complex manifold X, rankE = r, and dim X = n. Then +the Schur form Pλ(c(E, hE)) is positive for any partition λ ∈ Λ(k, r), k ≤ n +and k ∈ N. +References +[BC65] +Raoul Bott and S. S. Chern. Hermitian vector bundles and the equidistribution +of the zeroes of their holomorphic sections. Acta Math., 114:71–112, 1965. pages +3 +[BG71] +Spencer Bloch and David Gieseker. The positivity of the Chern classes of an +ample vector bundle. Invent. Math., 12:112–117, 1971. pages 2 +[BRT21] Sara C. Billey, Brendon Rhoades, and Vasu Tewari. Boolean product poly- +nomials, Schur positivity, and Chern plethysm. Int. Math. Res. Not. IMRN, +(21):16636–16670, 2021. pages 27 +[Cho75] +Man-Duen Choi. Completely positive linear maps on complex matrices. Linear +Algebra and its Applications, 10(3):285–290, 1975. pages 13 + +30 +XUEYUAN WAN +[DF22] +Simone Diverio and Filippo Fagioli. Pointwise universal Gysin formulae and ap- +plications towards Griffiths’ conjecture. The Annali della Scuola Normale Supe- +riore di Pisa, Classe di Scienze, XXIII(5):1597–1624, 2022. pages 3, 27, 28 +[DPS94] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider. Compact com- +plex manifolds with numerically effective tangent bundles. J. Algebraic Geom., +3(2):295–345, 1994. pages 2 +[Fag22a] Filippo Fagioli. A note on Griffiths’ conjecture about the positivity of Chern- +Weil forms . Differential Geometry and its Applications, 81, 2022. cvgmt preprint. +pages 3, 10, 11 +[Fag22b] Filippo Fagioli. Universal vector bundles, push-forward formulae and positivity +of characteristic forms. arXiv:2210.11157v1, 2022. pages 3 +[FH91] +William Fulton and Joe Harris. Representation theory, volume 129 of Graduate +Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings +in Mathematics. pages 22, 23 +[Fin22] +Siarhei Finski. On characteristic forms of positive vector bundles, mixed discrim- +inants, and pushforward identities. Journal of the London Mathematical Society, +106(2):1539–1579, 2022. pages 2, 3, 4, 7, 9, 10, 12, 24 +[FL83] +William Fulton and Robert Lazarsfeld. Positive polynomials for ample vector +bundles. Ann. of Math. (2), 118(1):35–60, 1983. pages 2, 17, 18, 22 +[Ful97] +William Fulton. Young tableaux, volume 35 of London Mathematical Society Stu- +dent Texts. Cambridge University Press, Cambridge, 1997. With applications to +representation theory and geometry. pages 5, 27, 29 +[Gri70] +Phillip A. Griffiths. Hermitian differential geometry, Chern classes, and posi- +tive vector bundles, pages 185–252. Princeton University Press, Princeton, 1970. +pages 2, 3, 17 +[Gul12] +Dincer Guler. On Segre forms of positive vector bundles. Canad. Math. Bull., +55(1):108–113, 2012. pages 3, 27 +[HK74] +Reese Harvey and Anthony Knapp. Positive (p,p) forms, Wirtinger’s inequality, +and currents, pages 43–62. 01 1974. pages 4, 10, 11 +[Kob87] +Shoshichi Kobayashi. Differential Geometry of Complex Vector Bundles. Prince- +ton University Press, Princeton, 1987. pages 5 +[Li21] +Ping Li. Nonnegative Hermitian vector bundles and Chern numbers. Math. Ann., +380(1-2):21–41, 2021. pages 3 +[RT19] +Julius Ross and Matei Toma. Universal vector bundles, push-forward formulae +and positivity of characteristic forms. arXiv:1905.13636v3, 2019. pages 3 +[Ste01] +John R. Stembridge. Multiplicity-free products of Schur functions. Ann. Comb., +5(2):113–121, 2001. pages 27 +[Xia22] +Jian Xiao. On the positivity of high-degree Schur classes of an ample vector +bundle. Sci. China Math., 65(1):51–62, 2022. pages 3 +Xueyuan Wan: Mathematical Science Research Center, Chongqing Uni- +versity of Technology, Chongqing 400054, China. +Email address: xwan@cqut.edu.cn + diff --git a/y9E2T4oBgHgl3EQfhgf_/content/tmp_files/load_file.txt b/y9E2T4oBgHgl3EQfhgf_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa0b8c956a5db08d33623b4d248649b5e889729a --- /dev/null +++ b/y9E2T4oBgHgl3EQfhgf_/content/tmp_files/load_file.txt @@ -0,0 +1,980 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf,len=979 +page_content='POSITIVITY OF SCHUR FORMS FOR STRONGLY DECOMPOSABLY POSITIVE VECTOR BUNDLES XUEYUAN WAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this paper, we define two types of strongly decomposable positivity, which are the generalizations of (dual) Nakano positivity, and are stronger than the decomposable positivity introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Finski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We give the criteria of strongly decomposable positivity of type I and type II, and show that the Schur forms of a strongly decomposably pos- itive vector bundle of type I are weakly positive, the Schur forms of a strongly decomposably positive vector bundle of type II are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' These answer a question of Griffiths affirmatively for strongly decom- posably positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' As a result, we can give an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Finski first proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposably positive vector bundles 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Connections and curvatures 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposable positivity 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Relation to decomposable positivity 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Positivity notions for differential forms 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposable positivity of type I 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A criterion of type I positivity 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Weak positivity of Schur forms 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposable positivity of type II 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A criterion of type II positivity 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Positivity of Schur forms 27 References 29 Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 32Q10, 32L10, 53C65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Schur forms, positivity, weak positivity, strongly decomposable positivity, (dual) Nakano positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Research of Xueyuan Wan is partially supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 12101093) and the Natural Science Foundation of Chongqing (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' CSTB2022NSCQ-JQX0008), the Scientific Research Foundation of the Chongqing University of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='03950v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='DG] 10 Jan 2023 2 XUEYUAN WAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Introduction Let (E, hE) be a Hermitian holomorphic vector bundle of rank r over a complex manifold X of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The Chern forms ci(E, hE) of degree 2i, 0 ≤ i ≤ r, and the total Chern form c(E, hE) are defined by c(E, hE) := r � i=0 ci(E, hE) := det � IdE + √−1 2π RE � , where RE ∈ A1,1(X, End(E)) denotes the Chern curvature of (E, hE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For any k ∈ N with 1 ≤ k ≤ n, let Λ(k, r) be the set of all the partitions of k by non-negative integers less than or equal to r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=', any element λ = (λ1, · · · , λk) ∈ Λ(k, r) satisfying r ⩾ λ1 ⩾ λ2 ⩾ · · · ⩾ λk ⩾ 0 and |λ| = k � i=1 λi = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Each partition λ ∈ Λ(k, r) gives rise to a Schur form by Pλ(c(E, hE)) := det(cλi−i+j(E, hE))1≤i,j≤k, which is a closed real (k, k)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The Schur forms contain the Chern forms and the signed Segre forms as special examples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=', P(k,0,··· ,0)(c(E, hE)) = ck(E, hE) and P(1,··· ,1,0,··· ,0)(c(E, hE)) = (−1)ksk(E, hE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In [Gri70, Page 129, Conjecture (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7)], Griffiths conjectured that the nu- merical positivity of Griffiths positive vector bundles (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) for a defini- tion), that is, if (E, hE) is a Griffiths positive vector bundle, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) � V P(c1, · · · , cs) > 0 where P(c1, · · · , cs) is a positive polynomial in the Chern classes c1, · · · , cs of any quotient bundle Q of E|V , V ⊂ X is any complex analytic subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Block-Gieseke [BG71] proved all Chern classes of an ample vector bundle satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1), Fulton-Lazarsfeld [FL83, Theorem I] extended Block-Gieseke’s result and proved all Schur polynomials are numerically positive for am- ple vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For nef vector bundles over compact Kähler manifolds, Demailly-Peternell-Schneider [DPS94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5] proved the numerical semi-positivity of all Schur polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Griffiths [Gri70, Page 247] also conjectured (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) holds on the level of the differential forms, which can be reformulated as follows, see [Fin22, Page 1541, Question of Griffiths].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 (Griffiths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let P ∈ R [c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr] be a non-zero non-negative linear combination of Schur polynomials of weighted degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Are the forms P � c1(E, hE), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr(E, hE) � weakly positive for any Griffiths positive vector bundle (E, hE) over a complex manifold X of dimension n, n ⩾ k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' POSITIVITY OF SCHUR FORMS 3 Recall that a real (k, k)-form u is called weakly positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non- negative) if u ∧ (√−1)(n−k)2β ∧ β > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) for any non-zero de- composable (k, 0)-form β = β1 ∧ · · · ∧ βn−k, where βi, 1 ≤ i ≤ n − k, are (1, 0)-forms, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 for the definitions of (weakly) positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non-negative) vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Griffiths [Gri70, Page 249] proved that the second Chern form of a Griffiths vector bundle is positive by using Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Guler [Gul12, Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] verified Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 for all signed Segre forms, and Diverio-Fagioli [DF22] showed the positivity of several other polynomials in the Chern forms of a Griffiths (semi)positive vector bundle by considering the pushforward of a flag bundle, including the later developments [Fag22a, Fag22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' See Xiao [Xia22] and Ross-Toma [RT19] for other related results of ample vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For Bott-Chern non-negative vector bundles, Bott-Chern [BC65, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5)] proved that all Chern forms are non-negative, Li [Li21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] extended Bott-Chern’s result and obtained all Schur forms are non- negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Later, Finski [Fin22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='15] proved the equivalence of Bott-Chern non-negativity and dual Nakano non-negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Moreover, using a purely algebraic method, Finski [Fin22, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4] proved that all Schur forms of a Nakano non-negative vector bundle are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For (dual) Nakano positive vector bundles, Finski [Fin22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] proved that all Schur forms are positive by the refinement of the determinantal formula of Kempf-Laksov on the level of differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' However, as pointed out by Finski [Fin22, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='18], the above algebraic method can be used to deal with the case of non-negativity, while for the positivity statement, it is not clear if one can refine the algebraic method because there is no similar criterion for (dual) Nakano positivity (see [Fin22, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='16]) and there is little known about the specific structure of the forms defined in [Fin22, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='83)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' This motivates the author to study the question of Griffiths (Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) by developing the purely algebraic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In [Fin22, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3], Finski introduced the definition of decomposably positive vector bundles, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, which is a generalization of both Nakano positivity and dual Nakano positivity, and coincides with Griffiths positivity for n · r ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' So it is natural to wonder if Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 holds for such positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this paper, we introduce two new notions of positivity of vector bundles, called strongly decomposable positivity of type I and type II, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4 and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' They fall in between (dual) Nakano positivity and decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Roughly speaking, (E, hE) is strongly decomposably positive of type I if, for any x ∈ X, there is a decomposition T 1,0 x X = Ux ⊕ Vx such that it is Nakano positive in the subspaces Ex⊗Ux and dual Nakano positive in the subspace Ex⊗Vx, and the cross curvature terms vanish, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Using the purely algebraic method, we answer Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 affirmatively for strongly decomposably positive vector bundles of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4 XUEYUAN WAN Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a strongly decomposably positive vector bundle over a complex manifold X, rankE = r, and dim X = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then the Schur form Pλ(c(E, hE)) is weakly positive for any partition λ ∈ Λ(k, r), k ≤ n and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From [HK74, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2], a real (k, k)-form u is non-negative if and only if u can be written as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) u = N � s=1 ( √ −1)k2αs ∧ αs for some (k, 0)-forms αs, 1 ≤ s ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='14), the Schur form has the following form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) Pλ(c(E, hE)) = � 1 2π �k � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' �2 ( √ −1)k2 � ρ,t,c,ϵ (−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' where ψρtcϵ is a (|ϵ|, k − |ϵ|)-form and is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) ψρtcϵ := � σ∈Sk qσt k� j=1 (Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' It is not clear to express (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence, it seems hard to prove that the Schur form Pλ(c(E, hE)) is a positive (k, k)-form by using the algebraic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' However, if (E, hE) is Nakano positive or dual Nakano positive, it is equivalent to A = 0 or B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For example, for A = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) gives ψρtcϵ1 = � σ∈Sk qσt k� j=1 Bρσ(j)cj, ϵ1 = (1, · · · , 1) and ψρtcϵ = 0 for any ϵ ̸= ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then the Schur form is given by Pλ(c(E, hE)) = � 1 2π �k � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' �2 ( √ −1)k2 � ρ,t,c ψρtcϵ1 ∧ ψρtcϵ1, which satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) because ψρtcϵ1 is a (k, 0)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' As a result, we can give an algebraic proof of the following positivity of Schur forms for (dual) Nakano positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3 (Finski [Fin22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a (dual) Nakano positive vector bundle of rank r over a complex manifold X of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then for any k ∈ N, k ⩽ n, and λ ∈ Λ(k, r), the (k, k)-form Pλ � c � E, hE�� is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Inspired by the definition of the strongly decomposable positivity of type I, it is natural to define the strongly decomposable positivity of type II by decomposing the vector bundle, which is the direct sum of Naknao positive and dual Nakano positive vector bundles point-wisely, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By POSITIVITY OF SCHUR FORMS 5 Littlewood-Richardson rule (see [Ful97, Chapter 5]), the Schur class of direct sum E ⊕ F can be given by Pλ(c(E ⊕ F)) = � µ,ν cλ µνPµ(c(E))Pν(c(F)), where cλ µν(≥ 0) is a Littlewood-Richardson coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this paper, by re- fining the above identity on the level of differential forms and using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3, we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a strongly decomposably positive vector bundle of type II over a complex manifold X, rankE = r, and dim X = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then the Schur form Pλ(c(E, hE)) is positive for any partition λ ∈ Λ(k, r), k ≤ n and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Comparing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4, it is natural to ask if the Schur forms are positive for a strongly decomposably vector bundle of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The article is organized as follows: In Section 2, we will define two types of strongly decomposably positive vector bundles, which are the generalizations of both Nakano positivity and dual Nakano positivity, and are stronger than decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In Section 3, we will recall the positivity notions for differential forms and show the positivity of the product of two positive forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In Section 4, we will give a criterion of a strongly decomposably positive vector bundle of type I, recall the definitions of Schur forms and Griffiths cone, then prove the weak positivity of Schur forms, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3 are established in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In Section 5, we will give a criterion of a strongly decomposably positive vector bundle of type II and prove the positivity of Schur forms, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4 is established in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The author would like to thank Siarhei Finski for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposably positive vector bundles This section will define two types of strongly decomposably positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Connections and curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this subsection, we will recall the definitions of the Chern connection and its curvature for a Hermitian holo- morphic vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' One can refer to [Kob87, Chapter 1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We will use the Einstein summation convention in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let π : (E, hE) → X be a Hermitian holomorphic vector bundle over a complex manifold X, rankE = r and dim X = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let ∇E be the Chern connection of (E, hE), which preserves the metric hE and is of (1, 0)-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' With respect to a local holomorphic frame {ei}1≤i≤r of E, one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) ∇Eei = θj i ej, 6 XUEYUAN WAN where θ = (θj i ) (j row, i column) is the connection form of ∇E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' More precisely, θj i = ∂hi¯kh ¯kj, where hi¯k := h(ei, ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In terms of matrix form, it is θ⊤ = ∂h · h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Considering e = (e1, · · · , er) as a row vector, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) can be written as ∇Ee = e · θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let RE = (∇E)2 ∈ A1,1(X, End(E)) denote the Chern curvature of (E, hE), and write RE = Rj iej ⊗ ei ∈ A1,1(X, End(E)), where R = (Rj i) (j row, i column) is the curvature matrix whose entries are (1, 1)-forms, {ei}1≤i≤r denotes the dual frame of {ei}1≤i≤r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The curvature matrix R = (Rj i) is given by Rj i = dθj i + θj k ∧ θk i = ¯∂θj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' If {˜ei}1≤i≤r is another local holomorphic frame of E with ˜ei = aj iej, then ˜e = e · a, where ˜e := (˜e1, · · · , ˜er) and a = (ai j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by �R the curvature matrix with respect to the local frame {˜ei}1≤i≤r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) �R = a−1 · R · a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let {zα}1≤α≤n be local holomorphic coordinates of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Write Rj i = Rj iα¯βdzα ∧ d¯zβ and denote Ri¯j := Rk i hk¯j = Ri¯jα¯βdzα ∧ d¯zβ, so that Ri¯jα¯β = Rk iα¯βhk¯j = −∂α∂¯βhi¯j + h ¯lk∂αhi¯l∂¯βhk¯j, where ∂α := ∂/∂zα and ∂¯β := ∂/∂¯zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' This subsection will define two types of strongly decomposably positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Firstly, we recall the following definitions of Nakano positive and dual Nakano positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1 ((Dual) Nakano positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian holomorphic vector bundle (E, hE) is called Nakano positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non-negative) if Ri¯jα¯βuiαujβ > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) for any non-zero element u = uiαei ⊗∂α ∈ E ⊗T 1,0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (E, hE) is called dual Naknao positive (non-negative) if Ri¯jα¯βv¯jαv¯iβ > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) POSITIVITY OF SCHUR FORMS 7 for any non-zero element v = v¯jα¯ej ⊗ ∂α ∈ E ⊗ T 1,0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In [Fin22, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='18], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Finski introduced the following new notion of positivity for vector bundles: decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 (Decomposably positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle � E, hE� is called decomposably non-negative if for any x ∈ X, there is a number N ∈ N and linear (respectively sesquilinear) forms l′ p : T 1,0 x X ⊗ Ex → C (respectively lp : T 1,0 x X ⊗ Ex → C), p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , N, such that for any v ∈ T 1,0 x X, ξ ∈ Ex, we have 1 2π � RE x (v, ¯v)ξ, ξ � hE = N � p=1 |lp(v, ξ)|2 + N � p=1 ��l′ p(v, ξ) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We say that it is decomposably positive if, moreover, � RE x (v, ¯v)ξ, ξ � hE ̸= 0 for v, ξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From the above definition, a (dual) Nakano positive vector bundle must be decomposably positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From [Fin22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='21], for n · r ≤ 6 decomposable positivity is equivalent to Griffiths positivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) Ri¯jα¯βvi¯vjξα¯ξβ > 0 for any non-zero ξ = ξα∂α ∈ T 1,0X and v = viei ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Decomposable positivity is strictly stronger than Griffiths positivity for all other n, r ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Next, we will introduce the notion of strongly decomposable positivity, which falls in between (dual) Nakano positivity and decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4 (Strongly decomposably positive of type I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is called a strongly decomposably positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non- negative) vector bundle of type I if, for any x ∈ X, there exists a decompo- sition T 1,0 x X = Ux ⊕ Vx such that Ri¯jα¯βuiαujβ > 0(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0), Ri¯jα¯βuiαv′jβ = 0, Ri¯jα¯βv¯jαv¯iβ > 0(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) for any non-zero elements u = uiαei⊗∂α ∈ Ex⊗Ux, v′ = v′jβej⊗∂β ∈ Ex⊗Vx and v = v¯jα¯ej ⊗ ∂α ∈ Ex ⊗ Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From the above definition, a strongly decomposably positive vector bundle of type I means that it is Nakano positive in the subspace Ex⊗ Ux, dual Nakano positive in the subspace Ex ⊗ Vx, and the cross curvature terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For any point x ∈ X, if Vx = {0} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Ux = {0}), then strongly decomposable positivity of type I is exactly Nakano positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' dual Nakano positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let π1 : (E1, hE1) → X1 be a Nakano positive vector bundle and π2 : (E2, hE2) → X2 be a dual Nakano positive vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by pi : X1 × X2 → Xi, i = 1, 2, the natural projection, then π : (p∗ 1E1 ⊕ p∗ 2E2, p∗ 1hE1 ⊕ p∗ 2hE2) → X1 × X2 8 XUEYUAN WAN is a strongly decomposably non-negative vector bundle of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4, a strongly decomposably positive vector bundle (E, hE) of type I means that there is a decomposition of holomorphic tangent bundle T 1,0 x X = Ux ⊕ Vx, such that (E, hE) is Nakano positive in Ex ⊗ Ux and dual Naknao positive in Ex ⊗ Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Naturally, one may define another strongly de- composable positivity of vector bundles by decomposing the vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' More precisely, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7 (Strongly decomposably positive of type II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We call (E, hE) a strongly decomposably positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non-negative) vector bundle of type II if, for any x ∈ X, there is an orthogonal decomposition of (Ex, hE|Ex), Ex = E1,x ⊕ E2,x, such that the Chern curvature RE x has the following form RE x = �RE x |E1,x 0 0 RE x |E2,x � , and Ri¯jα¯βuiαujβ > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) for any non-zero u = uiαei ⊗ ∂α ∈ E1,x ⊗ T 1,0 x X, Ri¯jα¯βvi¯βvj ¯α > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' ≥ 0) for any non-zero v = vi¯βei ⊗ ∂¯β ∈ E2,x ⊗ T 0,1 x X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A simple example of a strongly decomposably vector bundle of type II is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a Nakano positive vector bundle and (F, hF ) be a dual Nakano positive vector bundle over a complex manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then (E ⊕F, hE ⊕hF ) is a strongly decomposably positive vector bundle of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4 and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7, a strongly decomposably positive vector bundle is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='9 (Strongly decomposably positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bun- dle (E, hE) is called strongly decomposably positive if it is a strongly decom- posably positive vector bundle of type I or type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Similarly, one can define strongly decomposably negative (non-positive) vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Note that the dual of the Nakano positive (negative) vector bundle is dual Nakano negative (positive), so Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is a strongly decom- posably positive (non-negative) vector bundle of type I (type II) if and only if (E∗, hE∗) is a strongly decomposably negative (non-positive) vector bundle of type I (type II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a strongly decomposably positive vector bun- dle, and Q be a quotient bundle of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The curvature of the bundle Q is given by RQ = RE�� Q + C ∧ C ⊤ for some matrix C whose entries are (1, 0)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From the criteria of strongly decomposably positive vector bundles, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 and Theorem POSITIVITY OF SCHUR FORMS 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, and the above curvature formula of quotient bundles, the quotient bun- dle (Q, hQ) cease to be a strongly decomposably positive vector bundle in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Relation to decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From the equivalent descrip- tions of Nakano non-negative and dual non-negative due to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Finski [Fin22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='15 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='17], one has Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is decomposably non- negative if and only if the Chern curvature matrix has the following form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) R = −B ∧ B ⊤ + A ∧ A ⊤ with respect to a unitary frame, where A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' B) is a r × N matrix with (1, 0)-forms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (0, 1)-forms) as entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, (E, hE) is decomposably positive if for any x ∈ X, there is a number N ∈ N and linear (respectively sesquilinear) forms l′ p : T 1,0 x X ⊗ Ex → C (respectively lp : T 1,0 x X ⊗ Ex → C), p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , N, such that for any v ∈ T 1,0 x X, ξ ∈ Ex, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5) � RE x (v, ¯v)ξ, ξ � hE = N � p=1 |lp(v, ξ)|2 + N � p=1 ��l′ p(v, ξ) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We denote lp(v, ξ) = lip¯βvi¯ξβ, l′ p(v, ξ) = l′ ipαviξα, and set A = (Ajp) and B = (Bjp) by Ajp := Ajpαdzα = ljp¯αdzα, Bjp := Bjp¯βd¯zβ = l′ jpβd¯zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5) is equivalent to Ri¯jα¯β = N � p=1 lip¯βljp¯α + N � p=1 l′ ipαl′ jpβ = N � p=1 AjpαAipβ + N � i=1 Bjp¯βBip¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) With respect to a unitary frame, Rj iα¯β = Ri¯jα¯β and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) is equivalent to R = −B ∧ B ⊤ + A ∧ A ⊤, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' □ From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='12 and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, (E, hE) is decomposably pos- itive if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) holds and � RE x (v, ¯v)ξ, ξ � hE ̸= 0 for v, ξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, we have Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' If (E, hE) is a strongly decomposably positive vector bundle of type I or type II, then (E, hE) is decomposably positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 10 XUEYUAN WAN On the other hand, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, the two types of strongly decomposably positive vector bundles can not contain each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Both are the generalizations of (dual) Nakano positive vector bundles and are stronger than decomposable positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' One can refer to the following Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Decomposably positive Griffiths positive Nakano positive dual Nakano positive Strongly decomposably positive (type I) Strongly decomposably positive (type II) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Relations of several notions of positivity Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Note that Griffiths positivity, decomposable positivity, and (dual) Nakano positivity are stable under addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In particular, decompos- able positivity is spanned by Naknao positivity and dual Nakano positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence, it is also spanned by the strongly decomposable positivity of type I (type II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' So the strongly decomposable positivity of type I and type II are not stable under addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Positivity notions for differential forms In this section, we will recall positivity notions for differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For more details, one can refer to [Fag22a, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] and [HK74, Fin22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let V be a complex vector space of dimension n and let (e1, · · · , en) be a basis of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by (e1, · · · , en) the dual basis of V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let Λp,qV ∗ denote the space of (p, q)-forms, and Λp,p R V ∗ ⊂ Λp,pV ∗ be the subspace of real (p, p)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A form ν ∈ Λn,nV ∗ is called a non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' positive) volume form if ν = τ√−1e1 ∧ e1 ∧ · · · ∧ √−1en ∧ en for some τ ∈ R, τ ≥ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' τ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we set q = n − p, a (q, 0)-form β is called decomposable if β = β1 ∧ · · · ∧ βq for some β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , βq ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A real (p, p)-form u ∈ Λp,p R V ∗ is called weakly non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' weakly positive), if for every non-zero β ∈ Λq,0V ∗ decomposable, u ∧ (√−1)q2β ∧ ¯β is a non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' positive) volume form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' positive), if for every non-zero β ∈ Λq,0V ∗, u ∧ (√−1)q2β ∧ ¯β is a non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' positive) volume form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' POSITIVITY OF SCHUR FORMS 11 strongly non-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' strongly positive) if there are decompos- able forms α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , αN ∈ Λp,0V ∗ such that u = �N s=1(√−1)p2αs ∧αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let WPpV ∗, PpV ∗ and SPpV ∗ denote respectively the closed positive convex cones contained in Λp,p R V ∨ spanned by weakly non-negative, non-negative and strongly non-negative forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) SPpV ∗ ⊆ PpV ∗ ⊆ WPpV ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Note that the above two inclusions become equalities for p = 0, 1, n − 1, n, and the inclusions are strict for 2 ≤ p ≤ n − 2, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' [Fag22a, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='8] and [HK74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' If u is a positive (k, k)-form and v is a positive (l, l)-form, k + l ≤ n, then u ∧ v is a positive (k + l, k + l)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By [HK74, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3 (a)], u∧v is a non-negative (k+l, k+l)-form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' for any non-zero β ∈ Λn−k−l,0V ∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) u ∧ v ∧ ( √ −1)(n−k−l)2β ∧ ¯β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By [HK74, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2], v has the following form v = N � s=1 ( √ −1)l2αs ∧ αs for some (l, 0)-forms αs, 1 ≤ s ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' So u ∧ v ∧ ( √ −1)(n−k−l)2β ∧ ¯β = N � s=1 u ∧ ( √ −1)l2αs ∧ αs ∧ ( √ −1)(n−k−l)2β ∧ ¯β = N � s=1 u ∧ ( √ −1)l2αs ∧ β ∧ αs ∧ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Thus the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) holds if and only if u ∧ ( √ −1)l2αs ∧ β ∧ αs ∧ β = 0, 1 ≤ s ≤ N, which is equivalent to αs ∧ β = 0, 1 ≤ s ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Thus v ∧ ( √ −1)(n−k−l)2β ∧ ¯β = N � s=1 ( √ −1)l2αs ∧ β ∧ αs ∧ β = 0, which contradicts the positivity of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence u ∧ v ∧ ( √ −1)(n−k−l)2β ∧ ¯β > 0 for any non-zero β ∈ Λn−k−l,0V ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' u ∧ v is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' □ 12 XUEYUAN WAN Let X be a complex manifold of dimension n and denote by Ap,q(X) the space of all smooth (p, q)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A real (p, p)-form α ∈ Ap,p(X) is called weakly non-negative (weakly positive), non-negative (positive), or strongly non-negative (strongly positive) if for any x ∈ X, αx ∈ Λp,p R (T 1,0 x X)∗ is weakly non-negative (weakly positive), non-negative (positive), or strongly non-negative (strongly positive) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Strongly decomposable positivity of type I In this section, we will give a criterion of strongly decomposably positive vector bundles of type I and prove the weak positivity of Schur forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A criterion of type I positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this subsection, following S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Finski’s approach [Fin22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='17], we will give a criterion for the strongly decomposable positivity (non-negativity) of type I by using M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='- D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Choi’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a strongly decomposably non-negative vector bundle of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For any x ∈ X, there exists a decomposition T 1,0 x X = Ux ⊕ Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' One can take local holomorphic coordinates {z1, · · · , zn} around x such that Ux = spanC{∂1, · · · , ∂n0}, Vx = spanC{∂n0+1, · · · , ∂n}, where n0 := dim Ux and recall that ∂α := ∂/∂zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let {ei}1≤i≤r be a local holomorphic frame of E such that hi¯j(x) = hE(ei(x), ej(x)) = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' With respect to {zα}1≤α≤n and {ei}1≤i≤r, the Chern curvature matrix R = (Rj i) at x ∈ X has the following expression Rj i = Rj iα¯βdzα ∧ d¯zβ = n0 � α,β=1 Rj iα¯βdzα ∧ d¯zβ + n � α,β=n0+1 Rj iα¯βdzα ∧ d¯zβ + n0 � α=1 n � β=n0+1 Rj iα¯βdzα ∧ d¯zβ + n � α=n0+1 n0 � β=1 Rj iα¯βdzα ∧ d¯zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1) By assumption, (E, hE) is strongly decomposably non-negative of type I, so n0 � α=1 n � β=n0+1 Ri¯jα¯βuiαv′jβ = 0 for any �n0 α=1 uiαei ⊗ ∂α and �n β=n0+1 v′jβej ⊗ ∂β, which follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2) Ri¯jα¯β = 0, 1 ≤ α ≤ n0, n0 + 1 ≤ β ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By conjugation, one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) Ri¯jα¯β = Rj¯iβ ¯α = 0, n0 + 1 ≤ α ≤ n, 1 ≤ β ≤ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' POSITIVITY OF SCHUR FORMS 13 Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1), one has Rj i = n0 � α,β=1 Rj iα¯βdzα ∧ d¯zβ + n � α,β=n0+1 Rj iα¯βdzα ∧ d¯zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For the local frame {∂α}1≤α≤n, we define a local metric g around x by gα¯β = g(∂α, ∂β) := δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we define a linear map HV x : End(Vx) → End(Ex) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) HV x (∂α ⊗ dzγ) = Rj iα¯βg ¯βγej ⊗ ei = Rj iα¯γej ⊗ ei for any n0 + 1 ≤ α, γ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' With respect to the basis {∂α}n0+1≤α≤n, the matrix of ∂α ⊗ dzγ ∈ End(Ux) is Eαγ, which is the (n − n0) × (n − n0) matrix with 1 at the (α, γ)-component and zeros elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The matrix of Rj iα¯γej ⊗ ei ∈ End(Ex) is given by (Rj iα¯γ)1≤j,i≤r (j row, i column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In terms of matrices, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4) becomes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5) HV x (Eαγ) = (Rj iα¯γ)1≤j,i≤r = (Ri¯jα¯γ)1≤j,i≤r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then (HV x (Eαγ))n0+1≤α,γ≤n is a (n − n0) × (n − n0) block matrix with r × r matrices as entries, and (HV x (Eαγ))n0+1≤α,γ≤n = � (Ri¯jα¯γ)1≤j,i≤r � n0+1≤α,γ≤n (j α row, i γ column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Since (E, hE) is strongly decomposably non-negative of type I, so Ri¯jα¯βv¯jαv¯iβ ≥ 0 for any non-zero v = v¯jαei ⊗ ∂α ∈ Ex ⊗ Vx, which follows that the matrix (HV x (Eαγ))n0+1≤α,γ≤n is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By using [Cho75, Theorem 2 and Theorem 1], there exist (n − n0) × r matrices Vp, 1 ≤ p ≤ N1 (one can choose N1 = (n − n0) · r) such that HV x (Eαγ) = N1 � p=1 Vp ⊤ · Eαγ · Vp for any n0 + 1 ≤ α, γ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5) and considering the (j, i) entry, one has Rj iα¯β = N1 � p=1 (Vp ⊤ · Eαβ · Vp)j,i = N1 � p=1 (Vp)αj(Vp)βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 14 XUEYUAN WAN Hence n � α,β=n0+1 Rj iα¯βdzα ∧ d¯zβ = n � α,β=n0+1 N1 � p=1 (Vp)αj(Vp)βidzα ∧ d¯zβ = N1 � p=1 Ajp ∧ Aip, where Ajp := �n α=n0+1 (Vp)αjdzα, and one has � � n � α,β=n0+1 Rj iα¯βdzα ∧ d¯zβ � � 1≤j,i≤r = A ∧ A ⊤, where A = (Ajp) is a r × N1 matrix with (1, 0)-forms in V ∗ x as entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Similarly, by considering the linear map HU x : End(Ux) → End(E∗ x), HV x (∂α ⊗ dzγ) = Rj iα¯γei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' One can obtain that (HV x (Eαγ))1≤α,γ≤n0 = � (Ri¯jα¯γ)1≤j,i≤r � 1≤α,γ≤n0 (i α row, j γ column), which follows that � � n0 � α,β=1 Rj iα¯βdzα ∧ d¯zβ � � 1≤j,i≤r = −B ∧ B ⊤, where B = (Bjp) is a r × N2 (one can choose N2 = n0 · r) matrix with (0, 1)-forms in U ∗x as entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Thus, if (E, hE) is strongly decomposably non-negative of type I, then for any x ∈ X, the Chern curvature matrix at this point has the following form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) R = −B ∧ B ⊤ + A ∧ A ⊤ with respect to a unitary frame, where B is a r×N2 matrix with (0, 1)-forms as entries, A is a r × N1 matrix with (1, 0)-forms as entries, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7) spanC{B} ∩ spanC{A} = {0}, where {B} := {Bip, 1 ≤ i ≤ r, 1 ≤ p ≤ N2} and {A} := {Aip, 1 ≤ i ≤ r, 1 ≤ p ≤ N1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' It is noted that the above argument is independent of the choice of unitary frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' If ˜e = e · a is also a unitary frame, then a is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2), one has �R = a−1 · (−B ∧ B ⊤ + A ∧ A ⊤) · a = −a−1B ∧ a−1B ⊤ + a−1A ∧ a−1A ⊤, POSITIVITY OF SCHUR FORMS 15 which has the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Moreover, spanC{B} = spanC{a−1B} and spanC{A} = spanC{a−1A}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7) is equivalent to spanC{a−1B} ∩ spanC{a−1A} = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Conversely, we assume (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For any x ∈ X, taking local holomorphic coordinates {zα}1≤α≤n around x ∈ X such that spanC{dz1|x, · · · , dzn0|x} = spanC{B} and spanC{dzn0+1|x, · · · , dzn1|x} = spanC{A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we set Ux := spanC{∂1|x, · · · , ∂n0|x}, Vx = spanC{∂n0+1|x, · · · , ∂n|x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then Ux ⊕ Vx = T 1,0 x X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7), one can check that (E, hE) is strongly decomposably non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence (E, hE) is strongly decomposably non-negative of type I if and only if the Chern curvature matrix of (E, hE) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Next, we assume that (E, hE) is strongly decomposably positive of type I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=', it is strongly decomposably non-negative of type I and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='8) Ri¯jα¯βuiαujβ = 0 =⇒ uiα = 0, for all 1 ≤ i ≤ r, 1 ≤ α ≤ n0 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='9) Ri¯jα¯βv¯jαv¯iβ = 0 =⇒ v¯jα = 0, for all 1 ≤ j ≤ r, n0 + 1 ≤ β ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By the equivalent description of strongly decomposably non-negative of type I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=', (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7), one has Ri¯jα¯βuiαujβ = N2 � p=1 |Bip¯αuiα|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='8) is equivalent to the equation Bx = 0 has only zero solution, where B := (Bip¯α)p,iα = � � � � � B111 B112 · · Br1n0 B121 B112 · · Br2n0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' B1N21 B1N22 · · BrN2n0 � � � � � N2×rn0 is a N2 × rn0 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' This is also equivalent to the matrix rank(B) = rn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Similarly, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='9) is equivalent to the following matrix A := (Ajpα)p,jα = � � � � � A11(n0+1) A11(n0+2) · · Ar1n A12(n0+1) A11(n0+2) · · Ar2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A1N1(n0+1) A1N1(n0+2) · · ArN1n � � � � � N1×r(n−n0) 16 XUEYUAN WAN has rank r(n − n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In a word, we obtain Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is a strongly de- composably non-negative of type I if and only if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is strongly decomposably positive of type I if and only if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='7) hold, and rank(A) = r dim Vx, rank(B) = r dim Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' As a result, we obtain the following criteria of (dual) Nakano positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is Nakano posi- tive if and only if the Chern curvature matrix has the form R = −B ∧ B ⊤ with respect to some unitary frame, where B is a r × N matrix with (0, 1)-forms as entries, and rank(B) = rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A Hermitian vector bundle (E, hE) is dual Nakano positive if and only if the Chern curvature matrix has the form R = A ∧ A ⊤ with respect to some unitary frame, where A is a r × N matrix with (1, 0)-forms as entries, and rank(A) = rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Weak positivity of Schur forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In this subsection, we will prove the weak positivity of Schur forms for strongly decomposably positive vector bundles of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Schur forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let Λ(k, r) be the set of all the partitions of k by non- negative integers less than or equal to r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=', any element λ ∈ Λ(k, r) is a sequence r ⩾ λ1 ⩾ λ2 ⩾ · · · ⩾ λk ⩾ 0 satisfying |λ| = �k i=1 λi = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Each partition λ ∈ Λ(k, r) gives rise to a Schur polynomial Pλ ∈ Q [c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr] of degree k, defined as k × k determinant Pλ (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr) = det (cλi−i+j)1⩽i,j⩽k = ��������� cλ1 cλ1+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' cλ1+k−1 cλ2−1 cλ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' cλ2+k−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' cλk−k+1 cλk−k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' cλk ��������� , where by convention c0 = 1 and ci = 0 if i /∈ [0, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by Mr(C) and GLr(C) the vector spaces of r × r matrices and the general linear group of degree r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' A map P : Mr(C) → C is called GLr(C)- invariant if it is invariant under the conjugate action of GLr(C) on Mr(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' POSITIVITY OF SCHUR FORMS 17 Now we define the following GLr(C)-invariant function ci : Mr(C) → C, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , r by det (Ir + tX) = r � i=0 ti · ci(X), where Ir is the identity matrix in Mr(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then the graded ring of GLr(C)- invariant homogeneous polynomials on Mr(C), which we denote here by I(r) = �+∞ k=0 I(r)k, is multiplicatively generated by c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let (E, hE) be a Hermitian vector bundle, the i-th Chern form ci(E, hE) is defined by ci � E, hE� = ci �√−1 2π RE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For each λ ∈ Λ(k, r), the associated Schur form is then defined as Pλ(c(E, hE)) := Pλ(c1(E, hE), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr(E, hE)), which represents the Schur class Pλ(c(E)) := Pλ(c1(E), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr(E)) ∈ H2k(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Griffiths cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By [Gri70, Page 242, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6)], each P ∈ I(r)k can be written as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='10) P(B) = � σ,τ∈Sk � ρ∈[1,r]k pρστBρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k), where Bλµ, λ, µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , r are the components of the matrix B, Sk is the permutation group on k indices and [1, r] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' An element P ∈ I(r)k is called Griffiths non-negative if it can be expressed in the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='10) with pρστ = � t∈T λρt · qρσt¯qρτt, for some finite set T, some real numbers λρt ⩾ 0, and complex numbers qρσt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The Griffiths cone Π(r) ⊂ I(r) is defined as the cone of Griffiths non- negative polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='4 (Fulton-Lazarsfeld [FL83, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Let P = � λ∈Λ(k,r) aλ(P)Pλ (aλ(P) ∈ Q) be a non-zero weighted homogeneous polynomial in Q [c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then P lies in the Griffiths cone Π(r) if and only if each of the Schur coefficients aλ(P) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In particular, for each λ ∈ Λ(k, r), one has Pλ(B) = � σ,τ∈Sk � ρ∈[1,r]k pρστBρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k), 18 XUEYUAN WAN where pρστ = � 1≤i,j≤m � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' �2 aij(τ)aij(σ) with (aij(τ)) ∈ U(m), see [FL83, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote T = [1, m]2 and qσt := at(σ) for any t ∈ T, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='11) Pλ(B) = � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' �2 � σ,τ∈Sk � ρ∈[1,r]k �� t∈T qσtqτt � Bρσ(1)ρτ(1) · · · Bρσ(k)ρτ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Weak positivity of Schur forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' We assume that (E, hE) is a strongly decomposably positive vector bundle of type I over a complex manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='2, for any x ∈ X, there exists a decomposition T 1,0 x X = Ux ⊕ Vx such that the Chern curvature matrix R of (E, hE) has the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='12) R = −B ∧ B ⊤ + A ∧ A ⊤ with respect to a unitary frame, where B is a r ×N matrix with (0, 1)-forms in U ∗x as entries, A is a r × N matrix with (1, 0)-forms in V ∗ x as entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Moreover, rank(A) = r · dim Vx, rank(B) = r · dim Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For each λ ∈ Λ(k, r), by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='11), the Schur form Pλ(c(E, hE)) is given by Pλ(c(E, hE)) = �√−1 2π �k 1 (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' )2 � σ,τ∈Sk � ρ∈[1,r]k �� t∈T qσtqτt � k� j=1 Rρσ(j)ρτ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' the Chern curvature matrix satisfies Rρσ(j)ρτ(j) = (Bρτ(j)cj ¯ βjBρσ(j)cjαj + Aρτ(j)cjαjAρσ(j)cjβj)dzαj ∧ d¯zβj = N � cj=1 (Bρσ(j)cj ∧ Bρτ(j)cj + Aρτ(j)cj ∧ Aρσ(j)cj) = N � cj=1 � ϵj∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] (Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' POSITIVITY OF SCHUR FORMS 19 which follows that k� j=1 Rρσ(j)ρτ(j) = k� j=1 N � cj=1 � ϵj∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1] (Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj = � c∈[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='N]k � ϵ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1]k k� j=1 (Bρσ(j)cj ∧ Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj ∧ Aρσ(j)cj)1−ϵj = � c∈[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='N]k � ϵ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1]k k� j=1 (−1)ϵj+1(Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj ∧ (Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj = � c∈[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='N]k � ϵ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1]k (−1)|ϵ|+k(−1) k(k−1) 2 ( k� j=1 (Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj) ∧ ( k� j=1 (Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' where |ϵ| := �k j=1 ϵj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Recall that ρ ∈ [1, r]k, t ∈ T, c ∈ [1, N]k and ϵ ∈ [0, 1]k, we obtain that Pλ(c(E, hE)) = �√−1 2π �k 1 (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' )2 (−1) k(k−1) 2 � ρ,t,c,ϵ (−1)|ϵ|+k· ( � σ∈Sk qσt k� j=1 (Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj) ∧ ( � τ∈Sk qτt k� j=1 (Bρτ(j)cj)ϵj ∧ (Aρτ(j)cj)1−ϵj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we set (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='13) ψρtcϵ := � σ∈Sk qσt k� j=1 (Bρσ(j)cj)ϵj ∧ (Aρσ(j)cj)1−ϵj, which is a (|ϵ|, k − |ϵ|)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence Pλ(c(E, hE)) = � 1 2π �k � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' �2 ( √ −1)k2 � ρ,t,c,ϵ (−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='14) For any non-zero decomposable (n − k, 0)-form η = η1 ∧ · · · ∧ ηn−k, where ηi, 1 ≤ i ≤ n − k, are (1, 0)-forms, we assume that η1, · · · , ηi0 ∈ U ∗ x and ηi0+1, · · · , ηn−k ∈ V ∗ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we can take local holomorphic coordinates {zα}1≤α≤n around x ∈ X such that U ∗ x = spanC{dz1|x, · · · , dzn0|x}, with dzj|x = ηj, 1 ≤ j ≤ i0 and V ∗ x = spanC{dzn0+1|x, · · · , dzn|x}, with dzn0−i0+j|x = ηj, i0+1 ≤ j ≤ n−k, 20 XUEYUAN WAN and so ψρtcϵ can be written as the following form ψρtcϵ = � 1≤α1<···<α|ϵ|≤n0 n0+1≤β1<···<βk−|ϵ|≤n ψα1···α|ϵ| ¯β1···¯βk−|ϵ|dzα1 ∧ · · · ∧ dzα|ϵ| ∧ d¯zβ1 ∧ · · · ∧ d¯zβk−|ϵ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Then ( √ −1)k2 � ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ϵ (−1)|ϵ|+kψρtcϵ ∧ ψρtcϵ ∧ ( √ −1)(n−k)2η ∧ η = ( √ −1)k2 � ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='|ϵ|=n0−i0 (−1)n0−i0+k( √ −1)(n−k)2dz1 ∧ · · · ∧ dzi0∧ dzn0+1 ∧ · · · ∧ dzn−i0+n−k ∧ d¯z1 ∧ · · · ∧ d¯zi0 ∧ d¯zn0+1 ∧ · · · ∧ d¯zn−i0+n−k∧ � ψi0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0−i0+n−k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='¯ndzi0+1 ∧ · · · ∧ dzn0 ∧ d¯zn0−i0+n−k+1 ∧ · · · ∧ d¯zn� ∧ � ψi0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0−i0+n−k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='¯nd¯zi0+1 ∧ · · · ∧ d¯zn0 ∧ dzn0−i0+n−k+1 ∧ · · · ∧ dzn� = � ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='|ϵ|=n0−i0 |ψi0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='n0−i0+n−k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='¯n|2· ( √ −1)n2dz1 ∧ · · · ∧ dzn ∧ d¯z1 ∧ · · · ∧ d¯zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='15) which is a non-negative volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='14), the Schur form Pλ(c(E, hE)) is weakly non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='15), one knows that Pλ(c(E, hE)) ∧ ( √ −1)(n−k)2η ∧ ¯η = 0 if and only if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='16) ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯n = 0 for any ρ, t, c, |ϵ| = n0 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Now we take a special vector ϵ = (1, · · · , 1 � �� � n0−i0 , 0 · · · , 0) and denote j0 = n0 − i0, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='13), then ψρtcϵ = � σ∈Sk qσtBρσ(1)c1 ∧ · · · ∧ Bρσ(j0)cj0 ∧ Aρσ(j0+1)cj0+1 ∧ · · · ∧ Aρσ(k)ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='16), one has 0 = ψi0+1,··· ,n0,n0−i0+n−k+1,··· ,¯n = � τ1∈Sj0 � τ2∈Sk−j0 � σ∈Sk sgn(τ1)sgn(τ2)qσt· Bρσ(1)c1τ1(i0+1) · · · Bρσ(j0)cj0τ1(n0) · Aρσ(j0+1)cj0+1τ2(j0+n−k+1) · · · Aρσ(k)ckτ2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='17) POSITIVITY OF SCHUR FORMS 21 Since rank(A) = r · dim Vx, rank(B) = r · dim Ux, without loss of generality, we assume that the submatrices B′ = (Bc,iα)1≤c≤r dim Ux,1≤i≤r,1≤α≤dim Ux and A′ = (Ac,jα)1≤c≤r dim Vx,1≤j≤r,1≤α≤dim Vx of B and A are inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='17) and note that B′ c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='iα = Bic¯α and A′ c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='jα = Ajcα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' one has 0 = � τ1∈Sj0 � τ2∈Sk−j0 rn0 � c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='cj0=1 r(n−n0) � cj0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ck=1 � σ∈Sk sgn(τ1)sgn(τ2)qσt· B′ c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ρσ(1)τ1(i0+1) · · · B′ cj0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ρσ(j0)τ1(n0) · A′ cj0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ρσ(j0+1)τ2(j0+n−k+1) · · · A′ ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ρσ(k)τ2(n)· A′−1 lkβk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='ck · · · A′−1 lj0+1βj0+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='cj0+1B′−1 lj0βj0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='cj0 · · · B′−1 l1β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='c1 = � τ1∈Sj0 � τ2∈Sk−j0 � σ∈Sk sgn(τ1)sgn(τ2)qσt · δρσ(k)lkδτ2(n)βk · · · δρσ(j0+1)lj0+1· δτ2(j0+n−k+1)βj0+1δρσ(j0)lj0δτ1(n0)βj0 · · · δρσ(1)l1δτ1(i0+1)β1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='18) for any (β1, · · · , βj0) ∈ [1, n0]j0, (βj0+1, · · · , βk) ∈ [n0+1, n]n−j0 and (l1, · · · , lk) ∈ [1, r]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' By taking βs = � i0 + s 1 ≤ s ≤ j0, n − k + s j0 + 1 ≤ s ≤ k, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='18) becomes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19) � σ∈Sk qσtδρσ(1)l1 · · · δρσ(k)lk = 0 for any ρ, l ∈ [1, r]k and t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19) holds if and only if ψρtcϵ = 0 for any ρ, t, c, ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In fact, if ψρtcϵ = 0, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='16) holds, and follows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Conversely, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19) holds, then ψρtcϵ = � l∈[1,r]k ( � σ∈Sk qσtδρσ(1)l1 · · · δρσ(k)lk) k� j=1 Bljcj ϵj ∧ Aljcj 1−ϵj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For k ≤ r, one can take ρi = li = i for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Thus 0 = � σ∈Sk qσtδρσ(1)l1 · · · δρσ(k)lk = qId,t for any t ∈ T, which is a contradiction since (qId,t)t∈T ∈ U(m) is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Hence all (k, k)-Schur forms Pλ(c(E, hE)) are weakly positive for 22 XUEYUAN WAN any k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' In particular, all Chern forms ci(E, hE), 1 ≤ i ≤ r, are weakly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' For general k and r, we take l1, · · · , lk in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19) to be li = ρi, for 1 ≤ i ≤ k, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='19) implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='20) � ρ∈[1,r]k � σ∈Sk χλ(σ)δρ1ρσ(1) · · · δρkρσ(k) = 0, where χλ(σ) = Tr(qσt) = m � i=1 aii(σ) is the character of the representation φλ(σ) = (aij(σ)) ∈ U(m) corresponding to the partition λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' From [FL83, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='5)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='20) is equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='21) Pλ(Ir) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Here Pλ(•) denotes the invariant polynomial corresponding to the Schur function Pλ under the isomorphism I(r) ∼= Q(c1, · · · , cr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by x1, · · · , xr the Chern roots, which are defined by r � j=0 cjtj = (1 + tx1)(1 + tx2) · · · (1 + txr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Recall that the Schur polynomial is defined by Pλ (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr) = det � cλj−j+l � 1⩽j,l⩽k , where λ = (λ1, · · · , λk) ∈ Λ(k, r) is a partition satisfying k � i=1 λi = k and r ≥ λ1 ≥ · · · ≥ λk ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' Denote by λ′ the conjugate partition to the partition λ, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' [FH91, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='1, Page 45], then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='22) λ′ = (λ′ 1, · · · , λ′ r), with r � i=1 λ′ i = k and λ′ 1 ≥ · · · ≥ λ′ r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' The second Jacobi-Trudi identity (or Giambell’s formula) gives Pλ (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' , cr) = det � cλj−j+l � 1⩽j,l⩽k = sλ′(x1, · · · , xr), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='23) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' [FH91, Page 455, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='6)], where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='24) sλ′(x1, · · · , xr) := ���������� xλ′ 1+r−1 1 xλ′ 1+r−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' xλ′ 1+r−1 r xλ′ 2+r−2 1 xλ′ 2+r−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' xλ′ 2+r−2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' xλ′ r 1 xλ′ r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQfhgf_/content/2301.03950v1.pdf'} +page_content=' xλ′ r r ���������� � 1≤i